Step 1

The parent function of \(\displaystyle{P}{\left({x}\right)}=-{x}^{{{3}}}+{64}\ {i}{s}\ {y}={x}^{{{3}}}\), which passes through the points \(\displaystyle{\left(-{2},-{8}\right)},{\left(-{1},-{1}\right)},{\left({0},{0}\right)},{\left({1},{1}\right)}{\quad\text{and}\quad}{\left({2},{8}\right)}\).

\(\displaystyle{P}{\left({x}\right)}=-{x}^{{{3}}}+{64}\) is the graph of \(\displaystyle{y}={x}^{{{3}}}\) reflected across the x-axis and then translated up 64 units. Reflecting the points on \(\displaystyle{y}={x}^{{{3}}}\) across the x-axis gives (-2,8),(-1,1),(0,0),(1,-1) and (2,-8). Translating these points up 64 units then gives the points (-2,72), (-1,65),(0,64),(1,63), and (2,56).

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

\(\displaystyle{P}{\left({x}\right)}=-{x}^{{{3}}}+{64}\) Given function.

\(\displaystyle{0}=-{x}^{{{3}}}+{64}\) Substitute \(\displaystyle{P}{\left({x}\right)}={0}\)

\(\displaystyle{x}^{{{3}}}={64}\) Add \(\displaystyle{x}^{{{3}}}\) on both sides.

\(\displaystyle{x}=\sqrt{{{3}}}{\left\lbrace{64}\right\rbrace}\) Cube root both sides.

\(\displaystyle{x}={4}\) Simplify

The x-intercept of P(x) is then the point (4,0). The y-intercept is when \(\displaystyle{x}={0}\) which we already know is at the point (0,64). Plot the points, including the x-intercept, and then connect them with a smooth curve. Label the coordinates of the intercepts in your graph: