Problem 97 - 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. $$g(x)=(x-3)^{3}$$

Step-by-Step Solution

Answer

To graph \(g(x) = (x-3)^3\), begin by graphing the parent function \(f(x) = x^{3}\) and then shift all points on this graph three units to the right.

Step 1: Graph Standard Cubic Function

Start by plotting the standard cubic function \(f(x) = x^{3}\). For illustrative purposes, you can plot points like \((-2,-8)\), \((-1,-1)\), \((0,0)\), \((1,1)\), and \((2,8)\). Then connect these points to form a smooth curve.

Step 2: Understand the Transformation

The given function \(g(x) = (x-3)^3\) is a transformation of the function \(f(x) = x^{3}\). More specifically, it is a horizontal shift. When a constant is subtracted from \(x\), it results in a horizontal shift to the right by that constant (in this case 3 units).

Step 3: Transform the Graph

Now that we understand the transformation, shift every point from the original graph of \(f(x) = x^{3}\) three units to the right. This results in points like \((1,-8)\), \((2,-1)\), \((3,0)\), \((4,1)\), and \((5,8)\).

Step 4: Sketch the Transformed Graph

Finally, connect these points to form the graph of \(g(x) = (x-3)^3\). This graph should look the same as the original but shifted three units to the right.