Transformations of Functions (College Board AP® Precalculus): Exam Questions

The graph of $f$ is shifted to the left by 5 units to produce the graph of $g$. Which of the following gives an expression for $g\left(x\right)$ in terms of $f$?

The function $g$ is defined by $g\left(x\right)=-3f\left(x\right)$.

Which of the following describes the transformation that maps the graph of $f$ to the graph of $g$?

The function $f$ has domain $\left[-4,7\right]$. The function $g$ is defined by $g\left(x\right)=f\left(x\right)+5$.

What is the domain of $g$?

The function $f$ is given by $f\left(x\right)=2x^2-3x+1$.

The graph of which of the following functions is the image of the graph of $f$ after a vertical translation of the graph of $f$ by $4$ units?

The graph of $y=f\left(x\right)$, consisting of three line segments and a semicircle, is shown.

Which of the following is the transformed graph for $y=f\left(x-2\right)+3$ ?

The function $f$ is given by $f\left(x\right)=2x^2-x+4$. The graph of which of the following functions is the image of the graph of $f$ after a vertical dilation of the graph of $f$ by a factor of 3 ?

The functions $f$ and $g$ are defined for all real numbers such that $g\left(x\right)=f\left(3\left(x+2\right)\right)$. Which of the following sequences of transformations maps the graph of $f$ to the graph of $g$ in the same $xy$-plane?

The table gives values for a polynomial function $f$ at selected values of $x$. Let $g\left(x\right)=af\left(bx\right)+c$, where $a$, $b$, and $c$ are positive constants. In the $xy$-plane, the graph of $g$ is constructed by applying three transformations to the graph of $f$ in this order: a horizontal dilation by a factor of $2$, a vertical dilation by a factor of $2$, and a vertical translation by $3$ units upward. What is the value of $g(-8)$?

The function $g$ is defined by $g\left(x\right)=f\left(2x\right)$. The graph of $g$ is the graph of $f$ after which of the following transformations?

The function $g$ is defined by $g\left(x\right)=f\left(-x\right)$. The graph of $g$ is the graph of $f$ after a reflection over which of the following?

The function $f$ has domain $\left[-3,5\right]$ and range $\left[1,8\right]$. The function $g$ is defined by $g\left(x\right)=-2f\left(x+1\right)+3$. What is the range of $g$?

Let $f\left(x\right)=\sqrt{x}$. The function $g$ is defined by $g\left(x\right)=-2f\left(x-1\right)+5$. What is the value of $g\left(8\right)$?

The function $f$ has a relative minimum at the point $\left(3,-2\right)$. The function $g$ is defined by $g\left(x\right)=\frac{1}{2}f\left(x\right)$.

Which of the following are the coordinates of the relative minimum of the graph of $g$?

The function $g$ is defined by $g\left(x\right)=3f\left(2x-8\right)-5$. Which of the following describes the horizontal transformations that take the graph of $f$ to the graph of $g$?

The table gives values of a function $f$ at selected values of $x$.

The function $g$ is defined by $g\left(x\right)=-3f\left(\frac{1}{2}\left(x-4\right)\right)+7$.

What is the value of $g\left(8\right)$?

The function $f$ has domain $\left[-6,12\right]$. The function $g$ is defined by $g\left(x\right)=f\left(-2x+4\right)$.

What is the domain of $g$?

Let $f$ be a function with $f\left(2\right)=-1$ and $f\left(4\right)=3$. The graph of the function $g$ defined by $g\left(x\right)=af\left(x\right)+b$ passes through the points $\left(2,5\right)$ and $\left(4,-3\right)$. What is the value of $a+b$?

The function $f$ has domain $\left[-2,4\right]$. What is the domain of the function $g$ defined by $g\left(x\right)=f\left(3-2x\right)$?