/*! 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 114 Begin by graphing the cube root ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 114

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$r(x)=\frac{1}{2} \sqrt[3]{x-2}+2$$

Short Answer

Expert verified
The graph of the function \(r(x)=\frac{1}{2} \sqrt[3]{x-2}+2\) can be obtained from the graph of the cube root function \(f(x)=\sqrt[3]{x}\) by applying a horizontal shift to the right by 2 units, a vertical stretch by a factor of \(\frac{1}{2}\), and an upward vertical shift by 2 units. The transformed graph goes through the points (1,0), (2,2), (3,2.5) and so on.

Step by step solution

01

Graph the parent function

The first task is to plot the cube root function \(f(x) = \sqrt[3]{x}\). This function goes through the points (-1,-1), (0,0), (1,1) and so on
02

Apply horizontal shift

To plot the transformed function \(r(x) = \frac{1}{2}\sqrt[3]{x-2}+2\), a horizontal shift by 2 units to the right needs to be applied to the graph of the original function, because of the transformation \(x-2\) inside the cube root. This means that the function will now go through the points (1,-1), (2,0), (3,1) and so on.
03

Apply vertical stretch and shift

Now, apply the vertical transformations indicated by \(\frac{1}{2}\) and \(+2\) outside of the cube root function. The \(\frac{1}{2}\) before the cube root results in a vertical stretch by a factor of \(\frac{1}{2}\), while the \(+2\) results in a vertical shift upwards by 2 units. Thus, the graph now goes through the points (1,0), (2,2), (3,2.5) and so on.

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