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 .