Problem 100 - 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. $$h(x)=-(x-2)^{3}$$

Step-by-Step Solution

Answer

The graph of the given function \(h(x)=-(x-2)^{3}\) is a reflection of the standard cubic function \(f(x)=x^{3}\) in the x-axis, and then a horizontal shift of the result by 2 units to the right.

Step 1: Graph the Standard Cubic Function

First, graph the standard cubic function \(f(x) = x^3\). The graph will start from the bottom left quadrant, cross the origin and end in the upper right quadrant.

Step 2: Reflection in the X-axis

After graphing the standard function, apply a reflection in the x-axis. This is done by taking each point in the standard cubic function and reflecting it over the x-axis. In this case, you will end up with the graph upside down.

Step 3: Horizontal Shift

The final step is to shift the graph horizontally by 2 units to the right. Every point on the reflected graph is moved 2 units to the right to obtain the graph of \(h(x) = -(x-2)^3\). As a result, the graph will start from the upper left quadrant, move downwards and end at the bottom right quadrant.