CB 2026 User Manual

Question ID: 128c75e2

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function is defined by , where



. What is the product of



and



?

Correct Answer: 609

Rationale

The correct answer is . It’s given that the function is defined by , where . Substituting for in function yields

. This function can be rewritten as , or . Since , it follows that

. Substituting for in yields , or . Similarly, substituting for in

function yields . This function can be rewritten as , or . Since , it again

follows that . Substituting for in yields , or . Therefore,

and . Thus, the product of and is , or .

Question ID: 91e7ea5e

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The quadratic function h is defined as shown. In the xy-plane, the graph of intersects the x-axis at the points and , where t

is a constant. What is the value of t ?

Answer

A. 1

B. 2

C. 4

D. 8

Correct Answer: D

Rationale

Choice D is correct. It’s given that the graph of intersects the x-axis at and , where t is a constant. Since this graph

intersects the x-axis when or when , it follows that and . If , then . Adding

32 to both sides of this equation yields . Dividing both sides of this equation by 2 yields . Taking the square root

of both sides of this equation yields . Adding 4 to both sides of this equation yields . Therefore, the value of t is 8.

Choices A, B, and C are incorrect and may result from calculation errors.

Question ID: ebed7dc6

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

An auditorium has seats for people. Tickets to attend a show at the auditorium currently cost . For each increase to the ticket

price, fewer tickets will be sold. This situation can be modeled by the equation , where represents the

increase in ticket price, in dollars, and represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what

value of is the maximum of the graph?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. It’s given that the situation can be modeled by the equation , where represents the

increase in ticket price, in dollars, and represents the revenue, in dollars, from ticket sales. Since the coefficient of the term is negative, the

graph of this equation in the xy-plane opens downward and reaches its maximum value at its vertex. If a quadratic equation in the form

, where , , and are constants, is graphed in the xy-plane, the x-coordinate of the vertex is equal to . For the equation

, , , and . It follows that the x-coordinate of the vertex is , or . Therefore,

if the given equation is graphed in the

xy

-plane, the maximum of the graph occurs at an x-value of .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: a9084ca4

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The given function models the number of advertisements a company sent to its clients each year, where represents the number of years

since , and . If is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the

graph in this context?

Answer

A. The minimum estimated number of advertisements the company sent to its clients during the years was .

B. The minimum estimated number of advertisements the company sent to its clients during the years was .

C. The estimated number of advertisements the company sent to its clients in was .

D. The estimated number of advertisements the company sent to its clients in was .

Correct Answer: D

Rationale

Choice D is correct. The y-intercept of a graph in the xy-plane is the point where . For the given function , they

-intercept of the graph of

in the xy-plane can be found by substituting for in the equation , which gives . This is

equivalent to

, or . Therefore, the y-intercept of the graph of is . It’s given that the function



models the number of advertisements a company sent to its clients each year. Therefore, represents the estimated number of

advertisements the company sent to its clients each year. It's also given that represents the number of years since . Therefore,

represents years since , or . Thus, the best interpretation of the y-intercept of the graph of



is that the estimated number of

advertisements the company sent to its clients in was .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: 6abec9a8

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

What is the -intercept of the graph shown?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The y-intercept of a graph in the xy-plane is the point



on the graph where . At , the corresponding value of

is . Therefore, the y-intercept of the graph shownis .

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at , not .

Choice D is incorrect. This is the y-intercept of a graph in the xy-plane that intersects the y-axis at , not .

Question ID: b8f13a3a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

Function is defined by , where and are constants. In the xy-plane, the graph of has a y-intercept at

. The product of and is . What is the value of ?

Correct Answer: 20

Rationale

The correct answer is . It’s given that . Substituting



for in the equation yields

. It’s given that the y-intercept of the graph of is . Substituting for and for in the equation

yields , which is equivalent to , or . Adding to both sides of this

equation yields . It’s given that the product of and is , or . Substituting for in this equation yields .

Dividing both sides of this equation by yields .

Question ID: 1d3c5c95

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

An entomologist recommended a program to reduce a certain invasive beetle population in an area. The given function estimates this beetle

species' population years after , where . Which of the following is the best interpretation of in this context?

Answer

A. The estimated initial beetle population for this species and area in

B. The estimated beetle population for this species and area years after

C. The estimated percent decrease in the beetle population for this species and area each year after

D. The estimated percent decrease in the beetle population for this species and area every years after



Correct Answer: A

Rationale

Choice A is correct. For an exponential function in the form , where and are positive constants and , the initial value of

, or the value of when , is and the value of decreases by each time increases by . Therefore, the initial

value of the function , or the value of when , is . Therefore, the best interpretation of in this

context is the estimated initial beetle population for this species and area in .

Choice B is incorrect. The estimated beetle population for this species and area years after is , or approximately , not

.

Choice C is incorrect. The estimated percent decrease in the beetle population for this species and area each year after is ,

or , not .

Choice D is incorrect. The estimated percent decrease in the beetle population for this species and area every years after is

, or approximately , not .

Question ID: 788bfd56

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function is defined by



. What is the value of



?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The value of



is the value of



when .It's given that the function is defined by .

Substituting

for in this equation yields



. Since the positive square root of is , it follows that this equation can

be rewritten as

, or . Therefore, the value of is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the value of



, not .

Choice D is incorrect. This is the value of , not



.

Question ID: f89af023

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

A rectangular volleyball court has an area of 162square meters. If the length of the court is twice the width, what is the width of the court, in

meters?

Answer

A. 9

B. 18

C. 27

D. 54

Correct Answer: A

Rationale

Choice A is correct. It’s given that the volleyball court is rectangular and has an area of 162 square meters. The formula for the area of a

rectangle is , where A is the area, is the length, and w is the width of the rectangle. It’s also given that the length of the volleyball

court is twice the width, thus . Substituting the given value into the formula for the area of a rectangle and using the relationship

between length and width for this rectangle yields . This equation can be rewritten as . Dividing both sides of this

equation by 2 yields . Taking the square root of both sides of this equation yields . Since the width of a rectangle is a positive

number, the width of the volleyball court is 9 meters.

Choice B is incorrect because this is the length of the rectangle. Choice C is incorrect because this is the result of using 162 as the perimeter

rather than the area. Choice D is incorrect because this is the result of calculating w in the equation instead of

.

Question ID: e53add44

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The function S above models the annual salary, in dollars, of an employee n years after starting a job, where a is a constant. If the employee’s

salary increases by 4% each year, what is the value of a ?

Answer

A. 0.04

B. 0.4

C. 1.04

D. 1.4

Correct Answer: C

Rationale

Choice C is correct. A model for a quantity S that increases by a certain percentage per time period n is an exponential function in the form

, where I is the initial value at time for r% annual increase. It’s given that the annual increase in an employee’s

salary is 4%, so . The initial value can be found by substituting 0 for n in the given function, which yields . Therefore,

. Substituting these values for r and I into the form of the exponential function yields

, or . Therefore, the value of a in the given function is 1.04.

Choices A, B, and D are incorrect and may result from incorrectly representing the annual increase in the exponential function.

Question ID: 9654add7

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The revenue , in dollars, that a company receives from sales of a product is given by the function f above, where x is the unit price, in

dollars, of the product. The graph of in the xy-plane intersects the x-axis at 0 and a. What does a represent?

Answer

A. The revenue, in dollars, when the unit price of the product is $0

B. The unit price, in dollars, of the product that will result in maximum revenue

C. The unit price, in dollars, of the product that will result in a revenue of $0

D. The maximum revenue, in dollars, that the company can make

Correct Answer: C

Rationale

Choice C is correct. By definition, the y-value when a function intersects the x-axis is 0. It’s given that the graph of the function intersects the x-

axis at 0 and a, that x is the unit price, in dollars, of a product, and that , where , is the revenue, in dollars, that a company

receives from the sales of the product. Since the value of a occurs when , a is the unit price, in dollars, of the product that will result in a

revenue of $0.

Choice A is incorrect. The revenue, in dollars, when the unit price of the product is $0 is represented by , when . Choice B is incorrect.

