Problem 112 - Precalculus 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 function \(h(x)= \frac{1}{2} \sqrt[3]{x-2}\) is a cube root function that is shifted 2 units to the right and shrunk by a factor of 1/2 along the y-axis compared to the function \(f(x)= \sqrt[3]{x}\).

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

To understand how transformations will affect the function, start by graphing the root function. Plot the points by picking a range of x values, like x = -2, -1, 0, 1, and 2, and finding the corresponding y values. Cube root of these x-values will give us the respective y-values needed for the plot.

Step 2: Understand the Transformations

The function \(h(x)= \frac{1}{2} \sqrt[3]{x-2}\) is a transformation of \(f(x)= \sqrt[3]{x}\). The number 2 inside the cube root shifts the graph 2 units to the right (a horizontal translation), and the coefficient \(\frac{1}{2}\) outside the cube root shrinks the graph vertically by a factor of 1/2.

Step 3: Apply the Transformations and Graph \(h(x)\)

Apply the noted transformations to the points from the original cube root function graph. For every point on the graph of \(f(x)\), add 2 to the x-coordinate and multiply the y-coordinate by 1/2. This will give you the coordinates for the graph of \(h(x)\). Connect the points to form the graph of \(h(x)\).