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

Let $f\left(x\right)=\ln \left(x\right)$. The function $g$ is defined by $g\left(x\right)=0.5f\left(x-1\right)+4$.

Find the value of $g\left(3\right)$ as a decimal approximation. Show the work that leads to your answer.

The graph of the function $f$ has the following features:

• a vertical asymptote at $x=-3$

• a horizontal asymptote at $y=2$

• a $y$-intercept at $\left(0,6\right)$

The function $g$ is defined by $g\left(x\right)=f\left(x\right)-5$.

(i) Find the equation of the vertical asymptote of the graph of $g$.

(ii) Find the equation of the horizontal asymptote of the graph of $g$.

(iii) Find the coordinates of the $y$-intercept of the graph of $g$.

The function $f$ has range $\left[-2,7\right]$. The function $g$ is defined by $g\left(x\right)=-3f\left(x\right)+5$.

Determine the range of $g$. Show the work that leads to your answer.

The function $f$ has zeros at $x=-1$, $x=2$, and $x=6$. The function $g$ is defined by $g\left(x\right)=f\left(0.5\left(x-3\right)\right)$.

Determine the zeros of $g$. Show the work that leads to your answer.

The function $f$ has a maximum value of 8, which occurs at $x=-1$. The function $g$ is defined by $g\left(x\right)=2f\left(x+2\right)+5$.

Determine the maximum value of $g$ and the value of $x$ at which it occurs. Show the work that leads to your answer.

The function $f$ has domain $\left[-2.3,5.6\right]$. The function $g$ is defined by $g\left(x\right)=f\left(0.7x-1.5\right)$.

Determine the domain of $g$. Express the endpoints of the interval as decimal approximations. Show the work that leads to your answer.

The function $f$ has zeros at $x=-2$ and $x=4$, and $f\left(0\right)=-3$. The graph of $f$ in the $xy$-plane has a vertical asymptote at $x=6$.

The function $g$ is given by $g\left(x\right)=f\left(-2x\right)$.

(i) Find all zeros of $g$.

(ii) Find the equation of the vertical asymptote of the graph of $g$.

(iii) Find $g\left(0\right)$.

The function $g$ is given by $g\left(x\right)=-\frac{1}{2}f\left(3x+12\right)+5$.

Describe the additive and multiplicative transformations that map the graph of $f$ to the graph of $g$.

The function $f$ has a maximum value of $5$ at $x=4$. The function $g$ is given by

$g\left(x\right)=a·f\left(b\left(x+h\right)\right)+k$

where $a$, $b$, $h$, and $k$ are positive constants. The function $g$ has a maximum value of $11$ at $x=1$. The graph of $g$ passes through $\left(-1,3\right)$, which is the image of the point $\left(0,1\right)$ on the graph of $f$ under the transformation.

Find the values of $b$ and $h$. Show the work that leads to your answer.

Find the values of $a$ and $k$. Show the work that leads to your answer.

The function $f$ has domain $\left[-3.4,8.2\right]$ and range $\left[-1.5,4.3\right]$. The function $g$ is defined by $g\left(x\right)=-2f\left(0.6x+1.8\right)+3.5$.

(i) Find the domain of $g$. Express the endpoints of the interval as decimal approximations. Show the work that leads to your answer.

(ii) Find the range of $g$. Show the work that leads to your answer.