The unit price, in dollars, of the product that will result in maximum revenue is represented by the x-coordinate of the maximum of f. Choice D is

incorrect. The maximum revenue, in dollars, that the company can make is represented by the y-coordinate of the maximum of f.

Question ID: 263f9937

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

Growth of a Culture of Bacteria

Day

Number of bacteria per

milliliter at end of day

1

2

3

A culture of bacteria is growing at an exponential rate, as shown in the table above. At this rate, on which day would the number of bacteria per

milliliter reach ?

Answer

A. Day 5

B. Day 9

C. Day 11

D. Day 12

Correct Answer: D

Rationale

Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day 1 to day 2, the number of bacteria increased

from 2.5 × 105 to 5.0 × 105. At the end of day 3 there are 106 bacteria per milliliter. At the end of day 4, there will be106 × 2 bacteria per milliliter.

At the end of day 5, there will be (106 × 2) × 2, or 106 × (22) bacteria per milliliter, and so on. At the end of day d, the number of bacteria will be 10

6

× (2

d – 3

). If the number of bacteria per milliliter will reach 5.12 × 108 at the end of day d, then the equation

must hold. Since 5.12 × 108 can be rewritten as 512 × 106, the equation is equivalent to . Rewriting 512 as 29 gives d – 3 = 9, so d

= 12. The number of bacteria per milliliter would reach 5.12 × 108at the end of day 12.

Choices A, B, and C are incorrect. Given the growth rate of the bacteria, the number of bacteria will not reach 5.12× 108 per milliliter by the end of

any of these days.

Question ID: 926c246b

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The equation above estimates the global data traffic D, in terabytes, for the year that is t years after 2010. What is the best interpretation of the

number 5,640 in this context?

Answer

A. The estimated amount of increase of data traffic, in terabytes, each year

B. The estimated percent increase in the data traffic, in terabytes, each year

C. The estimated data traffic, in terabytes, for the year that ist years after 2010

D. The estimated data traffic, in terabytes, in 2010

Correct Answer: D

Rationale

Choice D is correct. Since t represents the number of years after 2010, the estimated data traffic, in terabytes, in 2010 can be calculated using

the given equation when . Substituting 0 for t in the given equation yields , or . Thus, 5,640

represents the estimated data traffic, in terabytes, in 2010.

Choice A is incorrect. Since the equation is exponential, the amount of increase of data traffic each year isn’t constant. Choice B is incorrect.

According to the equation, the percent increase in data traffic each year is 90%. Choice C is incorrect. The estimated data traffic, in terabytes, for

the year that is t years after 2010 is represented by D, not the number 5,640.

Question ID: 4dd4efcf

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

In the given quadratic function, and are constants. The graph of



in the xy-plane is a parabola that opens upward and has a vertex at

the point , where and are constants. If



and



, which of the following must be true?

I.

II.

Answer

A. I only

B. II only

C. I and II

D. Neither I nor II

Correct Answer: D

Rationale

Choice D is correct. It's given that the graph of



in the xy-plane is a parabola with vertex



. If , then for the graph of

, the point with an x-coordinate of and the point with an x-coordinate of have the same y-coordinate. In the xy-plane, a parabola is

a symmetric graph such that when two points have the same y-coordinate, these points are equidistant from the vertex, and the x-coordinate of

the vertex is halfway between the x-coordinates of these two points. Therefore, for the graph of , the points with x-coordinates and

are equidistant from the vertex, , and is halfway between and . The value that is halfway between and is , or .

Therefore,

. The equation defining can also be written in vertex form,



. Substituting for in this equation

yields



, or



. This equation is equivalent to



, or



. Since



, it follows that and . Dividing both sides of the equation by yields



, or . Since



, it's not true that



. Therefore, statement II isn't true. Substituting for in the equation

yields



, or



. Subtracting from both sides of this equation yields . If , then , or



. Since

could be any value less than , it's not necessarily true that



. Therefore, statement I isn't necessarily true. Thus, neither I nor II must be true.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: 252a3b3a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

Which of the following could be the equation of the graph shown in the xy-plane?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. Each of the given choices is an equation of the form



, where , , , and are positive

constants. In the xy-plane, the graph of an equation of this form has x-intercepts at , , and . The graph shown has x-

intercepts at , , and . Therefore, and . Of the given choices, only choices A and B have and . For an

equation in the form , if all values of that are less than or greater than correspond to negative y-values,

then the sum of all the exponents of the factors on the right-hand side of the equation is even. In the graph shown, all values of less than or

greater than correspond to negative y-values. Therefore, the sum of all the exponents of the factors on the right-hand side of the equation



must be even. For choice A, the sum of these exponents is , or , which is odd. For choice B, the sum of

these exponents is



, or , which is even. Therefore,



could be the equation of the graph shown.

Choice A is incorrect. For the graph of this equation, all values of less than correspond to positive, not negative, y-values.

Choice C is incorrect. The graph of this equation has x-intercepts at , , and , rather than x-intercepts at , , and

.

Choice D is incorrect. The graph of this equation has x-intercepts at , , and , rather than x-intercepts at , , and

.

Question ID: b39d74a0

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

x y

0 0

1 1

2 8

3 27

The table shown includes some values of x and their corresponding values of y. Which of the following graphs in the xy-plane could represent the

relationship between x and y ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. Each pair of values shown in the table gives the ordered pair of coordinates for a point that lies on the graph that represents

the relationship between x and y in the xy-plane: , , , and . Only the graph in choice B passes through the points listed in

the table that are visible in the given choices.

Choices A, C, and D are incorrect. None of these graphs passes through the point .

Question ID: d4950429

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

A rectangle has a length of units and a width of units. If the rectangle has an area of square units, what is the value of ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, units, by the given width,

units, yields



square units. If the rectangle has an area of square units, it follows that



, or



. Subtracting from both sides of this equation yields



. Factoring the left-hand side of this equation yields



. Applying the zero product property to this equation yields two solutions: and . Since is the rectangle’s

length, in units, which must be positive, the value of is .

Choice A is incorrect. This is the width, in units, of the rectangle, not the value of .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of .

Question ID: dd3b1e1a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

For the given function , the graph of



in the xy-plane passes through the point , where is a constant. What is the value of ?

Correct Answer: 17

Rationale

The correct answer is . It's given that the graph of in the xy-plane passes through the point , where is a constant. It follows

that equals . Substituting for in the given function yields , or . Therefore, the value of is .

Question ID: 35e05e19

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

A park ranger hung squirrel houses each in the shape of a right rectangular prism for fox squirrels. Each house has a height of inches. The

length of each house's base is inches, which is inch more than the width of the house's base. Which function gives the volume of each

house, in cubic inches, in terms of the length of the house's base?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct. The volume of a prism is equal to the area of its base times its height.It's given that the length of each house's base is

inches and that this length is inch more than the width, in inches, of the house's base. It follows that the width, in inches, of the house's base is

. The area of a rectangle is the product of its length and its width. Therefore, the area of the base of the house is square inches.

It's given that the height of each house is inches. Therefore, the function that gives the volume of each house, in cubic inches, in terms of

the length of the house's base is , or .

Choice B is incorrectand may result from conceptual or calculation errors.

Choice C is incorrectand may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 341ba5db

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

What is the minimum value of the given function?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. For a quadratic function defined by an equation of the form , where , , and are constants and

, the minimum value of the function is . In the given function, , , and . Therefore, the minimum value of the given

function is .

Choice A is incorrect. This is the value of for which the given function reaches its minimum value, not the minimum value of the function.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 18e35375

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function is defined by the given equation. For what value of does reach its minimum?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. It's given that , which can be rewritten as . Since the coefficient of the

-term is positive, the graph of



in the

xy

-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the

vertex is the value of such that



reaches its minimum. For an equation in the form , where , , and are constants,

the x-coordinate of the vertex is . For the equation , , , and . It follows that the x-coordinate of

the vertex is , or . Therefore, reaches its minimum when the value of is .

Alternate approach: The value of for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since

it’s given that , it follows that the two x-intercepts of the graph of in the xy-plane occur when and

, or at the points and . The midpoint between two points, and , is . Therefore, the

midpoint between



and



is , or . It follows that



reaches its minimum when the value of is .

