Question

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

Polynomial graphs
ANSWERED
asked 2021-08-01
Graph the polynomial by transforming an appropriate graph of the form \(\displaystyle{y}={x}^{{{n}}}\).
Show clearly all x- and y-intercepts. \(\displaystyle{P}{\left({x}\right)}={81}-{\left({x}-{3}\right)}^{{{4}}}\)

Expert Answers (1)

2021-08-02

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:
image
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)

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