PSAT: Advanced Math - Nonlinear Functions

What is the y-intercept of the graph shown?

A scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. Assuming exponential growth, the formula $P = C(2)^{rt}$ gives the number of bacteria in the growth medium, where $r$ and $C$ are constants and $P$ is the number of bacteria $t$ hours after the initial measurement. What is the value of $r$?

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?

The function $f$ is defined by $f(x) = a^x + b$, where $a$ and $b$ are constants and $a > 0$. In the $xy$-plane, the graph of $y = f(x)$ has a $y$-intercept at $(0, -25)$ and passes through the point $(2, 23)$. What is the value of $a + b$?

The area $A$, in square centimeters, of a rectangular cutting board can be represented by the expression $w(w + 9)$, where $w$ is the width, in centimeters, of the cutting board. Which expression represents the length, in centimeters, of the cutting board?

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 10 feet above the ground and reaches its maximum height of 1,034 feet above the ground 8 seconds after being launched.

Based on the model, what is the height, in feet, of the object above the ground 10 seconds after being launched?

Function $f$ is a quadratic function where $f(-20) = 0$ and $f(-4) = 0$. The graph of $y = f(x)$ in the xy-plane has a vertex at $(r, -64)$. What is the value of $r$?

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 $p$?

An investment account was opened with an initial value of $890. The value of the account doubled every 10 years. Which equation represents the value of the account $M(t)$, in dollars, $t$ years after the account was opened?

A submersible device is used for ocean research. The function $g(x) = -\frac{1}{55}(x + 19)(x - 35)$ gives the depth below the surface of the ocean, in meters, of the submersible device $x$ minutes after collecting a sample, where $x > 0$. How many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?

The area of a rectangular banner is 2,661 square inches. The banner's length $x$, in inches, is 24 inches longer than its width, in inches. Which equation represents this situation?

At the time that an article was first featured on the home page of a news website, there were 40 comments on the article. An exponential model estimates that at the end of each hour after the article was first featured on the home page, the number of comments on the article had increased by 190% of the number of comments on the article at the end of the previous hour.

Which of the following equations best represents this model, where $C$ is the estimated number of comments on the article $t$ hours after the article was first featured on the home page and $t \le 4$?

The graph of $y = f(x)$ is shown in the $xy$-plane. The value of $f(0)$ is an integer. What is the value of $f(0)$?

The function $g$ is defined by $g(x) = x^2 + 9$. For which value of $x$ is $g(x) = 25$?

A park ranger hung squirrel houses each in the shape of a right rectangular prism for fox squirrels. Each house has a height of 11 inches. The length of each house’s base is $x$ inches, which is 1 inch more than the width of the house’s base. Which function $V$ gives the volume of each house, in cubic inches, in terms of the length of the house’s base?

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, $x$?

The given equation models a company’s scheduled deliveries over 8 months, where $y$ is the estimated number of scheduled deliveries $x$ months after the end of May 2012, where $0 \le x \le 8$.

$y = -\dfrac{1}{4}x^{2} + 2x + 29$

Which statement is the best interpretation of the y-intercept of the graph of this equation in the $xy$-plane?

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?

The function $g$ is defined by $g(x) = (x + 14)(t - x)$, where $t$ is a constant. In the $xy$-plane, the graph of $y = g(x)$ passes through the point $(24, 0)$. What is the value of $g(0)$?

The graph models the number of active projects a company was working on $x$ months after the end of November 2012, where $0 \le x \le 6$. According to the model, what is the predicted number of active projects the company was working on at the end of November 2012?

The graph of the quadratic function $y = f(x)$ is shown. What is the vertex of the graph?

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?

The function $P$ models the population, in thousands, of a certain city $t$ years after 2005 and is given by $P(t)=290(1.04)^{(4/3)t}$. According to the model, the population is predicted to increase by $n\%$ every 18 months. What is the value of $n$?

A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population $p$ increased by 4% of the previous year's population. Which equation best represents this model, where $x$ is the number of years after 2005, for $x \le 5$?

$y = 576^{(2x + 2)}$

The graph of the given equation in the $xy$-plane has a $y$-intercept of $(r, s)$. Which of the following equivalent equations displays the value of $s$ as a constant, a coefficient, or the base?

