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Chapter 1: Problem 101

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$h(x)=\frac{1}{2} x^{3}$$

Short Answer

Expert verified
To obtain the graph of \(h(x) = \frac{1}{2}x^3\), begin by graphing the standard cubic function, \(f(x) = x^3\). Then transform this graph by 'shrinking' it vertically by a factor of 0.5, resulting in a wider and flatter version of the original cubic function.

Step by step solution

01

Graph the standard cubic function \(f(x) = x^3\)

Begin by setting up a diagram and axis representing values of x and f(x). This cubic function will have one root at x=0, and will pass through the origin as it increases to positive and negative infinity. The graph will extend upwards and to the right for positive x values and downwards and to the left for negative x values. It's going to look like an elongated 'S' rotating around the origin.
02

Understand the transformation

The transformation needed to convert \(f(x)\) to \(h(x)\) is a vertical shrink by a factor of 0.5. In other words, every y-coordinate in \(f(x)\) would be halved to yield the y-coordinates in the new function \(h(x)\). This is due to the multiplier factored into the cubic function, which 'shrinks' the function vertically by that amount.
03

Perform the transformation and obtain the graph of \(h(x)\)

Apply this transformation to the graph of \(f(x)\) to obtain the graph of \(h(x)\). Every point of y in \(f(x)\) will become half as high in \(h(x)\), creating a more 'flattened' version of the 'S' curve. If \(f(x)\) was drawn correctly, \(h(x)\) will resemble the same shape, only flatter and wider.

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