Problem 65 - Thomas Calculus

Question

Exercises \(57-66\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=1-x^{3}, \text { compressed horizontally by a factor of } 3 $$

Step-by-Step Solution

Answer

The new equation is \(y = 1 - 27x^3\).

Step 1: Identify the Transformation

The graph is compressed horizontally by a factor of 3. A horizontal compression by a factor can be achieved by modifying the input variable.

Step 2: Apply the Horizontal Compression

To compress the graph horizontally by a factor of 3, replace the variable \(x\) with \(3x\) in the function. Thus, the equation becomes \(y = 1 - (3x)^3\).

Step 3: Rewrite the Equation

Simplify the equation by expanding the expression \((3x)^3\). Therefore, \((3x)^3 = 27x^3\), making the equation \(y = 1 - 27x^3\).

Step 4: Conclusion

The compressed graph is represented by the equation \(y = 1 - 27x^3\), reflecting a horizontal compression by a factor of 3.

Key Concepts

Horizontal Compression

Horizontal compression is a transformation that affects the width of a graph. When a graph is compressed horizontally, it becomes narrower along the x-axis. The original width is decreased by a specific factor. In mathematical terms, this transformation is achieved by altering the x variable in the function.

• To compress a function horizontally by a factor of \(n\), replace the variable \(x\) with \(nx\).
• This effectively scales the x-values, making the graph narrower.

For example, in the function \(y = 1 - x^3\), applying a horizontal compression by a factor of 3 means replacing \(x\) with \(3x\). The new function becomes \(y = 1 - (3x)^3\). Simplifying gives you \(y = 1 - 27x^3\).This new equation represents a horizontally compressed version of the original cubic function. The graph appears three times narrower than the original in the horizontal direction.

Graph Transformation

Graph transformation involves altering the original appearance of a graph using mathematical operations. These transformations include translations, stretches, compressions, and reflections. With each transformation, the graph changes its position, size, or shape without altering its basic form. In the context of horizontal compression:

• The process focuses on modifying the input variables.
• This leads to a change along the x-axis while keeping the y-values unchanged.

Modifying the variable in a function to alter its graph is a common technique. It allows for: - Expanding or compressing the graph horizontally - Shifting the graph left, right, up, or down - Flipping the graph over the axes Every transformation of the graph maintains the essence of the shape. It simply becomes different in terms of position or size, while keeping its fundamental properties and behaviour intact.

Cubic Functions

Cubic functions are polynomial functions of degree three. They take the form \(y = ax^3 + bx^2 + cx + d\). These functions have distinct graphs that can appear curved, with regions that change direction. Because of the cubic term \(ax^3\), they can have unique characteristics such as inflection points where the graph changes concavity.For the function \(y = 1 - x^3\), the graph displays familiar cubic behaviour:

• The curve has one real root, where the function intersects the x-axis.
• It may also show points of inflection, changing the direction of curvature.

With transformations, like the horizontal compression, the shape of the cubic function graph adjusts accordingly. The primary shape remains a swooping curve passing through the x-axis at specific points. Understanding how these functions respond to graph transformations helps to grasp the broader concepts of mathematical transformations and their applications.