Problem 65 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x)=1 /\left(x^{2}+1\right)\). Using the same axes, draw the graphs of \(y=f(x), y=f(2 x)\), and \(y=f(x-2)+0.6\), all on the domain \([-4,4]\).

Step-by-Step Solution

Answer

Graph each transformation separately, then combine them on the same axes with distinct styles or colors.

Step 1: Understand the Parent Function

The parent function given is \( f(x) = \frac{1}{x^2 + 1} \). This represents a rational function where \(x^2 + 1 \) is in the denominator. As \(x\) increases or decreases from 0, the denominator increases, thus reducing the overall value of the function. The graph of this function is symmetric with respect to the y-axis due to its even nature.

Step 2: Graph the Parent Function

Plot the graph of \( y = \frac{1}{x^2 + 1} \) over the interval \([-4, 4]\). We can calculate several key points, such as the vertex at \( (0, 1) \) since \( f(0) = 1 \). As \( x \) approaches the boundaries of the interval, the value approaches zero but never touches the x-axis, indicating horizontal asymptotes at \( y = 0 \).

Step 3: Sketch the Graph of \(y = f(2x)\)

The function \( y = f(2x) \) involves a horizontal compression by a factor of 2. This makes the graph become narrower. To graph it, replace \( x \) with \( 2x \) in \( f(x) \), resulting in \( f(2x) = \frac{1}{(2x)^2 + 1} = \frac{1}{4x^2 + 1} \). This transformation doesn't affect the range, just the rate at which \( y \) approaches zero.

Step 4: Sketch the Graph of \(y = f(x-2) + 0.6\)

This transformation involves two steps: a horizontal shift to the right by 2 units and an upward shift by 0.6 units. Replace \( x \) with \( x-2 \) in \( f(x) \), giving \( f(x-2) = \frac{1}{(x-2)^2 + 1} \), and then shift that graph up by 0.6 units to find \( y = f(x-2) + 0.6 \). Plot this function over the same domain \([-4, 4]\), starting from the right-shifted position.

Step 5: Combine and Plot All Graphs on the Same Axes

With each transformation calculated, plot \( y = f(x) \), \( y = f(2x) \), and \( y = f(x-2) + 0.6 \) on the same graph paper or tool over the interval \([-4, 4]\). Ensure each graph reflects its transformation properly and distinguish each graph using different colors or styles.

Key Concepts

Rational Functions

A rational function is a type of function represented by a ratio of two polynomials. The function we are dealing with here, \( f(x) = \frac{1}{x^2 + 1} \), is a simple form where the numerator is a constant and the denominator is a quadratic expression. This type of function has specific characteristics:

• The numerator and the denominator determine the behavior of the graph.
• Rational functions often feature asymptotes—lines that the graph approaches but never touches.
• They can be used to model many real-world situations thanks to their dynamic nature.

In the given function, \( x^2 + 1 \) never becomes zero for real values of \( x \), ensuring that the function is defined for all real numbers. This keeps the graph smooth and continuous without breaks.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a graph as \( x \) approaches infinity or negative infinity. They indicate a value that the function will get infinitely close to but never actually reach. For the function \( f(x) = \frac{1}{x^2 + 1} \), the horizontal asymptote is at \( y = 0 \).

• As \( x \) becomes very large or very small, the value \( x^2 \) grows significantly, making the denominator large.
• This results in the value of the function approaching zero.

In graphical terms, although the curve gets very close to the x-axis for large values of \( |x| \), it never intersects it. Horizontal asymptotes can help us predict long-term trends within function graphs without calculating every point.

Horizontal and Vertical Shifts

Transformations like horizontal and vertical shifts change the location and shape of a graph without altering its overall structure. In this exercise:

• A horizontal shift is applied to the function \( y = f(x - 2) \), moving the graph to the right by 2 units.
• A vertical shift is applied to the function \( y = f(x-2) + 0.6 \), moving the graph up by 0.6 units.

Such transformations can be applied by modifying the equation of the function:

• Replacing \( x \) with \( x-2 \) results in the graph shifting right, as each value of \( x \) is adjusted by 2 units.
• Adding 0.6 to the function causes the entire graph to rise by that amount, without altering the horizontal asymptote.

This enables the transformation to maintain the graph's shape while changing its position.

Function Graphing

Graphing functions visually represents mathematical relationships, making them easier to analyze. For the presented exercise, different transformations are graphed together:

• Start by plotting the parent function \( y = \frac{1}{x^2 + 1} \), noting key features like symmetry and horizontal asymptotes.
• Graph \( y = f(2x) \) which compresses the graph horizontally by a factor of 2, making it appear narrower.
• Plot \( y = f(x-2) + 0.6 \) to observe combined horizontal and vertical shifts.

To effectively complete the graphing:

• Use graphing tools or graph paper, marking axes clearly to ensure accurate plotting.
• Consider using different colors or line styles to differentiate each transformation.

Function graphing becomes a powerful tool to visually interpret the effect of algebraic transformations and their real-world applications.