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Chapter 2: Problem 112

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

Short Answer

Expert verified
The function \(h(x)=\frac{1}{2} \sqrt[3]{x-2}\) results from the function \(f(x)=\sqrt[3]{x}\) by shifting the graph 2 units to the right and then halving all y-coordinates. The transformed graph retains the same overall shape as the original graph, but it is stretched vertically and shifted to the right.

Step by step solution

01

Graphing the basic function

Start by graphing the basic function \(f(x)=\sqrt[3]{x}\). This function looks like a curve that rises slowly for negative values of x, goes through the origin, and then rises more quickly for positive values of x.
02

Identify the transformations

The function \(h(x)=\frac{1}{2} \sqrt[3]{x-2}\) is derived from \(f(x)\) by applying a horizontal shift to the right by 2 units and a vertical stretch by a factor of 1/2. The horizontal shift is because of the 'x-2' inside the cube root, while the vertical stretch is due to the factor of '1/2' in front of the cube root.
03

Apply the transformations to graph h(x)

To graph \(h(x)\), start with the graph of \(f(x)\) and then shift every point on it 2 units to the right. This takes care of the horizontal shift. The vertical stretch means that the y-values are half of what they were in \(f(x)\), so they must be halved. We do this to the y-coordinates of every point on the graph after the horizontal shift.

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