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Chapter 2: Problem 18

Sketch the graph of \(y=x^{n}\) and each transformation. \(y=x^{5}\) (a) \(f(x)=(x+1)^{5}\) (b) \(f(x)=x^{5}+1\) (c) \(f(x)=1-\frac{1}{2} x^{5}\) (d) \(f(x)=-\frac{1}{2}(x+1)^{5}\)

Short Answer

Expert verified
The transformed graphs are created using shifts, scaling and reflection as explained in the steps. It's important to notice how these transformations affect the shape of the base function \(y=x^{5}\).

Step by step solution

01

Base Function Understanding

Before getting started with transformations, it's important to understand the base function. Here, the base function is \(y=x^{5}\) which has a distinct shape. It rises at both ends but more sharply for positive values of x.
02

Transforming \(f(x)=(x+1)^{5}\)

This is a horizontal shift of the base function. The graph of \(f(x)=(x+1)^{5}\) is the graph of \(y=x^{5}\) shifted 1 unit to the left.
03

Transforming \(f(x)=x^{5}+1\)

This is a vertical shift of the base function. The graph of \(f(x)=x^{5}+1\) is the graph of \(y=x^{5}\) shifted up by 1 unit.
04

Transforming \(f(x)=1-\frac{1}{2}x^{5}\)

This is both a vertical shift and a vertical stretch of the base function. The graph of \(f(x)=1-\frac{1}{2}x^{5}\) is the graph of \(y=x^{5}\) vertically stretched by a factor of \(\frac{1}{2}\) and shifted 1 unit upwards.
05

Transforming \(f(x)=-\frac{1}{2}(x+1)^{5}\)

This is a horizontal shift, vertical stretch, and reflection of the base function. The graph of \(f(x)=-\frac{1}{2}(x+1)^{5}\) is the graph of \(y=x^{5}\) shifted 1 unit to the left, vertically stretched by a factor of \(\frac{1}{2}\), and reflected in the x-axis.

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