Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{x^{2}}{x^{2}-4}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{x^{2}}{x^{2}-4}\) has intercepts at the origin (0,0), vertical asymptotes at x=2 and x=-2, a horizontal asymptote at y=1, and no holes.

Step 1: Determine the intercepts

For the y-intercept, set x to 0: \(f(0)=\frac{0^{2}}{0^{2}-4}=0\). Therefore, the y-intercept is at (0,0). For the x-intercept, set the numerator equal to zero and solve for x: \(x^{2}=0\). Therefore, the x-intercept is also at (0,0).

Step 2: Determine the vertical asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is not zero. Therefore, solve \((x^{2}-4) = 0\). Factoring gives \((x-2)(x+2)=0\), thus x=2 and x=-2 are the vertical asymptotes.

Step 3: Determine the horizontal asymptotes

If the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of their coefficients. Here, since degrees are equal (both are 2), there is a horizontal asymptote at \(y = \frac{1}{1}=1\).

Step 4: Check for holes

Holes occur when factors in the numerator and denominator cancel each other out. However, in this function, no factors are canceled, thus there are no holes.

Step 5: Graph the function

Using all the calculated values, draw the function \(f(x)=\frac{x^{2}}{x^{2}-4}\) by plotting the intercept at the origin (0,0), vertical asymptotes at x=2 and x=-2, and the horizontal asymptote at y=1. Confirm the graph with a graphing utility.

Key Concepts

Intercepts of a Rational Function

Understanding the intercepts of a rational function is crucial in sketching its graph. The points where the graph crosses the axes are known as intercepts. To find the y-intercept, we set the x-value to 0 and solve for y. In our given function, when we evaluate it at zero, \(f(0)=\frac{0^2}{0^2-4}\), the result is 0. Consequently, the y-intercept is the point \( (0,0) \).

To find the x-intercept(s), we set the y-value to 0, which means we need to find the root(s) of the numerator. The denominator does not affect the x-intercepts as long as it's not zero at the same x-values; those would be holes. For the exercise function \(f(x)=\frac{x^2}{x^2-4}\), solving \(x^2=0\) leads us to the same intercept, which is also \( (0,0) \).

When teaching students about intercepts, it's helpful to remind them that rational functions can have more than one x-intercept, and sometimes, they may not have any at all, depending on the numerator.

Vertical Asymptotes

The vertical asymptotes of a rational function are vertical lines that the graph approaches but never touches. They indicate where the function becomes undefined, which occurs when the denominator is zero provided the numerator isn't zero at the same points. To find these, set the denominator to zero and solve for x. For the example function, we solve \(x^2 - 4 = 0\), which factors to \( (x-2)(x+2)=0 \), yielding \(x=2\) and \(x=-2\). Therefore, the vertical asymptotes are at \(x=2\) and \(x=-2\).

It's important when graphing to draw dashed vertical lines at these asymptotes to clearly mark the