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of in the xy-plane.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: b5c43226

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

What is the y-intercept of the graph shown?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-

axis at the point . Therefore, the y-intercept of the graph shown is .



Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 50e40f08

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

If the given function is graphed in the xy-plane, where , what is the x-coordinate of an x-intercept of the graph?

Correct Answer: -6, 4

Rationale

The correct answer is either or . The x-intercepts of a graph in the xy-plane are the points where . Thus, for an x-intercept of the graph

of , . Substituting for



in the equation



yields . By the zero product

property, and . Subtracting from both sides of the equation yields . Adding to both sides of the

equation yields . Therefore, the x-coordinates of the x-intercepts of the graph of



are and . Note that -6 and 4 are

examples of ways to enter a correct answer.

Question ID: cfff8f8e

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

At the time of posting a video, a social media channel had subscribers. Each day for five days after the video was posted, the number of

subscribers doubled from the number the previous day. Which equation gives the total number of subscribers, , to the channel days after the

video was posted?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. It's given that each day for five days after a social media channel posted a video, the number of subscribers doubled from the

number the previous day. Since the number of subscribers doubled each day, the relationship between and can be represented by an

exponential equation of the form , where is the number of subscribers at the time of posting the video and the number of subscribers

to the channel increases by a factor of each day. It's given that at the time of posting the video, the channel had subscribers. Therefore,

. It's also given that the number of subscribers doubled, or increased by a factor of , from the number the previous day. Therefore, .

Substituting



for and for in the equation yields .

Choice A is incorrect. This equation gives the total number of subscribers to a channel for which the initial number of subscribers was and the

number increased each day by times the number from the previous day.

Choice C is incorrect. This equation gives the total number of subscribers to a channel for which the number of subscribers each day was half

the number from the previous day, rather than double the number.

Choice D is incorrect and may result from conceptual errors.

Question ID: 8462b105

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function gives the product of a number, , and a number that is more than . Which equation defines ?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It’s given that the function gives the product of a number, , and a number that is more than . A number that is

more than can be represented by the expression . Therefore, can be defined by the equation , or

.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: ce579859

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

A model estimates that at the end of each year from



to , the number of squirrels in a population was more than the number of

squirrels in the population at the end of the previous year. The model estimates that at the end of , there were squirrels in the

population. Which of the following equations represents this model, where is the estimated number of squirrels in the population years after

the end of and ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, , each year,

the model can be represented by an exponential equation of the form



, where is the estimated number of squirrels in the

population at the end of



, and the model estimates that at the end of each year, the number is more than the number at the end of the

previous year. Since the model estimates that at the end of each year, the number was more than the number at the end of the previous

year,

. Substituting for in the equation yields , which is equivalent to , or



. It’s given that the estimated number of squirrels at the end of was . This means that when , . Substituting

for and for in the equation



yields



, or . Dividing each side of this equation by yields .

Substituting

for in the equation



yields



.

Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was of, not



more than, the estimated number at the end of the previous year.

Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was



of, not



more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of , not the end of



, was .

Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of , not the end of



, was

.

Question ID: d139cf4b

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function is defined by the given equation. The function is defined by . Which expression represents the maximum value

of ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. It’s given that function is defined by and that . Substituting for in

yields , or . The maximum value of can be found by completing the square to

rewrite the equation defining



in the form , where the maximum value of the function is , which occurs when , and

is a negative constant. The equation is equivalent to , which can be rewritten as

, or . This equation is in the form , where

, , and . Thus, the maximum value of is .

Alternate approach: Since the function is a quadratic function, the maximum value of occurs at the value of that’s halfway between the

two zeros of the function. The zeros of function can be found by substituting for in the equation defining , which yields .

This equation can be rewritten as . By the zero product property, it follows that or . Subtracting from each

side of the equation yields . Dividing each side of this equation by yields . Therefore, the zeros of function



are and . The value that’s halfway between and can be found by calculating the average of and , which is , or . It follows

that the maximum of function



occurs when . Substituting for in the equation defining function yields

, which is equivalent to . Multiplying by in this equation to get a common denominator yields

, or , which is equivalent to . Thus, the maximum value of is . Since the

equation defining

is , the maximum value of is greater than the maximum value of . It follows that the maximum

value of

is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: a31417d1

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

From 2005 through 2014, the number of music CDs sold in the United States declined each year by approximately 15% of the number sold the

preceding year. In 2005, approximately 600 million CDs were sold in the United States. Of the following, which best models C, the number of

millions of CDs sold in the United States, t years after 2005?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. A model for a quantity C that decreases by a certain percentage per time period t is an exponential equation in the form

, where I is the initial value at time for r% annual decline. It’s given that C is the number of millions of CDs sold in the

United States and that t is the number of years after 2005. It’s also given that 600 million CDs were sold at time , so . This number

declines by 15% per year, so . Substituting these values into the equation produces , or .

Choice A is incorrect and may result from errors made when representing the percent decline. Choices C and D are incorrect. These equations

model exponential increases in CD sales, not exponential decreases.

Question ID: 9afe2370

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The population P of a certain city y years after the last census is modeled by the equation below, where r is a constant and is the population

when .

If during this time the population of the city decreases by a fixed percent each year, which of the following must be true?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The term (1 + r) represents a percent change. Since the population is decreasing, the percent change must be between 0%

and 100%. When the percent change is expressed as a decimal rather than as a percent, the percentage change must be between 0 and 1.

Because (1 + r) represents percent change, this can be expressed as 0 < 1 + r < 1. Subtracting 1 from all three terms of this compound inequality

results in –1 < r < 0.

Choice A is incorrect. If r < –1, then after 1 year, the population P would be a negative value, which is not possible. Choices C and D are incorrect.

For any value of r > 0, 1 + r > 1, and the exponential function models growth for positive values of the exponent. This contradicts the given

information that the population is decreasing.

Question ID: df71424b

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

10 8 6 4 2 2 4 6 8 10

x

10

8

6

4

2

2

4

y

O

The graph of



is shown, where , and , , and are constants. For how many values of does ?

Answer

A. Three

B. Two

C. One

D. Zero

Correct Answer: D

Rationale

Choice D is correct. Each point



on the graph of



in the xy-plane gives a value of and its corresponding value of , or . For

any value of for which , there is a corresponding point on the graph of



with a y-coordinate of . A point with a y-coordinate

of is a point where the graph intersects the x-axis. It's given that , where , , and are constants. In the xy-plane, the graph of

an equation of this form will lie entirely either above or below the horizontal line defined by . The part of the graph of shown lies

entirely below the horizontal line defined by , and thus the entire graph of



must lie below the line defined by . It follows

that the graph of will never intersect the x-axis. Therefore, there are zero values of x for which .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question ID: 203774bc

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The product of two positive integers is . If the first integer is greater than twice the second integer, what is the smaller of the two

integers?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. Let be the first integer and let be the second integer. If the first integer is greater than twice the second integer, then

. If the product of the two integers is , then . Substituting for in this equation results in

. Distributing the to both terms in the parentheses results in . Subtracting from both sides of this equation results

in . The left-hand side of this equation can be factored by finding two values whose product is , or , and

whose sum is . The two values whose product is and whose sum is are and . Thus, the equation

can be rewritten as , which is equivalent to , or . By the

zero product property, it follows that and . Subtracting from both sides of the equation yields

. Dividing both sides of this equation by yields . Since is a positive integer, the value of is not . Adding to both

sides of the equation yields . Substituting for in the equation yields . Dividing both sides of this

equation by results in . Therefore, the two integers are and , so the smaller of the two integers is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the larger of the two integers.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: c4cd5bcc

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

In the xy-plane, the y-coordinate of the y-intercept of the graph of the function f is c. Which of the following must be equal to c ?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct. A y-intercept is the point in the xy-plane where the graph of the function crosses the y-axis, which is where . It’s given

that the y-coordinate of the y-intercept of the graph of function f is c. It follows that the coordinate pair representing the y-intercept must be

. Therefore, c must equal .

Choices B, C, and D are incorrect because , , and would represent the y-value of the coordinate where , , and

, respectively.

