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Chapter 2: Problem 113

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 cube root function \(f(x) = \sqrt[3]{x}\) is graphed first. Then, the transformations are applied in order: horizontal shift 2 units to the left, vertical compression by a factor of 1/2, and downward shift of 2 units to arrive at the graph of the transformed function \(r(x) = \frac{1}{2}\sqrt[3]{x+2} - 2 \).

Step by step solution

01

Graph the Parent Function

First, graph the parent function \(f(x) = \sqrt[3]{x}\). Choose several values for x and compute the corresponding y values. Then plot these points on the graph and connect them to form a cube root graph.
02

Understand the Transformations

The function \(r(x) = \frac{1}{2}\sqrt[3]{x+2} - 2 \) is a transformation of the parent function. The '+2' inside the cube root shifts the graph 2 units to the left, the multiplication by 1/2 vertically compresses the graph, and the '-2' outside the cube root shifts the graph 2 units downwards.
03

Apply the Transformations

Start by applying the horizontal shift. Move all points of the parent function 2 units to the left. Then apply the vertical compression, shrinking the graph towards the x-axis by a factor of 1/2. Finally, shift the entire graph 2 units down.
04

Graph the Transformed Function

After all transformations have been applied, graph the transformed function \(r(x) = \frac{1}{2}\sqrt[3]{x+2} - 2 \). This is a cube root graph that has been shifted 2 units to the left, vertically compressed by a factor of 1/2, and shifted down 2 units. Use the coordinates of the points from the parent graph and the corresponding transformation for every point.

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