Problem 117 - 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

The graph of the function \(g(x)=\sqrt[3]{-x-2}\) can be obtained by graphing the function \(f(x)= \sqrt[3]{x}\), then reflecting this graph about the y-axis, and shifting it 2 units to the right.

Step 1: Graph the parent function

Start by graphing the parent function \(f(x)=\sqrt[3]{x}\). The cube root function appears as a curve that increases as \(x\), goes from left to right passing through the point (0,0), negative values of \(x\) give negative output while the positive ones give positive values.

Step 2: Identify the transformations

Observe that the given function \(g(x)=\sqrt[3]{-x-2}\) can be derived from the parent function \(f(x)=\sqrt[3]{x}\) through the transformations: reflection about the y-axis - because of the negative sign in front of \(x\), and horizontal shift - 2 units to the right.

Step 3: Apply the transformations to the graph

Finally, the graph of \(g(x)\) can be obtained by applying the transformations to the graph of \(f(x)\). Start by reflecting the graph of \(f(x)\) about the y-axis. Then shift this reflected graph 2 units to the right. Draw the curve. This new curve is the graph of \(g(x)=\sqrt[3]{-x-2}\).

Step 4: Verification

In order to confirm that you have the correct graph, you may want to choose specific points on \(f(x)\) and the corresponding points on \(g(x)\), to see if the transformations have been applied correctly. Choosing the point (1,1) on \(f(x)\) for example, through reflection it becomes (-1, 1) on \(g(x)\) and after a shift 2 units to the right it becomes (1,1). Likewise for many more points on \(f(x)\).