Problem 114 - 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. $$r(x)=\frac{1}{2} \sqrt[3]{x-2}+2$$

Step-by-Step Solution

Answer

The function \(r(x) = \frac{1}{2}\sqrt[3]{x-2}+2\) is the graph of the cube root function \(\sqrt[3]{x}\) which is shifted 2 units to the right, upward by 2 units, and vertically shrunken by a factor of 1/2.

Step 1: Identify the Parent Function

The parent function here is \(f(x) = \sqrt[3]{x}\). It is important to start by knowing the basic shape of the cube root function, which is known for its characteristic 'S' shape.

Step 2: Identify the Transformations

The given function \(r(x) = \frac{1}{2}\sqrt[3]{x-2}+2\) indicates two transformations: a horizontal shift and a vertical shift along with a vertical shrink. The term \(-2\) inside the cube root shifts the graph 2 units to the right. The term \(+2\) outside the cube root shifts the graph 2 units up. The coefficient \(\frac{1}{2}\) in front of the cube root function indicates a vertical shrink by a factor of \(\frac{1}{2}\).

Step 3: Graph the transformed function

First, start by sketching the graph of the parent function \(f(x) = \sqrt[3]{x}\), remembering the 'S' shape it has. Then, move the graph 2 units to the right to account for the \(x-2\) inside the cube root. This causes all the \(x\)-values to increase by 2. After, move the graph 2 units upward to account for the \(+2\) added outside the cube root, causing all \(y\)-values to increase by 2. Lastly, vertically shrink the graph by a factor of \(\frac{1}{2}\) using the coefficient in front of the cube root function, causing all \(y\)-values to be multiplied by \(\frac{1}{2}\). This will result in a new graph for the function \(r(x) = \frac{1}{2}\sqrt[3]{x-2}+2\)