Question ID: 68607eca

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

On April , there were views of an advertisement posted on a website. Every days after April , the number of views of the advertisement

had increased by of the number of views days earlier. The function gives the predicted number of views days after April . Which

equation defines ?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It’s given that on April , there were views of the advertisement. It’s also given that every days after April , the number

of views of the advertisement had increased by of the number of views days earlier. This situation can be represented by an exponential

function of the form , where is the number of views on April and every days after April , the number of views had

increased by of the number of views days earlier. It follows that , , and . Substituting for , for , and for in

the equation

yields , or .

Choice A is incorrect. This function gives the predicted number of views for an advertisement for which every days, the number of views was

, rather thanincreased by , of the number of views days earlier.

Choice B is incorrect. This function gives the predicted number of views for an advertisement for which every days, the number of views was

of the number of views days earlier, rather than an advertisement for which every days, the number of views had increased by of

the number of views days earlier.

Choice D is incorrect. This function gives the predicted number of views for an advertisement for which every days, rather than every days,

the number of views had increased by of the number of views days earlier, rather than days earlier.

Question ID: 78d5f91a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

For the function f defined above, what is the value of ?

Answer

A.

B.

C. 7

D. 11

Correct Answer: C

Rationale

Choice C is correct. Substituting for x in the given function f gives , which simplifies to

. This further simplifies to , or .

Choice A is incorrect and may result from correctly substituting for x in the function but incorrectly simplifying the resulting expression to

, or . Choice B is incorrect and may result from arithmetic errors. Choice D is incorrect and may result from

correctly substituting for x in the function but incorrectly simplifying the expression to , or 11.

Question ID: d675744f

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Which of the following is the graph in the xy-plane of the given equation?

A.

B.

C.

D.

Choice D is correct. The y-intercept of the graph of an equation is the point , where b is the value of y when . For the given equation,

when . It follows that the y-intercept of the graph of the given equation is . Additionally, for the given equation, the value of y

doubles for each increase of 1 in the value of x. Therefore, the graph contains the points , , , and . Only the graph

shown in choice D passes through these points.

Choices A and B are incorrect because these are graphs of decreasing, not increasing, exponential functions. Choice C is incorrect because the

value of y increases by a growth factor greater than 2 for each increase of 1 in the value of x.

Question ID: 5377d9cf

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

If , what is ?

Answer

A. –5

B. –2

C. 2

D. 5

Correct Answer: A

Rationale

Choice A is correct. Substituting –1 for x in the equation that defines f gives . Simplifying the expressions in

the numerator and denominator yields , which is equal to or –5.

Choices B, C, and D are incorrect and may result from misapplying the order of operations when substituting –1 for x.

Question ID: f880f910

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The area of a triangle is square centimeters. The length of the base of the triangle is centimeters greater than the height of the triangle.

What is the height, in centimeters, of the triangle?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The area,, of a triangle is given by the formula , where represents the length of the base of the triangle and

represents its height. It’s given that the area of a triangle is square centimeters and that the length of the base of this triangle is

centimeters greater than the height of the triangle. Let represent the height, in centimeters, of the triangle. It follows that the length of the base

of the triangle can be expressed as



. Substituting for , for , and for in the formula yields

, or . Multiplying both sides of this equation by yields . Applying the distributive property on

the right-hand side of this equation yields . Subtracting from both sides of this equation yields



. In

factored form, this equation is equivalent to . Applying the zero product property, it follows that or

. Subtracting from both sides of the equation yields . Adding to both sides of the equation yields

. Since represents the height of the triangle, it must be positive. Therefore, the height, in centimeters, of the triangle is.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 75915e3c

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

For the function f defined above, what is the value of ?

Answer

A. 9

B. 12

C. 18

D. 36

Correct Answer: C

Rationale

Choice C is correct. The value of is found by evaluating the expression when . Substituting 2 for x in the given equation

yields . Simplifying in the equation results in . Evaluating the right-hand side of the equation yields .

Therefore, the value of is 18.

Choice A is incorrect and may result from evaluating the expression as . Choice B is incorrect and may result from evaluating the expression

as . Choice D is incorrect and may result from evaluating the expression as .

Question ID: f44a29a8

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

An object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the square of the object’s speed, in

meters per second. What is the speed, in meters per second, of an object with a mass of 4 kilograms and kinetic energy of 18 joules?

Answer

A. 3

B. 6

C. 9

D. 36

Correct Answer: A

Rationale

Choice A is correct. It’s given that an object’s kinetic energy, in joules, is equal to the product of one-half the object’s mass, in kilograms, and the

square of the object’s speed, in meters per second. This relationship can be represented by the equation , where K is the kinetic

energy, m is the mass, and v is the speed. Substituting a mass of 4 kilograms for m and a kinetic energy of 18 joules for K results in the equation

, or . Dividing both sides of this equation by 2 yields . Taking the square root of both sides yields

and . Since speed can’t be expressed as a negative number, the speed of the object is 3 meters per second.

Choice B is incorrect and may result from computation errors. Choice C is incorrect. This is the value of rather than v. Choice D is incorrect.

This is the value of rather than v.

Question ID: 0121a235

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

x p(x)

The table above gives selected values of a polynomial function p. Based on the values in the table, which of the following must be a factor of p ?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. According to the table, when x is or 2, . Therefore, two x-intercepts of the graph of p are and .

Since and are x-intercepts, it follows that and are factors of the polynomial equation. This is because when

, the value of is 0. Similarly, when , the value of is 0. Therefore, the product is a factor of the

polynomial function p.

Choice A is incorrect. The factor corresponds to an x-intercept of , which isn’t present in the table. Choice B is incorrect. The factor

corresponds to an x-intercept of , which isn’t present in the table. Choice C is incorrect. The factors and correspond

to x-intercepts and , respectively, which aren’t present in the table.

Question ID: d71f6dbf

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The height, in feet, of an object x seconds after it is thrown straight up in the air can be modeled by the function .

Based on the model, which of the following statements best interprets the equation ?

Answer

A. The height of the object 1.4 seconds after being thrown straight up in the air is 1.64 feet.

B. The height of the object 1.64 seconds after being thrown straight up in the air is 1.4 feet.

C. The height of the object 1.64 seconds after being thrown straight up in the air is approximately 1.4 times as great as its initial height.

D. The speed of the object 1.4 seconds after being thrown straight up in the air is approximately 1.64 feet per second.

Correct Answer: A

Rationale

Choice A is correct. The value 1.4 is the value of x, which represents the number of seconds after the object was thrown straight up in the air.

When the function h is evaluated for x = 1.4, the function has a value of 1.64, which is the height, in feet, of the object.

Choices B and C are incorrect and may result from misidentifying seconds as feet and feet as seconds. Additionally, choice C may result from

incorrectly including the initial height of the object as the input x. Choice D is incorrect and may result from misidentifying height as speed.

Question ID: 20722644

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function is defined by . What is the value of ?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It's given that . Substituting for in this equation yields . This is equivalent to

, or



.

Choice A is incorrect. This is the value of , not .

Choice B is incorrect. This is the value of , not .

Choice D is incorrect. This is the value of , not .

Question ID: bba18ecb

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

When the quadratic function is graphed in the xy-plane, where , its vertex is . One of the x-intercepts of this graph is

. What is the other x-intercept of the graph?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. Since the line of symmetry for the graph of a quadratic function contains the vertex of the graph, the x-coordinate of the

vertex of the graph of



is the x-coordinate of the midpoint of its two x-intercepts. The midpoint of two points with x-coordinates and



has x-coordinate, where



. It′s given that the vertex is



and one of the x-intercepts is . Substituting for

and



for in the equation



yields . Multiplying each side of this equation by yields .

Adding

to each side of this equation yields . Therefore, the other x-intercept is .



Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 668f1863

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

Function is a quadratic function where and



. The graph of



in the xy-plane has a vertex at



. What is

the value of ?

Correct Answer: -12

Rationale

The correct answer is . It’s given that function is a quadratic function where



and . It follows that the graph of

in the xy-plane passes throughthe points and . When the graph of a quadratic function contains two points



and

, the x-coordinate of the vertex of the graph is the average of and .Therefore, the x-coordinate of the vertex of the graph of is

, or . It's given that the graph of in the xy-plane has a vertex at . It follows that the value of is



.

