Problem 105 - 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. $$h(x)=\frac{1}{2}(x-3)^{3}-2$$

Step-by-Step Solution

Answer

To graph the function \(h(x)=\frac{1}{2}(x-3)^{3}-2\), start with the base function \(f(x)=x^{3}\), then apply a horizontal shift of three units to the right, a vertical stretch by a factor of \(\frac{1}{2}\), and a vertical shift of two units downwards.

Step 1: Graph the standard cubic function

Graph the function \(f(x)=x^{3}\) on a set of axes. This is the base function which other functions will be transformed from.

Step 2: Identify the transformations

Note that the given function \(h(x)=\frac{1}{2}(x-3)^{3}-2\) involves three transformations of the base function. A horizontal shift three units to the right given by \((x-3)^{3}\), a vertical stretch by factor of \(\frac{1}{2}\), and a downward shift of two units given by \(-2\) at the end of the function.

Step 3: Apply the horizontal shift

Shift every point on the base function graph three units to the right. This transformation corresponds to the \((x-3)\) term in the function definition.

Step 4: Apply the vertical stretch

Next, stretch the function vertically by a factor of \(\frac{1}{2}\). This makes the function graph narrower. This transformation corresponds to the factor \(\frac{1}{2}\) in the function definition.

Step 5: Apply the vertical shift

Lastly, shift the entire function down by two units. This corresponds to the \(-2\) at the end of the function definition. Now the graph of the function \(h(x)=\frac{1}{2}(x-3)^{3}-2\) is complete.