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}

Armorikam

Armorikam

Answered question

2021-08-01

Graph the polynomial by transforming an appropriate graph of the form y=xn.
Show clearly all x- and y-intercepts. P(x)=81(x3)4

Answer & Explanation

tafzijdeq

tafzijdeq

Skilled2021-08-02Added 92 answers

Step 1
The parent function of P(x)=81(x3)4 is y=x4, which passed through the points (2,16),(1,1),(0,0),(1,1),and(2,16).
P(x)=(x3)4+81 is the graph of y=x4 reflected across the x-axis and then translated left 3 units right and up 81 units. Reflecting the points on y=x4 accross the x-axis given the points (2,16),(1,1),(0,0),(1,1),and(2,16). 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 P(x)=0:
P(x)=81(x3)4 Given function.
0=81(x3)4 Substitute P(x)=0
(x3)4=81 Add (x3)4
x3=±814 Take the 4th root of both sides.
x3=±3 Simplify
x=3±3 Add 3 on both sides
The y-intercept of P(x) are then x=33=0andx=3+3=6
The y-intercept is when x=0. From finding the x-intercept, we know y=0 when 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 y=x4 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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