Problem 60 - 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. $$ y=1+\frac{1}{x^{2}}, \quad \text { stretched horizontally by a factor of } 3 $$

Step-by-Step Solution

Answer

The stretched function is \( y = 1 + \frac{9}{x^2} \).

Step 1: Understanding the Horizontal Stretch

The problem states that the function should be stretched horizontally by a factor of 3. A horizontal stretch by a factor of \( a \) means replacing every \( x \) with \( \frac{x}{a} \) in the function equation. In this case, the horizontal stretch is by a factor of 3, so every \( x \) in the equation should be replaced by \( \frac{x}{3} \).

Step 2: Applying the Horizontal Stretch

Start with the original function: \[ y = 1 + \frac{1}{x^2} \]Now replace \( x \) with \( \frac{x}{3} \). This modifies the function as follows:\[ y = 1 + \frac{1}{\left(\frac{x}{3}\right)^2} \]

Step 3: Simplifying the New Equation

Simplify the new equation after substitution:\[ y = 1 + \frac{1}{\left(\frac{x}{3}\right)^2} = 1 + \frac{1}{\frac{x^2}{9}} \]Invert the fraction inside the expression:\[ y = 1 + \frac{9}{x^2} \]

Step 4: Confirming the Transformation

The transformed equation \( y = 1 + \frac{9}{x^2} \) represents the function stretched horizontally by a factor of 3 as required. The graph is now wider, because it takes longer to reach the same \( y \) values on the axis.

Key Concepts

Function Transformation

Function transformation is the process where we change the appearance or position of a graph for a given function. There are various ways to transform a function, such as translations, reflections, stretches, and compressions.
A function transformation lets us see what happens when we tweak the parameters in a function. By understanding these changes, you can predict how the graph of the function will look after transformation.

• Translations involve shifting the graph horizontally or vertically.
• Reflections flip the graph over a given axis.
• Stretches and compressions alter the graph's width or height, without changing its general shape.

With these tools, you can purposefully manipulate functions to fit specific criteria or constraints in mathematical modeling or real-world applications.

Graph Stretching

Graph stretching is a type of function transformation where the graph of a function is expanded or contracted either horizontally or vertically. This specific method involves multiplying or dividing the variable by a given factor.
Horizontal stretching occurs when all the x-values are divided by the stretch factor. For example, a stretch factor of 3 means for every point on the graph, the x-coordinate is expanded by this factor, making the graph appear wider.
In our example, by using a horizontal stretch on the function \(y = 1 + \frac{1}{x^2}\), we replace \(x\) with \(\frac{x}{3}\). This action transforms the function to \(y = 1 + \frac{9}{x^2}\), which effectively stretches the graph horizontally, spreading it out over a wider area.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world systems or phenomena. It provides a framework to analyze and predict behaviors within these models.

• Models use various functions and transformations to describe complex systems simply.
• By applying transformations like stretches, you can adjust the model to better fit empirical data or experimental results.

In the context of our exercise, the mathematical model changes by incorporating a horizontal stretch. This change might mirror how certain processes take longer to complete without altering their qualitative behavior, akin to understanding reaction times in physics or scaling in economics. Such models offer insights that are crucial for decision making, optimization, and achieving a deeper understanding of underlying systems.