Question ID: 70753f99

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function f is defined by . The graph of f in the xy-plane is a parabola. Which of the following intervals contains the

x-coordinate of the vertex of the graph of f ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The graph of a quadratic function in the xy-plane is a parabola. The axis of symmetry of the parabola passes through the

vertex of the parabola. Therefore, the vertex of the parabola and the midpoint of the segment between the two x-intercepts of the graph have the

same x-coordinate. Since , the x-coordinate of the vertex is . Of the shown intervals, only the

interval in choice B contains –2. Choices A, C, and D are incorrect and may result from either calculation errors or misidentification of the graph’s

x-intercepts.

Question ID: 5e63b9cf

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

1 2 3 4 5 6 7 8 9 10

x

1

2

3

4

5

6

7

8

9

10

y

O

The graph shows a marble's height above the ground , in inches, seconds after it started moving on an elevated track of a marble run. Which

of the following is the best interpretation of the -intercept of the graph?

Answer

A. The marble's height was inches above the ground seconds after it started moving.

B. The marble's height was inches above the ground when it started moving.

C. The marble's minimum height was inches above the ground.

D. The marble's minimum height was inches above the ground.

Correct Answer: B

Rationale

Choice B is correct. The y-intercept of a graph is the point at which the graph intersects the y-axis. The graph shown intersects the y-axis at the

point . Therefore, the y-intercept of the graph is . It’s given that is the height of the marble above the ground, in inches, and is the

number of seconds after the marble started moving. It follows that the marble's height was inches above the ground seconds after it started

moving. Therefore, the best interpretation of the y-intercept of the graph is that the marble’s height was inches above the ground when it

started moving.

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question ID: 6676f055

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

An engineer wanted to identify the best angle for a cooling fan in an engine in order to get the greatest airflow. The engineer discovered that the

function above models the airflow , in cubic feet per minute, as a function of the angle of the fan , in degrees. According to the model,

what angle, in degrees, gives the greatest airflow?

Answer

A. –0.28

B. 0.28

C. 27

D. 880

Correct Answer: C

Rationale

Choice C is correct. The functionfis quadratic, so it will have either a maximum or a minimum at the vertex of the graph. Since the coefficient of

the quadratic term(–0.28)is negative, the vertex will be at a maximum. The equation f( ) = –0.28( – 27)2 + 880 is given in vertex form, so the

vertex is at = 27. Therefore,an angle of 27 degreesgives the greatest airflow.

Choices A and B areincorrect and may be the result ofmisidentifyingwhich value in a quadratic equation in vertex form represents the

vertex.Choice D is incorrect.This choice identifies the maximum value of f( ) rather than the value of for which f( ) is maximized.

Question ID: dd8ac009

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

Time (years) Total amount (dollars)

Sara opened a savings account at a bank. The table shows the exponential relationship between the time , in years, since Sara opened the

account and the total amount , in dollars, in the account. If Sara made no additional deposits or withdrawals, which of the following equations

best represents the relationship between and ?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. It’s given that the relationship between and is exponential. The table shows that the value of increases as the value of

increases. Therefore, the relationship between and can be represented by an increasing exponential equation of the form ,

where

and are positive constants. The table shows that when , . Substituting for and for in the equation

yields



, which is equivalent to



, or



. Substituting for in the equation

yields



. The table also shows that when , . Substituting for and for in the equation

yields



, or



. Dividing both sides of this equation by yields .

Subtracting

from both sides of this equation yields . Substituting for in the equation yields



. Therefore, of the choices, choice B best represents the relationship between and .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 58dcc59f

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the width. If the area of the garden will be

104square feet, what will be the length, in feet, of the garden?

Rationale

The correct answer is 13. Let w represent the width of the rectangular garden, in feet. Since the length of the garden will be 5 feet longer than the

width of the garden, the length of the garden will be feet. Thus the area of the garden will be . It is also given that the area of

the garden will be 104 square feet. Therefore, , which is equivalent to . Factoring this equation results

in . Therefore, and . Because width cannot be negative, the width of the garden must be 8 feet. This

means the length of the garden must be feet.

Question ID: 84dd43f8

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

For the function , , and for each increase in by , the value of decreases by . What is the value of ?

Correct Answer: 3.44, 86/25

Rationale

The correct answer is . It’s given that



and that for each increase in by , the value of



decreases by . Because the

output of the function decreases by a constant percentage for each -unit increase in the value of , this relationship can be represented by an

exponential function of the form



, where represents the initial value of the function and represents the rate of decay,

expressed as a decimal. Because



, the value of must be . Because the value of



decreases by



for each -unit increase

in , the value of must be , or . Therefore, the function can be defined by . Substituting for in this

function yields

, which is equivalent to , or . Either or



may be entered as the correct

answer.

Alternate approach: It’s given that and that for each increase in by , the value of



decreases by . Therefore, when



,

the value of



is , or , of , which can be expressed as . Since



, the value of is .

Similarly, when



, the value of



is



of , which can be expressed as . Since



, the value of



is . Either or



may be entered as the correct answer.

Question ID: 59d1f4b5

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The equation above models the number of members,M, of a gym t years after the gym opens. Of the following, which equation models the

number of members of the gym q quarter years after the gym opens?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct. In 1 year, there are 4 quarter years, so the number of quarter years, q, is 4 times the number of years, t ; that is, . This

is equivalent to , and substituting this into the expression for M in terms of t gives .

Choices B and D are incorrect and may be the result of incorrectly using instead of . (Choices B and D are nearly the same since

is equivalent to , which is approximately .) Choice C is incorrect and may be the result of incorrectly using



and unnecessarily dividing 0.02 by 4.

Question ID: 281a4f3b

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled each year will be

2%less than the number of students enrolled the year before. Which of the following functions models the relationship between the number of

students enrolled, , and the number of years after 2015, x?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. Because the change in the number of students decreases by the same percentage each year, the relationship between the

number of students and the number of years can be modeled with a decreasing exponential function in the form , where

is the number of students, a is the number of students in 2015, r is the rate of decrease each year, and x is the number of years since 2015.

It’s given that 3,000 students were enrolled in 2015 and that the rate of decrease is predicted to be 2%, or 0.02. Substituting these values into the

decreasing exponential function yields , which is equivalent to .

Choices A, B, and C are incorrect and may result from conceptual errors when translating the given information into a decreasing exponential

function.

Question ID: 100030d9

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

A rubber ball bounces upward one-half the height that it falls each time it hits the ground. If the ball was originally dropped from a distance of

20.0 feet above the ground, what was its maximum height above the ground, in feet, between the third and fourth time it hit the ground?

Rationale

The correct answer is 2.5. After hitting the ground once, the ball bounces to feet. After hitting the ground a second time, the

ball bounces to feet. After hitting the ground for the third time, the ball bounces to feet. Note that 2.5 and 5/2

are examples of ways to enter a correct answer.

Question ID: d84a514a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function gives a company’s predicted annual revenue, in dollars, years after the company started selling jewelry

online, where . What is the best interpretation of the statement

“

is approximately equal to

”

in this context?

Answer

A. years after the company started selling jewelry online, its predicted annual revenue is approximately dollars.

B. years after the company started selling jewelry online, its predicted annual revenue will have increased by a total of approximately

dollars.

C. When the company’s predicted annual revenue is approximately dollars, it is times the predicted annual revenue for the previous

year.

D. When the company’s predicted annual revenue is approximately dollars, it is greater than the predicted annual revenue for the

previous year.

Correct Answer: A

Rationale

Choice A is correct. It’s given that the function gives a company’s predicted annual revenue, in dollars, years after the company started

selling jewelry online. Since the value of is the value of when , it follows that

“

is approximately equal to

”

means

that is approximately equal to when . Therefore, the best interpretation of the given statement is that years after the

company started selling jewelry online, its predicted annual revenue is approximately dollars.

Choice B is incorrect. The function gives the predicted annual revenue, not the predicted increase in annual revenue.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect. In the given function, represents the number of years after the company started selling jewelry online, not the percent

increase in revenue from the previous year.

