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

Step-by-Step Solution

Answer

To graph the function \(g(x)=\sqrt[3]{x}+2\), take the graph of the cube root function \(f(x) = \sqrt[3]{x}\), and shift it upwards by 2 units. Each point (x,y) on \(f(x)\) will transform to (x,y+2) on \(g(x)\).

Step 1: Graphing the cube root function

To graph the function \(f(x) = \sqrt[3]{x}\) consider various values of x. The cubic function's graph starts from negative infinity, crosses the origin (0,0) and goes to positive infinity. It is symmetric with respect to the origin.

Step 2: Understanding the transformation

The given function \(g(x)=\sqrt[3]{x}+2\), differs from \(f(x)=\sqrt[3]{x}\) by the term +2. This represents a vertical shift of the graph upwards by 2 units. In graph transformation, adding a constant to the function results in the graph shifting vertically by that amount.

Step 3: Applying the transformation

To graph \(g(x)=\sqrt[3]{x}+2\), start with the graph of \(f(x)=\sqrt[3]{x}\), and shift every point 2 units upwards. This means, each point \((x, y)\) on the graph of \(f\), will now be represented as \((x, y+2)\) on the graph of \(g\).