College Algebra, Chapter 4, 4.2, Section 4.2, Problem 6

Sketch the graph of each function by applying sketch transformations on the standard form $y = x^n$. Indicate all $x$ and $y$-intercept on each graph.
a.) $P(x) = x^4 - 16$
b.) $Q(x) = (x+2)^4$
c.) $R(x) = (x+2)^4 - 16$
d.) $S(x) = -2(x+2)^4$

a.) The graph of $P(x) = x^4 - 16$ is the graph of $y = x^4$ that is shifted 16 units downward. Its $y$-intercept is -16 and the $x$-intercept are 2 and $-2$.



b.) The graph of $Q(x) = (x + 2)^4$ is the graph of $y = x^4$ that is shifted 2 units to the left. Its $y$-intercept is 16 and $x$-intercept is $-2$.




c.) The graph of $R(x) = (x+2)^4 - 16$ is the graph of $y = x^4$ that is shifted 2 units to the left and the result is shifted 16 units downward. To solve for $y$-intercept, we set $x = 0$

$
\begin{equation}
\begin{aligned}
y &= (0 + 2)^4 - 16\\
\\
y &= 16 - 16 \\
\\
y &= 0
\end{aligned}
\end{equation}
$


To solve for $x$-intercept, we set $y = 0$

$
\begin{equation}
\begin{aligned}
0 &= (x + 2)^4 - 16\\
\\
(x + 2)^4 &= 16\\
\\
x + 2 &= \pm \sqrt[4]{16}\\
\\
x + 2 &= \pm 2\\
\\
x &= -2 \pm 2\\
\\
x &= 0 \quad \text{ and } \quad x = -4
\end{aligned}
\end{equation}
$





d.) The graph $S(x) = -2(x + 2)^4$ is the graph of $y = x^4$ that is shifted 2 units to the left then the result is reflected about $x$-axis and stretched vertically but a factor of 2. Its $y$-intercept is $-32$ and $x$-intercept is $-2$.