Question ID: c7a187a7

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

If the given function is graphed in the xy-plane, where , what is an x-intercept of the graph?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct.It's given that



. The x-intercepts of a graph in the xy-plane are the points where . Thus, for an x-intercept of

the graph of function , . Substituting for in the equation



yields . Factoring the

right-hand side of this equation yields . By the zero product property, and . Subtracting from

both sides of the equation yields . Adding to both sides of the equation yields . Therefore, the x-

intercepts of the graph of



are



and . Of these two x-intercepts, only



is given as a choice.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: ebe4bde0

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The quadratic function graphed above models a particular measure of plant diversity as a function of the elevation in a region of Switzerland.

According to the model, which of the following is closest to the elevation, in meters, at which plant diversity is greatest?

Answer

A. 13,500

B. 3,000

C. 1,250

D. 250

Correct Answer: C

Rationale

Choice C is correct. Each point on the graph represents the elevation x, in meters, and the corresponding measure of plant diversity y in a

region of Switzerland. Therefore, the point on the graph with the greatest y-coordinate represents the location that has the greatest measure of

plant diversity in the region. The greatest y-coordinate of any point on the graph is approximately 13,500. The x-coordinate of that point is

approximately 1,250. Therefore, the closest elevation at which the plant diversity is the greatest is 1,250 meters.

Choice A is incorrect. This value is closest to the greatest y-coordinate of any point on the graph and therefore represents the greatest measure

of plant diversity, not the elevation where the greatest measure of plant diversity occurs. Choice B is incorrect. At an elevation of 3,000 meters

the measure of plant diversity is approximately 4,000. Because there are points on the graph with greater y-coordinates, 4,000 can’t be the

greatest measure of plant diversity, and 3,000 meters isn’t the elevation at which the greatest measure of plant diversity occurs. Choice D is

incorrect. At an elevation of 250 meters, the measure of plant diversity is approximately 11,000. Because there are points on the graph with

greater y-coordinates, 11,000 can’t be the greatest measure of plant diversity and 250 meters isn’t the elevation at which the greatest measure of

plant diversity occurs.

Question ID: 635f54ee

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The surface area of a cube is , where a is a positive constant. Which of the following gives the perimeter of one face of thecube?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. A cube has 6 faces of equal area, so if the total surface area of a cube is , then the area of one face is .

Likewise, the area of one face of a cube is the square of one of its edges; therefore, if the area of one face is , then the length of one edge

of the cube is . Since the perimeter of one face of a cube is four times the length of one edge, the perimeter is .

Choice A is incorrect because if the perimeter of one face of the cube is , then the total surface area of the cube is ,

which is not . Choice C is incorrect because if the perimeter of one face of the cube is 4a, then the total surface area of the cube is

, which is not . Choice D is incorrect because if the perimeter of one face of the cube is 6a, then the total surface

area of the cube is , which is not .

Question ID: e1391dd6

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

According to Moore’s law, the number of transistors included on microprocessors doubles every 2 years. In 1985, a microprocessor was

introduced that had 275,000 transistors. Based on this information, in which of the following years does Moore’s law estimate the number of

transistors to reach 1.1million?

Answer

A. 1987

B. 1989

C. 1991

D. 1994

Rationale

Choice B is correct. Let x be the number of years after 1985. It follows that represents the number of 2-year periods that will occur within an

x-year period. According to Moore’s law, every 2 years, the number of transistors included on microprocessors is estimated to double. Therefore,

x years after 1985, the number of transistors will double times. Since the number of transistors included on a microprocessor was 275,000,

or .275 million, in 1985, the estimated number of transistors, in millions, included x years after 1985 can be modeled as . The year

in which the number of transistors is estimated to be 1.1 million is represented by the value of x when . Dividing both sides

of this equation by .275 yields , which can be rewritten as . Since the exponential equation has equal bases on each side, it

follows that the exponents must also be equal: . Multiplying both sides of the equation by 2 yields . Therefore, according

to Moore’s law, 4 years after 1985, or in 1989, the number of transistors included on microprocessors is estimated to reach 1.1 million.

Alternate approach: According to Moore’s law, 2 years after 1985 (in 1987), the number of transistors included on a microprocessor is estimated

to be , or 550,000, and 2 years after 1987 (in 1989), the number of transistors included on microprocessors is estimated to be

, or 1,100,000. Therefore, the year that Moore’s law estimates the number of transistors on microprocessors to reach 1.1 million is

1989.

Choices A, C, and D are incorrect. According to Moore’s law, the number of transistors included on microprocessors is estimated to reach

550,000 in 1987, 2.2 million in 1991, and about 6.2 million in 1994.

Question ID: d46da42c

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function f is defined as shown. Which of the following graphs in the xy-plane could be the graph of ?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. For the quadratic function , the ver tex of the graph is , and because the term is positive, the vertex

is the minimum of the function. Choice D is the only option that meets these conditions.

Choices A, B, and C are incorrect. The vertex of each of these graphs doesn’t correspond to the minimum of the given function.

Question ID: 5bf0f84a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The function above models the height h, in feet, of an object above ground tseconds after being launched straight up in the air. What does the

number 72 represent in the function?

Answer

A. The initial height, in feet, of the object

B. The maximum height, in feet, of the object

C. The initial speed, in feet per second, of the object

D. The maximum speed, in feet per second, of the object

Correct Answer: A

Rationale

Choice A is correct. The variable t represents the seconds after the object is launched. Since , this means that the height, in feet, at 0

seconds, or the initial height, is 72 feet.

Choices B, C, and D are incorrect and may be the result of misinterpreting the function in context.

Question ID: 70ebd3d0

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The function N defined above can be used to model the number of species of brachiopods at various ocean depths d, where d is in hundreds of

meters. Which of the following does the model predict?

Answer

A. For every increase in depth by 1 meter, the number of brachiopod species decreases by 115.

B. For every increase in depth by 1 meter, the number of brachiopod species decreases by 10%.

C. For every increase in depth by 100 meters, the number of brachiopod species decreases by 115.

D. For every increase in depth by 100 meters, the number of brachiopod species decreases by 10%.

Correct Answer: D

Rationale

Choice D is correct. The function N is exponential, so it follows that changes by a fixed percentage for each increase in d by 1. Since d is

measured in hundreds of meters, it also follows that the number of brachiopod species changes by a fixed percentage for each increase in

ocean depth by 100 meters. Since the base of the exponent in the model is 0.90, which is less than 1, the number of brachiopod species

decreases as the ocean depth increases. Specifically, the number of brachiopod species at a depth of meters is 90% of the number of

brachiopod species at a depth of d meters. This means that for each increase in ocean depth by 100 meters, the number of brachiopod species

decreases by 10%.

Choices A and C are incorrect. These describe situations where the number of brachiopod species are decreasing linearly rather than

exponentially. Choice B is incorrect and results from interpreting the decrease in the number of brachiopod species as 10% for every 1-meter

increase in ocean depth rather than for every 100-meter increase in ocean depth.

Question ID: de39858a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function is defined by , where and are positive constants. The graph of in the -plane passes through the

points ( , ) and ( , ). What is the value of ?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It’s given that the function is defined by and that the graph of in the xy-plane passes through the

points



and



. Substituting for and for in the equation

 

yields , or



.

Subtracting from both sides of this equation yields



. Substituting for and for



in the equation



yields

. Subtracting from both sides of this equation yields , which can be rewritten as . Taking the square root of both

sides of this equation yields



and , but because it’s given that is a positive constant, must equal . Because the value of is

and the value of is , the value of is , or .

Choice A is incorrect and may result from finding the value of rather than the value of .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from correctly finding the value of as , but multiplying it by the y-value in the first ordered pair rather than

by the value of .

Question ID: 1178f2df

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The table shows three values of and their corresponding values of , where



and is a quadratic function. What is the y-

coordinate of the y-intercept of the graph of



in the xy-plane?

