College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 64

Below is the graph of $f$. Draw the graph of the following functions.

$\begin{equation} \begin{aligned} &\text{a.) } y = f(x -2) &&& &\text{b.) } y = -f(x)\\ \\ &\text{c.) } y = 3- f(x) &&& &\text{d.) } y = \frac{1}{2} f(x) - 1\\ \\ &\text{e.) } y = f^{-1} (x) &&& &\text{f.) } y = f(-x) \end{aligned} \end{equation}$




a.) The graph of $y = f(x - 2)$ is obtained by shifting the given graph two units to the right



b.) The graph of $y = -f(x)$ is obtained by reflecting the given graph about $x$-axis



c.) The graph of $y = 3 - f(x)$ is obtained by reflecting the given graph about $x$-axis then the result is shifted 3 units upward.



d.) The graph of $\displaystyle y = \frac{1}{2} f(x) - 1$ is obtained by shrinking the given graph vertically by a factor of $\displaystyle \frac{1}{2}$, then the result is shifted one unit downward.



e.) To obtain the graph of $f^{-1} (x)$, we reflect the given graph about the line $y = x$



f.) The graph of $y = f(-x)$ is obtained by reflecting the given graph about $y$-axis