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Chapter 7: Problem 5

Begin with the function \(f(x)=x^{4}\) a. Create a new function \(g(x)\) by vertically stretching \(f(x)\) by a factor of \(6 .\) b. Create a new function \(h(x)\) by vertically compressing \(f(x)\) by a factor of \(\frac{1}{2}\). c. Create a new function \(j(x)\) by first vertically stretching \(f(x)\) by a factor of 2 and then reflecting it across the \(x\) -axis.

Short Answer

Expert verified
g(x) = 6x^4; h(x) = \frac{1}{2}x^4; j(x) = -2x^4

Step by step solution

01

Vertical Stretch for g(x)

To vertically stretch the function \(f(x) = x^4\) by a factor of 6, multiply the entire function by 6. Therefore, \(g(x) = 6f(x)\). Substituting \(f(x)\) into the equation, we get: \[ g(x) = 6x^4 \]
02

Vertical Compression for h(x)

To vertically compress the function \(f(x) = x^4\) by a factor of \(\frac{1}{2}\), multiply the entire function by \(\frac{1}{2}\). Therefore, \(h(x) = \frac{1}{2}f(x)\). Substituting \(f(x)\) into the equation, we get: \[ h(x) = \frac{1}{2}x^4 \]
03

Stretch and Reflect for j(x)

First, vertically stretch the function \(f(x) = x^4\) by a factor of 2, giving \(2x^4\). Then, reflect this new function across the x-axis by multiplying by -1. Therefore, \(j(x) = -2f(x)\). Substituting \(f(x)\) into the equation, we get: \[ j(x) = -2x^4 \]

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Key Concepts

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Vertical Stretch
Vertical stretch affects how

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