Problem 41 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(s(x)=\frac{6}{x^{2}-5 x-6}\)

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = -1\) and \(x = 6\), horizontal asymptote at \(y = 0\), y-intercept at \((0, -1)\).

Step 1: Factor the Denominator

The first step is to factor the quadratic expression in the denominator: \[ x^2 - 5x - 6. \] Identify two numbers that multiply to -6 and add to -5. These numbers are -6 and 1. Thus, the factorization is:\[ x^2 - 5x - 6 = (x - 6)(x + 1). \] So the function becomes:\[ s(x) = \frac{6}{(x - 6)(x + 1)}. \]

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero, assuming the numerator isn't zero at the same points. From the factored form \((x - 6)(x + 1)\), set each factor equal to zero:- \(x - 6 = 0\), so \(x = 6\).- \(x + 1 = 0\), so \(x = -1\).Thus, there are vertical asymptotes at \(x = 6\) and \(x = -1\).

Step 3: Determine the Horizontal Asymptote

The horizontal asymptote of a rational function \(\frac{p(x)}{q(x)}\) depends on the degrees of \(p(x)\) and \(q(x)\). Here, \(p(x)\) is a constant (degree 0), and \(q(x)\) is quadratic (degree 2), so the horizontal asymptote is \(y = 0\) because the degree of the denominator is greater than the degree of the numerator.

Step 4: Find Intercepts

For the y-intercept, set \(x = 0\):\[ s(0) = \frac{6}{(0 - 6)(0 + 1)} = \frac{6}{-6} = -1. \]The y-intercept is at \((0, -1)\). For x-intercepts, set the numerator to zero. Since the numerator is 6, which is never zero, there are no x-intercepts.

Step 5: Sketch the Graph

Use the information:- Vertical asymptotes at \(x = 6\) and \(x = -1\)- Horizontal asymptote at \(y = 0\)- Y-intercept at \((0, -1)\)Sketch the graph by plotting these points and asymptotes. The graph approaches the asymptotes but does not intersect them.

Step 6: Use a Graphing Device

To confirm, use a graphing calculator or software. Enter the function \(s(x)=\frac{6}{x^{2}-5x-6}\) and check the graph matches your sketch with vertical asymptotes at \(x = 6\) and \(x = -1\), horizontal asymptote at \(y = 0\), and a y-intercept at \((0, -1)\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a rational function approaches but never touches or crosses. They occur where the denominator of the function equals zero, provided the numerator doesn't also equal zero at that same point. This function, \(s(x)=\frac{6}{x^{2}-5x-6}\), has a denominator \((x - 6)(x + 1)\). By setting each factor to zero:

• \(x - 6 = 0\) gives \(x = 6\)
• \(x + 1 = 0\) gives \(x = -1\)

Thus, there are vertical asymptotes at \(x = 6\) and \(x = -1\). When you sketch the graph of a rational function, these asymptotes tell us where the function "shoots off" towards infinity.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) becomes very large or very small. For the rational function \(\frac{p(x)}{q(x)}\), the horizontal asymptote is determined by the degrees of the numerator and the denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If the degrees are equal, divide the coefficients of the highest degree terms.
• If the numerator's degree is higher, there is no horizontal asymptote.

In our function, the numerator is constant (degree 0), and the denominator is quadratic (degree 2). So the horizontal asymptote is \(y = 0\). As \(x\) heads towards infinity or negative infinity, the graph levels off along this line.

Intercepts

Intercepts are points where the graph intersects the axes:
**Y-Intercept**: This is found by setting \(x = 0\) in the function \(s(x)\). Calculating for \(s(0)\), we get:\[s(0) = \frac{6}{(0 - 6)(0 + 1)} = \frac{6}{-6} = -1\]Thus, the y-intercept is at \((0, -1)\). It shows where the graph crosses the y-axis.
**X-Intercepts**: These occur where the function \(s(x)\) is zero, which requires setting the numerator to zero. However, since the numerator here is 6, a constant never equal to zero, there are no x-intercepts. This means the graph never crosses the x-axis.

Graphing Rational Functions

Graphing rational functions involves combining all the information about intercepts, asymptotes, and the behavior of the function at these critical points. To graph \(s(x) = \frac{6}{(x - 6)(x + 1)}\):

• Draw vertical asymptotes as dashed lines at \(x = 6\) and \(x = -1\).
• Draw the horizontal asymptote at \(y = 0\).
• Plot the y-intercept at \((0, -1)\).

Now, sketch the curve of the rational function. The graph should get very close to the asymptotes, forming branches that either increase or decrease without crossing these lines.
Using a graphing tool can confirm your sketch. Simply input the function into a calculator or software to see how closely your sketch matches the actual graph. This practice solidifies your understanding of how rational functions behave and how to accurately predict and depict their graphs.