Problem 49 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x^{2}+x-6}{x^{2}-1}$$

Step-by-Step Solution

Answer

For the function \(f(x)=\frac{x^{2}+x-6}{x^{2}-1}\), the x-intercepts are at x = -3 and x = 2, the y-intercept is at y = 6, the vertical asymptotes are at x = -1 and x = 1, and the horizontal asymptote is at y = 1.

Step 1: Find the Intercepts

The x-intercepts are found by setting the numerator equal to zero, because wherever the numerator is zero, the function is zero. So for this function, we have to solve \(x^{2}+x-6=0\). The y-intercept is found by setting x = 0 in the function.

Step 2: Find the Asymptotes

The vertical asymptotes are found by setting the denominator equal to zero and solving for x, because the function approaches infinity where the denominator is zero. Therefore, for this function the equation is \(x^{2}-1=0\). To find any horizontal asymptote, compare the degrees in the numerator and denominator. If the degrees are equal (which is the case here), the horizontal asymptote is the ratio of the coefficients of the highest degree terms.

Step 3: Sketch the Graph

After creating a table of values around the intercepts and asymptotes from steps 1 and 2, we plot these points along with the intercepts and asymptotes. By joining these points we get the graph of the function.

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a rational function occur where the denominator is equal to zero, leading the function to approach infinity or negative infinity. For the given function \( f(x)=\frac{x^{2}+x-6}{x^{2}-1} \), we can find the vertical asymptotes by solving the equation of the denominator: \[ x^2 - 1 = 0 \] Solving for \( x \), we have: \[ x^2 = 1 \] \[ x = \pm 1 \] This means the function will have vertical asymptotes at \( x = 1 \) and \( x = -1 \). At these points, the function is undefined because the denominator becomes zero.

• Think of vertical asymptotes as invisible lines that the graph approaches but never touches or crosses.
• The graph will get closer and closer to these lines but never actually reach them.

It's essential to correctly identify vertical asymptotes as they are a crucial element in graphing rational functions and understanding their behavior.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches infinity or negative infinity. They are determined by comparing the degrees of the numerator and denominator of the rational function. For our function \( f(x)=\frac{x^{2}+x-6}{x^{2}-1} \), the degrees of both the numerator and the denominator are the same (degree 2). This configuration means there might be a horizontal asymptote.To find this asymptote, take the ratio of the leading coefficients (the coefficients of the highest degree terms):

• Numerator's leading coefficient: 1
• Denominator's leading coefficient: 1

Therefore, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

• This line \( y = 1 \) represents the value the function approaches as \( x \) goes towards infinity.
• Unlike a vertical asymptote, the graph can actually touch a horizontal asymptote. However, it will eventually level off near this line.

Understanding horizontal asymptotes helps in predicting the end behavior of rational functions and provides a sense of their long-term trends.

Graphing Intercepts

Intercepts are critical points where the graph intersects the axes. They help in plotting the rough shape of a function's graph.**X-Intercepts**: To find x-intercepts, set the numerator of the function equal to zero and solve, as these are the values where the function equals zero. For \( f(x)=\frac{x^{2}+x-6}{x^{2}-1} \):\[ x^2 + x - 6 = 0 \]Using the quadratic formula or factoring, we find:\[ (x-2)(x+3) = 0 \]So the x-intercepts are \( x = 2 \) and \( x = -3 \).

• At these x-values, the graph will cross the x-axis.

**Y-Intercept**: Find the y-intercept by setting \( x = 0 \) in the function:\[ f(0) = \frac{0^2 +0 -6}{0^2 -1} = \frac{-6}{-1} = 6 \]This means the graph crosses the y-axis at \( y = 6 \).

• Intercepts serve as anchor points when plotting the graph and aid in the formation of the complete graph.
• These points are where the graph truly 'interacts' with the coordinate axes.

By accurately finding intercepts, you can form a clearer picture of a rational function's graph and trace your graph through crucial points on the axes.