/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Sketch the graph of \(y=x^{n}\) ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 15

Sketch the graph of \(y=x^{n}\) and each transformation. \(y=x^{3}\) (a) \(f(x)=(x-4)^{3}\) (b) \(f(x)=x^{3}-4\) (c) \(f(x)=-\frac{1}{4} x^{3}\) (d) \(f(x)=(x-4)^{3}-4\)

Short Answer

Expert verified
The graph of function (a) is the graph of \(y=x^{3}\) shifted 4 units to the right. The graph of function (b) is the graph of \(y=x^{3}\) shifted 4 units down. The graph of function (c) is the graph of \(y=x^{3}\) reflected over the x-axis and shrunk vertically by a factor of \(1/4\). The graph of function (d) is the graph of \(y=x^{3}\) shifted 4 units to the right and 4 units down.

Step by step solution

01

Function (a) \(f(x)=(x-4)^{3}\)

This is a transformation of the function \(y=x^{3}\), with a horizontal shift of 4 units to the right. So, the graph of \(f(x)=(x-4)^{3}\) is the graph of \(y=x^{3}\) shifted 4 units to the right.
02

Function (b) \(f(x)=x^{3}-4\)

This is a transformation of \(y=x^{3}\) with a vertical shift of 4 units down. Thus, the graph of \(f(x)=x^{3}-4\) is the graph of \(y=x^{3}\) shifted 4 units down.
03

Function (c) \(f(x)=-\frac{1}{4} x^{3}\)

This is a transformation of the function \(y=x^{3}\) that includes a reflection about the x-axis and a vertical shrink by a factor of \(1/4\). Therefore, the graph of \(f(x)=-\frac{1}{4} x^{3}\) is the graph of \(y=x^{3}\) reflected over the x-axis and shrunk vertically by a factor of \(1/4\).
04

Function (d) \(f(x)=(x-4)^{3}-4\)

This is a transformation of the function \(y=x^{3}\) that includes a horizontal shift of 4 units to the right and a vertical shift of 4 units down. Therefore, the graph of \(f(x)=(x-4)^{3}-4\) is the graph of \(y=x^{3}\) shifted 4 units to the right and 4 units down.

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