Problem 112 - 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. $$h(x)=\frac{1}{2} \sqrt[3]{x-2}$$

Step-by-Step Solution

Answer

The graph of \(h(x)\) is a cube root function with half the vertical scale and shifted 2 units to the right from the graph of \(f(x)\).

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

This function produces a curve with a shape that increases slowly for negative x, goes through the origin, and then increases more steeply for positive x. The curve is symmetric about the origin.

Step 2: Apply the vertical scaling.

The function \(h(x)\) takes the cube root function and scales it vertically by a factor of 1/2, this operation is represented by the equation \(h(x) = 1/2 f(x)\). Taking each vertical coordinate in the original graph and multiplying it by 1/2. This squishes the graph toward the x-axis.

Step 3: Apply the horizontal shifting

Lastly, the function \(h(x)\) replaces \(x\) with \(x - 2\), which has the effect of shifting the graph 2 units to the right. This operation is represented by the equation \(h(x) = f(x - 2)\), which means replace every occurrence of \(x\) with \(x - 2\) in the original function \(f(x)\). It results in each point on the graph of \(f(x)\) being moved 2 units to the right to produce the graph of \(h(x)\).