Problem 98 - 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. $$g(x)=(x-2)^{3}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = (x - 2)^{3}\) is a shift of the graph of \(f(x) = x^{3}\) by 2 units to the right.

Step 1: Graph the Standard Cubic Function

The cubic function \(f(x) = x^{3}\) is the most simple in its family. This will be graphed first. The graph of this function shows that as \(x\) becomes larger, \(f(x)\) also becomes larger. Similarly, as \(x\) becomes smaller, \(f(x)\) also becomes smaller. The graph cuts through the origin (0,0).

Step 2: Understand the Transformation

The function \(g(x) = (x - 2)^{3}\) is a transformation of the function \(f(x) = x^{3}\). The value inside the parentheses, \(x - 2\), tells us that this is a horizontal shift or translation. In this case, it's a shift of 2 units to the right.

Step 3: Graph the Transformed Function

Starting from the graph of \(f(x) = x^{3}\), shift every point on this graph 2 units to the right to graph \(g(x) = (x - 2)^{3}\).