The function $f(w) = 6w^2$ gives the area of a rectangle, in square feet $(\text{ft}^2)$, if its width is $w$ ft and its length is 6 times its width. Which of the following is the best interpretation of $f(14) = 1{,}176$?

Function $f$ is defined by $f(x) = -a^x + b$, where $a$ and $b$ are constants. In the $xy$-plane, the graph of $y = f(x) - 15$ has a $y$-intercept at $\left(0, -\frac{99}{7}\right)$. The product of $a$ and $b$ is $\frac{65}{7}$. What is the value of $a$?

A sample of a certain isotope takes 29 years to decay to half its original mass. The function $s(t) = 184(0.5)^{\tfrac{t}{29}}$ gives the approximate mass of this isotope, in grams, that remains $t$ years after a 184-gram sample starts to decay. Which statement is the best interpretation of $s(87) = 23$ in this context?

The function $f$ is defined by $f(x) = 270(0.1)^x$. What is the value of $f(0)$?

$f(x) = \dfrac{a - 19}{x} + 5$

In the given function $f$, $a$ is a constant. The graph of function $f$ in the $xy$-plane, where $y = f(x)$, is translated 3 units down and 4 units to the right to produce the graph of $y = g(x)$. Which equation defines function $g$?

What is the x-coordinate of the x-intercept of the graph shown?

At the time of posting a video, a social media channel had 53 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, $n$, to the channel $d$ days after the video was posted?

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

A.

B.

C.

D.

A machine launches a softball from ground level. The softball reaches a maximum height of 51.84 meters above the ground at 1.8 seconds and hits the ground at 3.6 seconds. Which equation represents the height above ground $h$, in meters, of the softball $t$ seconds after it is launched?

The function $f$ is defined by $f(x) = x^3 + 15$. What is the value of $f(2)$?

The table shows three values of $x$ and their corresponding values of $y$, where $y = f(x) + 4$ and $f$ is a quadratic function.

What is the $y$-coordinate of the $y$-intercept of the graph of $y = f(x)$ in the $xy$-plane?

$f(t) = 5t - 2t^2$

The function $f$ is defined by the given equation. The function $g$ is defined by $g(t) = f(t) + 3$. Which expression represents the maximum value of $g(t)$?

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?

The number of bacteria in a liquid medium doubles every day. There are 44,000 bacteria in the liquid medium at the start of an observation. Which represents the number of bacteria, $y$, in the liquid medium $t$ days after the start of the observation?

The exponential function $g$ is defined by $g(x) = 19 \cdot a^{x}$, where $a$ is a positive constant. If $g(3) = 2{,}375$, what is the value of $g(4)$?

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 $a$?

A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function $h$ estimates that the maximum height reached after each time the ball hits the ground is 85% of the maximum height reached after the previous time the ball hit the ground. Which equation defines $h$, where $h(n)$ is the estimated maximum height of the ball after it has hit the ground $n$ times and $n$ is a whole number greater than 1 and less than 10?

The function $f$ is defined by $f(x) = a\sqrt{x + b}$, where $a$ and $b$ are constants. In the $xy$-plane, the graph of $y = f(x)$ passes through the point $(-24, 0)$, and $f(24) < 0$. Which of the following must be true?

A function $p$ estimates that there were 2,000 animals in a population in 1998. Each year from 1998 to 2010, the function estimates that the number of animals in this population increased by 3% of the number of animals in the population the previous year. Which equation defines this function, where $p(x)$ is the estimated number of animals in the population $x$ years after 1998?

What is an $x$-coordinate of an $x$-intercept of the graph of $y = 3(x - 14)(x + 5)(x + 4)$ in the $xy$-plane?

The $y$-intercept of the graph of $y = x^2 + 31$ in the $xy$-plane is $(0, y)$. What is the value of $y$?

The function f is defined by $f(x) = (x + 3)(x + 1)$. 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 ?

If $f(x)=\dfrac{x^{2}-6x+3}{x-1}$, what is $f(-1)$?

The function $g$ is defined by $g(x) = x(x - 2)(x + 6)^2$. The value of $g(7 - w)$ is $0$, where $w$ is a constant. What is the sum of all possible values of $w$?