Problem 114 - 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

The graph of function \(r(x)\) starts from the point (2,2) and continues to rise at a slower rate than \(f(x)\) due to being vertically compressed by a factor of 1/2.

Step 1: Graph the cube root function

Begin by graphing the cube root function \(f(x) = \sqrt[3]{x}\). This function is a fairly simple function to graph as it increases and decreases at a steady rate. The graph will start from negative infinity, passing through the origin (0,0) and then rise to positive infinity.

Step 2: Apply the horizontal shift

The graph will be horizontally shifted by two units to the right in comparison with the graph of \(f(x)\). This means that every point on the original \(f(x)\) graph will be moved two units to the right. Thus \(f(x) = \sqrt[3]{x}\) becomes \(f(x) = \sqrt[3]{x-2}\). The graph starts from the point (2,0) now after this shift.

Step 3: Apply the vertical compression transformation

The graph would be vertically compressed by a factor of 1/2. This effectively reduces the vertical y-values of each point on the curve of the graph by half while leaving the x-values unchanged. So, \(f(x) = \sqrt[3]{x-2}\) turns into \(f(x) = \frac{1}{2}\sqrt[3]{x-2}\).

Step 4: Apply the vertical shift

Finally, the function will be vertically shifted up by two units which means every point on the graph of \(f(x) = \frac{1}{2} \sqrt[3]{x-2}\) is moved two units upwards, resulting in the graph of \(r(x) = \frac{1}{2} \sqrt[3]{x-2} + 2\). The graph now correctly represents \(r(x)\).