Question

In Exercises \(49-52\), consider the graph of \(f(x)=x^{3}\). Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(f\) is vertically shrunk by a factor of \(\frac{1}{3}\).

Short answer

The equation of the given function after being vertically shrunk by a factor of 1/3 is \(g(x) = \frac{1}{3}x^{3}\).

Step-by-step solution

Identify the Vertical Shrink Factor

Given that the graph of \(f\) which is \(f(x) = x^{3}\) is vertically shrunk by a factor of \(\frac{1}{3}\), we note that the shrink factor is \(\frac{1}{3}\).

Apply the Vertical Shrink to the Original Function

We apply a vertical shrink to the function by multiplying its output by \(\frac{1}{3}\). The new function, \(g(x)\), is thus given by \(g(x) = \frac{1}{3}f(x)\). Substituting \(f(x) = x^{3}\) into the equation gives \(g(x) = \frac{1}{3}x^{3}\).

Verification by Graphing

To verify the shrink factor, graph both \(f(x) = x^{3}\) and \(g(x) = \frac{1}{3}x^{3}\) using a graphing utility. Observe that the graph of \(g(x)\) should be a vertically shrunk version of \(f(x)\) by a factor of \(\frac{1}{3}\).

Key concepts

Rigid and Nonrigid Transformations

Understanding the distinction between rigid and nonrigid transformations is essential in graphing and analyzing functions. A rigid transformation involves moving the graph of a function around the coordinate plane without changing its shape or size. This includes translations (shifting left/right/up/down) and rotations. Imagine picking up a piece of graphed paper and moving it; the image drawn on it does not stretch or shrink—this is akin to a rigid transformation.

A nonrigid transformation, on the other hand, alters the size or shape of a graph. This can include vertical stretches and shrinks, as well as horizontal compressions or extensions. For instance, when you photocopy a picture and resize it, that's similar to applying a nonrigid transformation. The function's general form remains the same, but the dimensions change. In the context of the provided exercise, the vertical shrink by a factor of \(\frac{1}{3}\) represents a nonrigid transformation.

Graphing Polynomial Functions

Polynomial functions like \(f(x) = x^3\) are fundamental in mathematics, and graphing them helps us understand their behavior. The graph of a basic third-degree polynomial, such as our function \(f(x)\), reveals a smooth curve known as a cubic parabola. It descends steeply, crosses the origin, and then ascends steeply again. Since any real number is a valid input for \(x\), we refer to its domain as all real numbers.

To visualize changes, like a vertical shrink, we first need to grasp the original function's graph. Mastering graphing polynomial functions empowers students to predict and comprehend how transformations will impact the overall graph. Remember, the end behavior of the polynomial, how the function acts as \(x\) approaches infinity or negative infinity, remains unchanged despite nonrigid transformations.

Function Transformation Verification

After applying transformations to an original function, it's important to verify that the changes made reflect the desired transformation. In the context of our exercise, the student's task was to apply a vertical shrink to \(f(x) = x^3\) and to verify it graphically. The transformed function \(g(x) = \frac{1}{3}x^3\) still needs to maintain the fundamental properties of the cubic function while displaying a vertical shrink.

To verify, we compare key points from the original function with those of the transformed function, looking at their \(y\)-values; these should be exactly one third of the original for any \(x\)-value. Moreover, graphing both functions presents a visual confirmation that \(g(x)\) is indeed a vertically shrunk version of \(f(x)\). Such verification is vital to ensure understanding not just memorization of the transformation process.

Graphing Utility Usage

Utilizing graphing utilities can significantly simplify the process of verifying transformations. These digital tools allow students to swiftly and accurately plot functions and their transformations, providing visual evidence to support algebraic work. In the exercise, graphing both \(f(x) = x^3\) and \(g(x) = \frac{1}{3}x^3\) enables us to see the effect of the vertical shrink by comparing the two graphs side by side.

Students should grow comfortable with using such graphing utilities, not only for verification but also for exploring the properties of different functions and transformations. When inputting functions, pay close attention to parentheses and other details that could alter the intended graph. Simply put, mastering graphing utilities will greatly enhance a student's ability to analyze and understand the behavior of functions in a dynamic way.