Problem 62 - Essential Calculus

Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=\frac{x^{2}}{x^{2}-9} $$

Step-by-Step Solution

Answer

The graph of the equation \(y=\frac{x^{2}}{x^{2}-9}\) has vertical asymptotes at \(x=-3\) and \(x=3\), a horizontal asymptote at \(y=1\), and a y-intercept at (0,0). The graph isn't symmetric with respect to either axis. These features are confirmed with a graphing utility.

Step 1: Find the Vertical Asymptotes

The vertical asymptotes occur at the values of \(x\) for which the denominator of the function becomes zero, while the numerator is not zero. Set the denominator equal to zero and solve for \(x\). Hence, \(x^{2}-9=0\), we get \(x=3\) and \(x=-3.\)

Step 2: Find the Horizontal Asymptote

The horizontal asymptote for a rational function is determined by the ratio of the leading coefficients of the numerator and the denominator. In this case, the horizontal asymptote is \(y=1\) as the degrees of the numerator and denominator are equal and the ratio of the leading coefficients is 1.

Step 3: Find the Intercepts

The x-intercepts are found by setting \(y=0\) and solving for \(x\), which in this case there aren't any as \(x^2\) is never negative. The y-intercept is found by setting \(x=0\), which results in \(y = 0\).

Step 4: Check for Symmetry

Replace \(x\) with \(-x\) and simplify the equation. If the equation remains the same, it's symmetric with respect to the y-axis. Here, the function is not symmetric.

Step 5: Sketch the Graph

Plot the asymptotes, \(x=-3\), \(x=3\) and \(y=1\). Also plot the y-intercept at (0,0). Then draw the graph of the function using these values, approaching the asymptotes as \(x\) approaches \(3\) or \(-3\) from either side.

Step 6: Verify with a Graphing Utility

Plug the function into a graphing utility to verify your sketch. The graph should match your sketch.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes represent points where the function is undefined. These occur when the denominator of the rational function equals zero, but the numerator is not zero at the same point.
For the function given, \( y = \frac{x^2}{x^2-9} \), the denominator becomes zero when \( x^2 - 9 = 0 \). Solving this equation, we get two solutions: \( x = 3 \) and \( x = -3 \).

• At these values, the function is undefined, and as \( x \) approaches each asymptote, the function's value tends to infinity or negative infinity.
• On a graph, vertical lines at \( x = 3 \) and \( x = -3 \) indicate these boundaries.

Recognizing vertical asymptotes helps you understand the behavior of the function near those points, guiding you in sketching the graph.

Horizontal Asymptote

The horizontal asymptote of a rational function provides insight into the behavior of the function as \( x \) approaches infinity or negative infinity. For the function \( y = \frac{x^2}{x^2-9} \), the degrees of the numerator and denominator are the same.
In such cases, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.

• For this function, both leading coefficients are 1, giving a horizontal asymptote at \( y = 1 \).
• This means as \( x \) goes towards infinity in either direction, the value of \( y \) approaches 1.

Understanding the horizontal asymptote helps predict long-term behavior on the graph, illustrating how it "levels out" at \( y = 1 \).

Symmetry

Symmetry in a function's graph can simplify graphing by reducing the amount of computation needed. A function may exhibit various types of symmetry, such as symmetry with respect to the y-axis (even functions) or origin (odd functions).
For the function \( y = \frac{x^2}{x^2-9} \), checking for symmetry involves replacing \( x \) with \( -x \) to see if the equation remains unchanged.

• Substituting \( -x \) into the function does not yield an identical function, which indicates the graph is not symmetric around the y-axis.
• Also, it's not symmetric around the origin as no such identity transformation occurs.

Identifying symmetry can make graphing much easier, but in this case, the lack of symmetry means you must sketch without this shortcut.

Intercepts

Intercepts are important points that help locate the function's position relative to axes. The x-intercepts are points where the graph crosses the x-axis (\( y = 0 \)) and the y-intercept is where the graph crosses the y-axis (\( x = 0 \)).
For \( y = \frac{x^2}{x^2-9} \), you will find:

• x-intercepts: Setting \( y = 0 \) gives \( x = 0 \) for a rational function, leading to \( x^2 = 0 \), but in this function form, no real solutions exist as the function never actually touches the x-axis.
• y-intercept: By setting \( x = 0 \), the function becomes \( y = \frac{0}{0-9} \). Hence, the y-intercept is at (0, 0).

Locating intercepts assists in confirming where the graph starts and finishes against the axes, providing anchor points for sketching.