Problem 54 - Thomas Calculus

Question

Exercises \(51-60\) 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 new equation is \( y = 1 + \frac{9}{x^2} \).

Step 1: Identify the Impact on the Function

The original function given is \( y = 1 + \frac{1}{x^2} \). When a graph is stretched horizontally by a factor of \(3\), it affects the \(x\)-component of the function.

Step 2: Adjust the Variable in the Denominator

To stretch the graph horizontally by a factor of \(3\), replace \(x\) with \(x/3\) in the denominator of the function. This modifies the expression \(\frac{1}{x^2}\) to \(\frac{1}{(x/3)^2}\).

Step 3: Simplify the Modified Expression

Simplify the modified expression: \(\frac{1}{(x/3)^2}\). This results in \(\frac{1}{x^2/9} = \frac{9}{x^2}\).

Step 4: Write the New Stretched Function

Combine the results of the modifications into the new function. Hence, the equation for the stretched graph is \( y = 1 + \frac{9}{x^2} \).

Key Concepts

Horizontal Stretch

A horizontal stretch affects the x-values of a function. When we stretch a function horizontally by a factor of 3, we are expanding the graph away from the y-axis. This means that each x-coordinate will be divided by 3, making the graph appear wider. It is important to apply the stretching factor correctly to the function.

• In our example, the original function is given by: \( y = 1 + \frac{1}{x^2} \).
• To achieve a horizontal stretch by a factor of 3, replace \(x\) with \(x/3\).
• This modification changes the formula to \( y = 1 + \frac{1}{(x/3)^2} \).

By simplifying \( \frac{1}{(x/3)^2} \), we arrive at \( \frac{9}{x^2} \). So, the stretched function becomes \( y = 1 + \frac{9}{x^2} \). It’s essential to understand how the graph is manipulated in this process.

Graph Transformation

Graph transformation covers different ways to alter the appearance of a graph on the coordinate plane. These transformations can include stretching, compressing, translating (shifting), or reflecting.

• In general, multiplying a function by a number greater than 1 will compress it, while multiplying by a number between 0 and 1 stretches it horizontally or vertically.
• For the function \( y = 1 + \frac{1}{x^2} \), a horizontal stretch changes how the graph extends along the x-axis.
• This results in a slower approach of the function towards the x-axis as \(x\) moves away from the origin.

Graph transformations provide a powerful toolkit for analyzing and predicting how changing certain parameters will impact the overall shape and position of a graph.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They can have vertical asymptotes and horizontal asymptotes, which guide the behavior of the graph as it approaches specific values of \(x\).

• Our base function, \( y = 1 + \frac{1}{x^2} \), is an example of a rational function.
• It has a vertical asymptote at \( x = 0 \) because \( \frac{1}{x^2} \) becomes undefined there.
• The horizontal asymptote is \( y = 1 \) as \( x \) tends to infinity, meaning the graph will get close to this line but never touch it.

Understanding rational functions includes studying their asymptotes, intercepts, and how they react to transformations. In this scenario, the transformation adjusts the distance from these asymptotes, particularly through the application of a horizontal stretch.