Problem 113 - 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. $$r(x)=\frac{1}{2} \sqrt[3]{x+2}-2$$

Step-by-Step Solution

Answer

To graph \(r(x) = 0.5 \cdot \sqrt[3]{x+2} - 2\), start from the graph of \(f(x) = \sqrt[3]{x}\), shift it left by 2 units, down by 2 units and shrink it vertically by a factor of 0.5.

Step 1: Graph the cube root function

Firstly, plot the cube root function \(f(x) = \sqrt[3]{x}\). This function has a graph that passes through the origin (0,0) and rises slowly, it also curves in the 2nd and 4th quadrants.

Step 2: Apply horizontal shift

The function \(r(x) = 0.5 \cdot \sqrt[3]{x+2}\) introduces a horizontal shift. The \(+2\) inside the cubed root affects the x-values, shifting them to the left by 2 units, since it's a positive value inside the function. This results to a new function \(r_1(x) = 0.5 \cdot \sqrt[3]{x}.\)

Step 3: Apply vertical shift

The function \(r(x) = 0.5 \cdot \sqrt[3]{x+2} - 2\) introduces a vertical shift. The \(-2\) outside the cubed root function affects the y-values, shifting them down by 2 units. This results to the final function \(r(x) = 0.5 \cdot \sqrt[3]{x} - 2\).

Step 4: Apply vertical shrink

Lastly, notice the presence of the \(0.5\) factor that multiplies the cube root function which causes vertical shrink by a factor of 0.5. This means that the y-values of \(f(x) = \sqrt[3]{x+2}\) should be halved, producing the final function \(r(x) = 0.5 \cdot \sqrt[3]{x+2} - 2.\)