Problem 111 - 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. $$h(x)=\frac{1}{2} \sqrt[3]{x+2}$$

Step-by-Step Solution

Answer

The function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\) is the cubic root function horizontally shifted 2 units to the left and vertically shrunk by a factor of 1/2. The graph will intercept the y axis at 0 and will increase in the positive direction with a flatter slope than the parent function.

Step 1: Graph the Parent Function

Begin by graphing the parent function, \(f(x)=\sqrt[3]{x}\). The cubic function crosses the origin (0,0) and increases in the positive direction. The slope of the function gradually flattens as x increases and steepens as x decreases.

Step 2: Apply Horizontal Shift

Next, apply a horizontal shift to the function. The expression \(x+2\) under the cubic root indicates a shift to the left by 2 units. To do this, subtract 2 from the x-coordinates of each point on the original function.

Step 3: Apply Vertical Shrink

Next, apply a vertical compression of the function. The coefficient \( \frac{1}{2} \) in front of the cubic root in our function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\) shrinks the graph vertically by a factor of 1/2. To achieve this, multiply the y-coordinates of the horizontally shifted graph by 1/2.

Step 4: Graph the Transformed function

Finally, graph the transformed function using the new coordinates obtained from steps 2 and 3. The resulting graph will represent the function \(h(x)=\frac{1}{2} \sqrt[3]{x+2}\).