Problem 116 - College Algebra Essentials

Question

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt[3]{x-2}$$

Step-by-Step Solution

Answer

To graph \(h(x) = -\sqrt[3]{x - 2}\), shift the graph of the cube root function \(\sqrt[3]{x}\) 2 units to the right and then reflect it over the x-axis.

Step 1: Graph \(f(x) = \sqrt[3]{x}\)

To begin with, it's necessary to understand the basic graph of a cube root function. This function gets steeper for positive x and shallower for negative x. It passes through the origin (0,0).

Step 2: Identify the Transformations

The function to be graphed, \(h(x) = -\sqrt[3]{x-2}\), is a transformation of the cube root function. \$h(x)\$ is obtained by shifting \(f(x)\) to the right by two units, and then reflecting it across the x-axis.

Step 3: Apply the Horizontal Translation

The term inside the cube root function, \(x - 2\), is causing a shift to the right by 2. Each point on the graph of \(f(x)\) should be moved 2 units to the right.

Step 4: Apply the Reflection

The negative sign in front of the cube root function causes a reflection of the graph across the x-axis. Each point on the graph should be reflected across the x-axis.

Step 5: Graph the Transformed Function

After applying the shift and reflection, plot the transformed points to get the graph of \(h(x) = -\sqrt[3]{x-2}\). The graph will have the same shape as the original cube root function, but shifted and reflected.