Correct Answer: -2112

Rationale

The correct answer is



. It's given that is a quadratic function. It follows that can be defined by an equation of the form

, where , , and are constants.It's also given that the table shows three values of and their corresponding values of ,

where

. Substituting



for



in this equation yields



. This equationrepresents a

quadratic relationship between

and , where



is either the maximum or the minimum value of , which occurs when . For quadratic

relationships between

and , the maximum or minimum value of occurs at the value of halfway between any two values of that have the

same corresponding value of . The table shows that x-values of and correspond to the same y-value, . Since is halfway between

and , the maximum or minimum value of occurs at an x-valueof . The table shows that when , . It follows that and

. Subtracting from both sides of the equation yields . Substituting for and for in the equation



yields



, or



. The value of can be found by substituting any x-value and its

corresponding y-value for and , respectively, in this equation. Substituting for and for in this equation yields



, or



. Subtracting from both sides of this equation yields



, or



. Dividing both sides of this equation

by

yields



. Substituting for , for , and for in the equation yields



.

The



y

-intercept of the graph of



in the xy-plane is the point on the graph where . Substituting for in the equation

yields



, or



. This is equivalent to



, so the y-intercept of the

graph of



in the xy-plane is . Thus,the y-coordinate of they

-intercept of the graph of



in thexy

-plane is



.

Question ID: 45df91ee

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

If the given function is graphed in the xy-plane, where , what is the y-intercept of the graph?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct. The x-coordinate of any y-intercept of a graph is . Substituting for in the given equation yields . Since

any nonzero number raised to the

power is , this gives , or . The y-intercept of the graph is, therefore, the point

.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 97158b3a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The area A, in square centimeters, of a rectangular painting can be represented by the expression , where is the width, in

centimeters, of the painting. Which expression represents the length, in centimeters, of the painting?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It's given that the expression



represents the area, in square centimeters, of a rectangular painting, where is

the width, in centimeters, of the painting. The area of a rectangle can be calculated by multiplying its length by its width. It follows that the length,

in centimeters, of the painting is represented by the expression .

Choice A is incorrect. This expression represents the width, in centimeters, of the painting, not its length, in centimeters.

Choice B is incorrect. This is the difference between the length, in centimeters, and the width, in centimeters, of the painting, not its length, in

centimeters.

Choice D is incorrect. This expression represents the area, in square centimeters, of thepainting, not its length, in centimeters.

Question ID: 84e8cc72

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an

elevated surface. The model indicates the object has an initial height of feet above the ground and reaches its maximum height of feet

above the ground seconds after being launched. Based on the model, what is the height, in feet, of the object above the ground seconds

after being launched?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. It's given that a quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds,

after the object is launched off an elevated surface. This quadratic function can be defined by an equation of the form

,

where

is the height of the object seconds after it was launched, and , , and are constants such that the function reaches its

maximum value, , when . Since the model indicates the object reaches its maximum height of feet above the ground seconds

after being launched, reaches its maximum value, , when . Therefore, and . Substituting for and

for in the function yields . Since the model indicates the object has an initial height of

feet above the ground, the value of is when . Substituting for and for in the equation

yields , or . Subtracting from both sides of this equation yields . Dividing

both sides of this equation by yields . Therefore, the model can be represented by the equation .

Substituting for in this equation yields , or . Therefore, based on the model, seconds

after being launched, the height of the object above the ground is

feet.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 0ad5012e

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The given equation models a company’s scheduled deliveries over months, where is the estimated number of scheduled deliveries months

after the end of May , where . Which statement is the best interpretation of the y-intercept of the graph of this equation in the xy-

plane?

Answer

A. At the end of May , the estimated number of scheduled deliveries was .

B. At the end of May , the estimated number of scheduled deliveries was .

C. At the end of June , the estimated number of scheduled deliveries was .

D. At the end of June , the estimated number of scheduled deliveries was .

Correct Answer: B

Rationale

Choice B is correct. The y-intercept of a graph in the xy-plane is the point where . For the given equation, the y-intercept of the graph in the

xy

-plane can be found by substituting for in the equation, which yields , or . Therefore, the y-intercept of

the graph is

. It’s given that is the estimated number of scheduled deliveries months after the end of May . Therefore,

represents months after the end of May , or the end of May . Thus, the best interpretation of the y-intercept of the graph of this

equation in the xy-plane is that at the end of May , the estimated number of scheduled deliveries was .

Choice A is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question ID: dba7432e

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

x f(x)

0 5

1

2

3

The table above gives the values of the function f for some values of x. Which of the following equations could define f ?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. Each choice has a function with coefficient 5 and base 2, so the exponents must be analyzed. When the input value of x

increases, the output value of f(x) decreases, so the exponent must be negative. An exponent of –x yields the values in the table: ,

, , and .

Choices A and B are incorrect and may result from choosing equations that yield an increasing, rather than decreasing, output value of f(x) when

the input value of x increases. Choice C is incorrect and may result from choosing an equation that doesn’t yield the values in the table.

Question ID: 0aaef7aa

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The function is defined by . What is the value of when is equal to ?

Answer

A.

B.

C.

D.

Correct Answer: A

Rationale

Choice A is correct. It's given that . Substituting for in this equation yields . Dividing each side of this equation by

yields . Taking the cube root of each side of this equation yields . Therefore, when is equal to , the value of is .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: b7cd6ca6

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The equation



gives the estimated number of employees at a restaurant, where is the number of years since the restaurant

opened. Which of the following is the best interpretation of the number

in this context?

Answer

A. The estimated number of employees when the restaurant opened

B. The increase in the estimated number of employees each year

C. The number of years the restaurant has been open

D. The percent increase in the estimated number of employees each year

Correct Answer: A

Rationale

Choice A is correct. For an exponential function of the form , where and are constants, the initial value of the function—that is,

the value of the function when

—

is and the value of the function increases by a factor of each time increases by . Since the function

gives the estimated number of employees at a restaurant and is the number of years since the restaurant opened, the best

interpretation of the number

in this context is the estimated number of employees when , or when the restaurant opened.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question ID: 6d9e01a2

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The given equation defines the function . For what value of does reach its minimum?

Correct Answer: 25/4, 6.25

Rationale

The correct answer is . The given equation can be rewritten in the form , where , , and are constants. When

, is the value of for which



reaches its minimum. The given equation can be rewritten as , which is

equivalent to



. This equation can be rewritten as ,

or



, which is equivalent to . Therefore, , so the value of for which

reaches its minimum is . Note that 25/4 and 6.25 are examples of ways to enter a correct answer.

Question ID: 9f2ecade

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The function h is defined above, where a, b, and c are integer constants. If the zeros of the function are 6, and 7, what is the value of c ?

Rationale

The correct answer is 210. Since , 6, and 7 are zeros of the function, the function can be rewritten as .

Expanding the function yields . Thus, , , and . Therefore, the value of c is 210.

Question ID: d45572cc

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The graph of is shown in the xy-plane. The value of is an integer. What is the value of ?

Correct Answer: 3

Rationale

The correct answer is . The value of is the value of on the graph of in the xy-plane that corresponds with . It's given that

the value of is an integer. For the graph of shown, when , the corresponding integer value of is . Therefore, the value of

is .

Question ID: 79ba511a

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function is defined by . What is the value of ?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. The value of is the value of



when . Substituting for in the given function yields , or



, which is equivalent to . Therefore, the value of is .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of



when , rather than .

Choice D is incorrect and may result from conceptual or calculation errors.

Question ID: 50418728

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The graph of the polynomial function , where , is shown. The y-intercept of the graph is . What is the value of ?

Correct Answer: -3

Rationale

The correct answer is. The y-intercept of the graph of a function in the xy-plane is the point where the graph crosses the y-axis. The graph of

thepolynomial function shown crosses the y-axis at the point . It's given that the y-intercept of the graph is



. Thus, the value of is

.

Question ID: f5e8ccf1

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

The function f is defined above. Which of the following is NOT an x-intercept of the graph of the function in the xy-plane?

Answer

A.

B.

C.

D.

Correct Answer: B

Rationale

Choice B is correct. The graph of the function f in the xy-plane has x-intercepts at the points , where . Substituting 0 for

in the given equation yields . By the zero product property, if , then

, , or . Solving each of these linear equations for x, it follows that , , and ,

respectively. This means that the graph of the function f in the xy-plane has three x-intercepts: , , and . Therefore,

isn’t an x-intercept of the graph of the function f.

