Step 1

The parent function of \(\displaystyle{P}{\left({x}\right)}={81}-{\left({x}-{3}\right)}^{{{4}}}\ {i}{s}\ {y}={x}^{{{4}}}\), which passed through the points \(\displaystyle{\left(-{2},{16}\right)},{\left(-{1},{1}\right)},{\left({0},{0}\right)},{\left({1},{1}\right)},{\quad\text{and}\quad}{\left({2},{16}\right)}.\)

\(\displaystyle{P}{\left({x}\right)}=-{\left({x}-{3}\right)}^{{{4}}}+{81}\) is the graph of \(\displaystyle{y}={x}^{{{4}}}\) reflected across the x-axis and then translated left 3 units right and up 81 units. Reflecting the points on \(\displaystyle{y}={x}^{{{4}}}\) accross the x-axis given the points \(\displaystyle{\left(-{2},-{16}\right)},{\left(-{1},-{1}\right)},{\left({0},{0}\right)},{\left({1},-{1}\right)},{\quad\text{and}\quad}{\left({2},-{16}\right)}\). Translating these points right 3 units and up 81 units then gives the points (1,65),(2,80),(3,81),(4,80), and (5,65).

The x-intercept of P(x) is when \(\displaystyle{P}{\left({x}\right)}={0}\):

\(\displaystyle{P}{\left({x}\right)}={81}-{\left({x}-{3}\right)}^{{{4}}}\) Given function.

\(\displaystyle{0}={81}-{\left({x}-{3}\right)}^{{{4}}}\) Substitute \(\displaystyle{P}{\left({x}\right)}={0}\)

\(\displaystyle{\left({x}-{3}\right)}^{{{4}}}={81}\) Add \(\displaystyle{\left({x}-{3}\right)}^{{{4}}}\)

\(\displaystyle{x}-{3}=\pm\sqrt[4]{81}\) Take the 4th root of both sides.

\(\displaystyle{x}-{3}=\pm{3}\) Simplify

\(\displaystyle{x}={3}\pm{3}\) Add 3 on both sides

The y-intercept of P(x) are then \(\displaystyle{x}={3}-{3}={0}{\quad\text{and}\quad}{x}={3}+{3}={6}\)

The y-intercept is when \(\displaystyle{x}={0}\). From finding the x-intercept, we know \(\displaystyle{y}={0}\) when \(\displaystyle{x}={0}\) so the y-intercept is at (0,0).

Plot the points and then connect them with a smoth curve. Label the coordinates of the intercepts in your graph:

Answer: See the explanation for the graph. To make the graph, reflect the graph of \(\displaystyle{y}={x}^{{{4}}}\) across the x-axis and then translate right 3 units and up 81 units. Label the intercepts at (0,0) and (6,0)