Problem 48 - 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. \(r(x)=\frac{4 x^{2}}{x^{2}-2 x-3}\)

Step-by-Step Solution

Answer

x-intercept and y-intercept at (0,0); Vertical asymptotes at x = 3 and x = -1; Horizontal asymptote at y = 4.

Step 1: Find the x-intercepts

To find the x-intercepts of the function, set the numerator of the rational function equal to zero. \(4x^2 = 0\)Solving for \(x\), we find that the x-intercepts are at \(x = 0\). So, the function has one x-intercept at the point (0, 0).

Step 2: Find the y-intercept

To find the y-intercept, substitute \(x = 0\) into the function:\(r(0) = \frac{4(0)^2}{0^2 - 2(0) - 3} = 0\).This confirms that the y-intercept is also at the origin (0, 0).

Step 3: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero, and the numerator is not zero. Set the denominator equal to zero and solve:\(x^2 - 2x - 3 = 0\)Factoring gives us \( (x - 3)(x + 1) = 0 \).So, the vertical asymptotes are at \(x = 3\) and \(x = -1\).

Step 4: Find Horizontal Asymptotes

The horizontal asymptotes can be determined by the degrees of the polynomials in the numerator and the denominator. Here, both the numerator and the denominator are of degree 2. Divide the leading coefficients:\(\lim_{x \to \infty} \frac{4x^2}{x^2} = 4\).Thus, the horizontal asymptote is \(y = 4\).

Step 5: Sketch the graph

With this information, we have:- x-intercept and y-intercept at (0,0)- Vertical asymptotes at \(x = 3\) and \(x = -1\)- Horizontal asymptote at \(y = 4\).Plot these points and lines, understand the behavior near the intercepts and asymptotes: As \(x\) approaches the vertical asymptotes \(x = -1\) and \(x = 3\), the graph approaches infinity or negative infinity. As \(x\) moves towards infinity, the graph approaches the line \(y = 4\).

Step 6: Confirm with graphing device

Use a graphing calculator or online graphing tool to input the function \(r(x) = \frac{4x^2}{x^2 - 2x - 3}\). Verify that the plotted graph matches your hand sketch, particularly at intercepts and asymptotes.

Key Concepts

Intercepts

Intercepts are key points where the graph of a function crosses the coordinate axes. These are crucial to determining the shape and behavior of a graph. In a rational function like \( r(x) = \frac{4x^2}{x^2 - 2x -3} \), we first look for the x-intercepts by setting the numerator equal to zero. Here, we solve \( 4x^2 = 0 \), which gives the solution \( x = 0 \). Therefore, the x-intercept is at the origin (0, 0).

To find the y-intercept, we substitute \( x = 0 \) into the function. This also results in \( r(0) = 0 \), reaffirming that the y-intercept is at the same point, (0, 0).

• X-intercept: Set numerator = 0, solve for x.
• Y-intercept: Substitute x = 0 into function, solve for y.

Together, these intercepts confirm that our graph will pass through the origin.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. In rational functions, we often deal with vertical and horizontal asymptotes. For the function \( r(x) = \frac{4x^2}{x^2 - 2x - 3} \), finding these asymptotes helps us understand the graph's behavior at infinity and near undefined points.

**Vertical Asymptotes**
Vertical asymptotes occur at points where the denominator is zero, and the numerator is not zero. Solve \( x^2 - 2x - 3 = 0 \) by factoring to get \((x-3)(x+1)=0\), which gives the values \( x = 3 \) and \( x = -1 \). These are our vertical asymptotes.

**Horizontal Asymptotes**
For horizontal asymptotes, compare the degree of the numerator to the degree of the denominator. Since both are degree 2, we use their leading coefficients to find \( y = 4 \). This indicates that as \( x \) approaches infinity, the function value approaches 4.

• Vertical asymptotes: Where denominator = 0.
• Horizontal asymptote: Ratio of leading coefficients if degrees are equal.

These asymptotes help enclose different parts of the graph and guide its curved trajectory.

Graphing Techniques

Graphing a rational function involves plotting intercepts and asymptotes, as they provide critical information on the structure of the graph. With \( r(x) = \frac{4x^2}{x^2 - 2x - 3} \), understanding the intercepts and asymptotes allows us to visualize the function.

**Plot the Intercepts and Asymptotes**
Begin by marking the x-intercept and y-intercept at (0, 0). Next, draw the vertical asymptotes at \( x = 3 \) and \( x = -1 \) as dashed lines, indicating where the graph will not be defined. Then, sketch the horizontal asymptote at \( y = 4 \).

**Sketch the Curve**
Consider the behavior near the asymptotes: the graph will reach towards infinity as it approaches \( x = 3 \) and \( x = -1 \). Where \( x \) flows towards infinity, the graph converges towards the line \( y = 4 \).

• Plot intercepts and asymptotes: Mark crucial points and lines.
• Sketch the curve: Draw based on key features and asymptotic behavior.

This step-by-step method helps in creating an accurate representation of the rational function, giving insight into its dynamics.