Problem 101 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.