Problem 51 - College Algebra and Calculus: An Applied Approach

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 stretched by a factor of 4 .

Step-by-Step Solution

Answer

The function \(f(x)=x^{3}\) vertically stretched by a factor of 4 yields the function \(f(x) = 4x^{3}\).

Step 1: Understand vertical stretching

In a vertical stretch, every y-coordinate of the original function is multiplied by the given factor. For example, if you have the function \(f(x)=x^{3}\) and a vertical stretching factor of 4, the y-coordinate of every point will be 4 times its original value. Therefore, the new function would be \(f(x) = 4x^{3}\).

Step 2: Verify result with a graphing utility

To ensure the derived function \(f(x) = 4x^{3}\) is indeed a vertical stretch of \(f(x) = x^{3}\) by a factor of 4, one can compare their graphs using a graphing utility tool. After graphing, it will be seen that the graph of the derived function is steeper, meaning it stretches vertically compared to \(f(x) = x^{3}\). Thus, confirming that the function \(f(x) = 4x^{3}\) is the original function \(f(x) = x^{3}\) vertically stretched by a factor of 4.

Key Concepts

Rigid and Nonrigid Transformations

When studying functions and their graphs, it's important to understand the concepts of rigid and nonrigid transformations. These transformations alter the appearance of a graph in various ways.

Rigid transformations include translations, which shift the entire graph of a function to a different position without changing its shape or orientation, and rotations around a point.

On the other hand, nonrigid transformations include stretches and compressions, which actually change the shape of the graph. Vertical stretching, like in the exercise with the function \(f(x)=x^{3}\), is a type of nonrigid transformation that multiplies all the y-coordinates of the original graph by a certain factor. This changes the graph's steepness or rate of increase (or decrease).

Understanding these types of transformations is crucial for interpreting changes in the behavior of the graphs of functions and making accurate predictions based on those graphs.

Graphing Utility

A graphing utility is an essential tool for students and mathematicians alike to visualize functions and their transformations. These utilities can be software programs, online platforms, or even features included in advanced calculators.

With a graphing utility, one can easily enter a function, such as \(f(x)=x^{3}\), and apply transformations to see their effects. For example, when using a graphing utility to verify vertical stretching, you will immediately notice the changes in the graph's appearance after the transformation.

Benefits of Using a Graphing Utility

• Immediate visualization of complex functions.
• Ability to compare the original function with its transformed counterpart.
• Confirmation of algebraic manipulations through visual representation.
• Helpful in understanding the concept of continuity and limits.

The graphing utility proves to be an indispensable resource for students to experiment with functions and solidify their understanding of mathematical concepts.

Cubic Functions

Cubic functions are a category of polynomial functions represented by the general form \(f(x)=ax^{3}+bx^{2}+cx+d\), where \(a, b, c\), and \(d\) are constants, and \(a \eq 0\). The simplest cubic function is \(f(x)=x^{3}\), which serves as the parent function for all cubic functions.

Cubic functions are known for their distinctive 'S' shaped curve, called a 'cubic curve.' Their graphs can provide invaluable insights into critical points, inflections, and end behavior. Understanding cubic functions is vital in calculus, physics, engineering, and economics because these functions often model natural phenomena and various types of growth.

Characteristics of Cubic Functions

• They have one or three real roots.
• There is always one point of inflection.
• The end behavior: as \(x\) approaches infinity, \(f(x)\) approaches infinity if \(a>0\) and negative infinity if \(a<0\).

Getting familiar with cubic functions and their transformations enables students to tackle a wide range of problems and build a strong foundation for advanced mathematical concepts.