Problem 107 - 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. $$g(x)=\sqrt[3]{x}+2$$

Step-by-Step Solution

Answer

To graph \(g(x)=\sqrt[3]{x}+2\) we take original function \(f(x) = \sqrt[3]{x}\) and vertically shift it upward by 2 units.

Step 1: Sketch the Original Function

Start sketching the original cube root function, \(f(x) = \sqrt[3]{x}\). Make sure to note key feature: passing through origin and behaviour on positive and negative axis.

Step 2: Understanding the Transformation

The function \(g(x)=\sqrt[3]{x}+2\) is a vertical shift of \(f(x) = \sqrt[3]{x}\). The '+2' adds a shift of 2 units up.

Step 3: Graphing the Transformed Function

Use the transformations discussed to sketch \(g(x) = \sqrt[3]{x} + 2\) by shifting the graph of \(f(x) = \sqrt[3]{x}\) two units upward. The characteristics of the function will be the same as \(f(x) = \sqrt[3]{x}\), but the entire graph will be 2 units above the original graph.