Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=2 \sqrt[3]{x-2}+1\)

Step-by-Step Solution

Answer

The function \(y=2 \sqrt[3]{x-2}+1\) is the result of transforming the parent function \(f(x)=\sqrt[3]{x}\) right by 2 units, up by 1 unit, and a vertical stretch by a factor of 2. The graph of \(y\) looks like a sideways 'S' or 'Z', shifted right by 2 units, stretched vertically by a factor of 2, and shifted up by 1 unit.

Step 1: Identify and Understand the Transformations

Firstly, recognize that the given function \(y=2 \sqrt[3]{x-2}+1\) has the form \(y=a \sqrt[3]{x-h}+k\), which indicates that there are three transformations: a horizontal translation (h units right if h>0), a vertical translation (k units up if k>0), and a vertical stretch by factor of a. In this case, \(h=2\), \(k=1\), and \(a=2\), so the function is transformed 2 units to the right, 1 unit up, and vertically stretched by a factor of 2.

Step 2: Apply the Transformations to the Parent Function

Start from the parent function \(f(x)=\sqrt[3]{x}\). Apply the transformations in the given order:1. Horizontal translation right by 2 units: Translates every point on \(f(x)=\sqrt[3]{x}\) 2 units to the right, transforming it to \(f(x)=\sqrt[3]{x-2}\).2. Vertical stretch by a factor of 2: Multiplies the y-coordinate of every point on \(f(x)=\sqrt[3]{x-2}\) by 2, transforming it to \(f(x)=2 \sqrt[3]{x-2}\).3. Vertical translation up by 1 unit: Translates every point on \(f(x)=2 \sqrt[3]{x-2}\) 1 unit up, transforming it to \(f(x)=2 \sqrt[3]{x-2}+1\).

Step 3: Sketch the Transformed Graph

To sketch the graph, start from the basic graph of \(f(x)=\sqrt[3]{x}\), which looks like a sideways 'S' or 'Z'. Remember that horizontal translations do not change the shape of the graph, and vertical transformations only stretch or shrink the graph vertically. After applying each transformation in order, the graph of \(y\) should retain the basic shape of \(f(x)\), but be shifted right by 2 units, stretched vertically by a factor of 2, and shifted up by 1 unit. Plot a few points if needed to confirm the shape.

Step 4: Verify with a Graphing Utility

After sketching the graph by hand, use any graphing utility or software to plot the function \(y=2 \sqrt[3]{x-2}+1\). Compare the graph obtained from the software with the sketch drawn by hand to verify the sketch. The two should match.

Key Concepts

Horizontal Translation

A horizontal translation is a transformation that shifts the graph of a function left or right on the coordinate plane. This adjustment happens when we change the variable inside the function. For a function like

• \( y = \sqrt[3]{x} \), shifting it leads to \( y = \sqrt[3]{x-h} \)

and here, the value of \( h \) determines the direction and magnitude of the movement:

• If \( h > 0 \), the graph moves \( h \) units to the right.
• If \( h < 0 \), the graph moves \( |h| \) units to the left.

In our task, the function \( y = \sqrt[3]{x} \) is shifted by 2 units to the right, resulting in \( y = \sqrt[3]{x-2} \). This transformation doesn't alter the shape of the graph; it simply changes the graph's position horizontally.

Vertical Stretch

A vertical stretch is another type of transformation that affects the graph's appearance on the y-axis. It makes the graph either taller and narrower, or shorter and wider. This happens through multiplication of the function's output by a constant factor \( a \).

• If \( a > 1 \), the graph stretches and becomes taller.
• If \( 0 < a < 1 \), the graph compresses and becomes shorter.

Using our example, moving from \( \sqrt[3]{x-2} \) to \( 2\sqrt[3]{x-2} \), where \( a = 2 \), doubles the y-values of every point on the graph. This vertical stretch alters the steepness but doesn't change the original "sideways S" shape of the graph. The resulting curve is more elongated vertically, making it appear taller.

Vertical Translation

Vertical translation moves the graph up or down without affecting its shape. This occurs when a constant \( k \) is added to the function, resulting in

• \( y = \sqrt[3]{x-2} + k \)

Here, \( k \) indicates the direction and distance of the shift:

• If \( k > 0 \), the graph shifts \( k \) units upwards.
• If \( k < 0 \), the graph shifts \( |k| \) units downwards.

For the given function \( y = 2\sqrt[3]{x-2} + 1 \), the addition of 1 shifts the entire graph one unit up. This is the final transformation applied, modifying the graph's vertical position but not its shape or steepness. The combination of this last step with previous transformations finalizes the position of the graph on the coordinate plane.