Alternate approach: Substitution may be used. Since by definition an x-intercept of any graph is a point in the form where k is a constant,

and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function f.

Substituting for x and 0 for in the given equation yields , or . Therefore,

the point doesn’t lie on the graph of the function f and can’t be an x-intercept of the graph.

Choices A, C, and D are incorrect because each of these points is an x-intercept of the graph of the function f in the xy-plane. By definition, an x-

intercept is a point on the graph of the form , where k is a constant. Substituting for x and 0 for in the given equation yields

, or . Since this is a true statement, the point lies on the graph of the function f and is an

x-intercept of the graph. Performing similar substitution using the points and also yields the true statement , illustrating

that these points also lie on the graph of the function f and are x-intercepts of the graph.

Question ID: 1fe10d97

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The given function models the population of Lowell years after a census. Which of the following functions best models the population of

Lowell months after the census?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. It’s given that the function models the population of Lowell years after a census. Since there are months in a year,

months after the census is equivalent to years after the census. Substituting for in the equation



yields

. Therefore, the function that best models the population of Lowell months after the census is

.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: b73ee6cf

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the

following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. Stating that the population will increase each year by 3% from the previous year is equivalent to saying that the population

each year will be 103% of the population the year before. Since the initial population is 50,000, the population after tyears is given by

50,000(1.03)t. It follows that the equation 60,000 = 50,000(1.03)t can be used to estimate the number of years it will take for the population to

reach 60,000.

Choice A is incorrect. This equation models how long it will take the population to decrease from 60,000 to 50,000, which is impossible given the

growth factor. Choice B is incorrect and may resultfrom misinterpreting a 3% growth as growth by a factor of 3. Additionally, this equation

attempts to model how long it will take the population to decrease from 60,000 to 50,000. Choice C is incorrect and may result

frommisunderstanding how to model percent growthby multiplying the initial amount by a factor greater than 1.

Question ID: 3918e8bc

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

An object is kicked from a platform. The equation represents this situation, where is the height of the object above the

ground, in meters, seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?

Answer

A.

B.

C.

D.

Correct Answer: D

Rationale

Choice D is correct. It’s given that the equation represents this situation, where is the height, in meters, of the object

seconds after it is kicked. It follows that the height, in meters, from which the object was kicked is the value of when . Substituting for

in the equation yields , or . Therefore, the object was kicked from a height of meters.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: 7eed640d

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If

is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph?

Answer

A. The initial height of the projectile

B. The maximum height of the projectile

C. The time at which the projectile reaches its maximum height

D. The time at which the projectile hits the ground

Correct Answer: D

Rationale

Choice D is correct. The positive x-intercept of the graph of is a point for which . Since models the height

above the ground, in feet, of the projectile, a y-value of 0 must correspond to the height of the projectile when it is 0 feet above ground or, in other

words, when the projectile is on the ground. Since x represents the time since the projectile was launched, it follows that the positive x-intercept,

, represents the time at which the projectile hits the ground.

Choice A is incorrect and may result from misidentifying the y-intercept as a positive x-intercept. Choice B is incorrect and may result from

misidentifying the y-value of the vertex of the graph of the function as an x-intercept. Choice C is incorrect and may result from misidentifying the

x-value of the vertex of the graph of the function as an x-intercept.

Question ID: 43926bd9

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

For the exponential function f, the table above shows several values of x and their corresponding values of , where a is a constant greater

than 1. If k is a constant and , what is the value of k ?

Rationale

The correct answer is 8. The values of for the exponential function f shown in the table increase by a factor of for each increase of 1 in

x. This relationship can be represented by the equation , where b is a constant. It’s given that when , .



Substituting 2 for x and for into yields . Since , it follows that . Thus, an

equation that defines the function f is . It follows that the value of k such that can be found by solving the

equation , which yields .

Question ID: f25a34aa

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The area of a triangle is equal to square centimeters. The length of the base of the triangle is centimeters, and the height of the

triangle is centimeters. What is the value of ?

Correct Answer: 110

Rationale

The correct answer is . The area of a triangle is equal to one half of the product of the length of the base of the triangle and the height of the

triangle. It's given that the length of the base of the triangle is centimeters and the height of the triangle is centimeters;

therefore, its area is square centimeters. It's also given that the area of the triangle is equal to square centimeters.

Therefore, it follows that . This equation can be rewritten as , or .

Subtracting from both sides of this equation yields . Adding to both sides of this equation yields . Therefore, the

value of is .

Question ID: a58232b7

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

The functions and are defined by the given equations, where . Which of the following equations displays, as a constant or coefficient,

the minimum value of the function it defines, where ?

I.

II.

Answer

A. I only

B. II only

C. I and II

D. Neither I nor II

Correct Answer: D

Rationale

Choice D is correct. A function defined by an equation in the form , where , , and are positive constants and , has a

minimum value of

. It's given that function is defined by , which is equivalent to .

Substituting for in this equation yields , or . Therefore, the minimum value of is , so

doesn't display its minimum value as a constant or coefficient. It's also given that function is defined by

. Substituting for in this equation yields , or . Therefore, the minimum value of is

, so doesn't display its minimum value as a constant or coefficient. Therefore, neither I nor II displays, as a

constant or coefficient, the minimum value of the function it defines, where

.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question ID: a7711fe8

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

What is the minimum value of the functionf defined by



?

Answer

A.

B.

C. 2

D. 4

Correct Answer: A

Rationale

Choice A is correct. The given quadratic function f is in vertex form, , where is the vertex of the graph of

in the xy-plane. Therefore, the vertex of the graph of is . In addition, the y-coordinate of the vertex represents either the

minimum or maximum value of a quadratic function, depending on whether the graph of the function opens upward or downward. Since the

leading coefficient of f (the coefficient of the term ) is 1, which is positive, the graph of opens upward. It follows that at

, the minimum value of the function f is .

Choice B is incorrect and may result from making a sign error and from using the x-coordinate of the vertex. Choice C is incorrect and may result

from using the x-coordinate of the vertex. Choice D is incorrect and may result from making a sign error.

Question ID: 1a722d7d

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Hard

Question

Let the functionp be defined as , wherec is a constant. If , what is the value of ?

Answer

A. 10.00

B. 10.25

C. 10.75

D. 11.00

Correct Answer: D

Rationale

Choice D is correct. The value of p(12) depends on the value of the constant c, so the value of c must first be determined. It is given that p(c) =

10. Based on the definition of p, it follows that:

This means that for all values of x. Therefore:

Choice A is incorrect. It is the value of p(8), not p(12). Choices B and C are incorrect. If one of these values were correct, then x = 12 and the

selected value of p(12) could be substituted into the equation to solve for c. However, the values of c that result from choices B and C each result

in p(c) < 10.

Question ID: ee05c84e

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Easy

Question

The function f is defined above. What is the value of ?

Answer

A. 250

B. 500

C. 750

D. 2,000

Correct Answer: C

Rationale

Choice C is correct. Adding the like terms x and yields the equation . Substituting 20 for x yields

. The product is equal to 25, and the difference is equal to 30. Substituting these values

in the given equation gives , and multiplying 25 by 30 results in



.

Choices A, B, and D are incorrect and may result from conceptual or computational errors when finding the value of .

Question ID: 5c00c2c1

Assessment Test Domain Skill Difficulty

SAT Math Advanced Math Nonlinear functions Medium

Question

There were no jackrabbits in Australia before 1788 when 24 jackrabbits were introduced. By 1920 the population of jackrabbits had reached 10

billion. If the population had grown exponentially, this would correspond to a 16.2% increase, on average, in the population each year. Which of

the following functions best models the population of jackrabbits t years after 1788?

Answer

A.

B.

C.

D.

Correct Answer: C

Rationale

Choice C is correct. This exponential growth model can be written in the form , where is the population t years after

1788, A is the initial population, and r is the yearly growth rate, expressed as a decimal. Since there were 24 jackrabbits in Australia in 1788,

. Since the number of jackrabbits increased by an average of 16.2% each year, . Therefore, the equation that best models

this situation is .

Choices A, B, and D are incorrect and may result from misinterpreting the form of an exponential growth model.