College Algebra, Chapter 4, Chapter Review, Section Review, Problem 12

Graph the polynomial $P(x) = 81 - (x - 3)^4$ by transforming an appropriate graph of the form $y = x^n$ and show all the $x$ and $y$ intercepts clearly.

The graph of $P(x) = 81 - (x - 3)^4$ is obtained from the graph of $y = x^4$ that is shifted $3$ units to the right and reflected about the $x$ axis. Then, the result is shifted $81$ units upward.

To find the $x$ intercept, we set $y = 0$, so


$
\begin{equation}
\begin{aligned}

0 =& 81 - (x - 3)^4
\\
\\
(x - 3)^4 =& 81
\\
\\
x - 3 =& \pm \sqrt[4]{81}
\\
\\
x - 3 =& \pm 3
\\
\\
x =& 3 \pm 3

\end{aligned}
\end{equation}
$


We have,

$x = 0$ and $x = 6$

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


$
\begin{equation}
\begin{aligned}

P(0) =& 81 - (0 - 3)^4
\\
\\
=& 81 - (81)
\\
\\
=& 0

\end{aligned}
\end{equation}
$


The $y$ intercept is .