Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=-\sqrt[3]{x}, \quad y_{3}=-2 \sqrt[3]{x}$$

Step-by-Step Solution

Answer

Graph are transformations of \(y = \sqrt[3]{x}\); \(y_2\) is a reflection, and \(y_3\) is a reflection with vertical stretch.

Step 1: Sketch the Parent Function

The parent function for these transformations is \( y = \sqrt[3]{x} \). This cubic root function passes through the origin (0,0), has rotational symmetry about the origin, and is defined for all real x. It looks like an elongated S-shape with points plotted and connected smoothly across the x-axis.

Step 2: Sketch the Reflection for y_2

The function \( y_2 = -\sqrt[3]{x} \) is a reflection of the parent function \( y = \sqrt[3]{x} \) across the x-axis. To sketch this transformation, take each point on \( y_1 \) and reflect it downward. This means multiplying the y-values of \( y_1 \) by -1.

Step 3: Sketch the Vertical Stretch and Reflection for y_3

The function \( y_3 = -2\sqrt[3]{x} \) involves both reflection and vertical stretch. First, reflect the parent function \( y_1 = \sqrt[3]{x} \) across the x-axis as in \( y_2 \). Then, stretch the reflected graph vertically by a factor of 2. Multiply each of the reflected y-values by 2, which stretches the graph away from the x-axis, creating a steeper curve.

Step 4: Verify with Graphing Technology

Use a graphing calculator to input \( y_1 = \sqrt[3]{x} \), \( y_2 = -\sqrt[3]{x} \), and \( y_3 = -2\sqrt[3]{x} \). Select an appropriate viewing window, such as x: [-10, 10] and y: [-10, 10], to see the transformations accurately. Compare these graphs on the calculator with the manually sketched ones to ensure correctness.

Key Concepts

Cubic Root Function

The cubic root function, represented as \( y = \sqrt[3]{x} \), is a fundamental concept in functions and graph transformations. This function creates a graph with a distinct shape that is often compared to an elongated 'S'.
The graph of the cubic root function passes through the origin (0,0) and extends infinitely in both directions along the x-axis. It is unique because every real number has a cubic root, meaning the function is defined for all real \( x \), unlike square root functions which only accommodate non-negative inputs.
Some key characteristics of the cubic root graph include:

• Its symmetry about the origin, meaning if you rotate the graph 180 degrees around the origin, it will coincide with itself.
• The ability of the graph to cross the x-axis and continue smoothly from negative to positive values.
• Its gentle slope near the origin which becomes steeper as \( x \) moves away from zero in either direction.

Reflection Across the X-Axis

Reflecting a graph across the x-axis is a simple yet powerful transformation. For the function \( y_2 = -\sqrt[3]{x} \), reflection means that every point on the cubic root graph \( y_1 = \sqrt[3]{x} \) is mirrored about the x-axis.
To perform this transformation, you take the y-values from the original graph and multiply them by -1. This flips the graph upside down, placing what was above the x-axis below it and vice versa.
The reflection across the x-axis retains the overall shape of the cubic root function but inverts it vertically. So, while initially the graph rises from left to right through the origin, the reflected function \( y_2 \) will instead fall from left to right.

Vertical Stretching

Vertical stretching involves altering the steepness of a graph, making it appear taller or flatter. With the function \( y_3 = -2\sqrt[3]{x} \), you can see a vertical stretch applied along with a reflection.
This process is accomplished by multiplying the y-values of \( y_2 = -\sqrt[3]{x} \) by 2. As a result:

• The graph becomes steeper because every point's distance from the x-axis doubles.
• The graph rises and falls more sharply compared to the non-stretched version \( y_2 \).

With vertical stretching, it’s important to notice how it affects the graph's angle and feel. Although the graph's shape and direction (due to reflection) remain similar, the stretch intensifies the rate at which the function changes as \( x \) moves away from zero.