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Chapter 3: Problem 53

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}$$

Short Answer

Expert verified
Y-intercept: (0, -2); vertical asymptotes at x = 3, x = -1; horizontal asymptote at y = 3.

Step by step solution

01

Find the x-intercepts

To find the x-intercepts of the function, set the numerator equal to zero and solve for \(x\). The numerator is \(3x^2 + 6\). Set it equal to zero:\[3x^2 + 6 = 0\]Solving for \(x\), we get:\[3x^2 = -6\]\[x^2 = -2\]Since \(x^2 = -2\) does not have real solutions, there are no x-intercepts for this function.
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the function \(r(x)\):\[r(0) = \frac{3(0)^2 + 6}{(0)^2 - 2(0) - 3} = \frac{6}{-3} = -2\]Thus, the y-intercept is \((0, -2)\).
03

Find the vertical asymptotes

The vertical asymptotes occur where the denominator is zero. Solve \(x^2 - 2x - 3 = 0\). Factoring gives:\[(x - 3)(x + 1) = 0\]Setting each factor to zero gives:\[x - 3 = 0 \quad \text{implies} \quad x = 3\]\[x + 1 = 0 \quad \text{implies} \quad x = -1\]So, the vertical asymptotes are at \(x = 3\) and \(x = -1\).
04

Find the horizontal asymptote

The degrees of the numerator and denominator are equal (both degree 2). Thus, the horizontal asymptote is the ratio of the leading coefficients:\[\text{Horizontal asymptote: } y = \frac{3}{1} = 3\]
05

Sketch the graph

Based on the asymptotes and intercepts, sketch the graph. Indicate the vertical asymptotes at \(x = 3\) and \(x = -1\), a horizontal asymptote at \(y = 3\), and the y-intercept at \((0, -2)\). Note that there are no x-intercepts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intercepts
Intercepts are the points where a graph crosses the axes, and they provide crucial information for graphing functions. For the rational function \( r(x) = \frac{3x^2 + 6}{x^2 - 2x - 3} \), we look for both x-intercepts and y-intercepts.

To find **x-intercepts**, set the numerator of the rational function to zero, because rational functions are zero when their numerators are zero. In this case:
  • \(3x^2 + 6 = 0\) solves to \( x^2 = -2 \), which has no real solutions since no real number squared results in a negative number.
  • This means there are no x-intercepts for this function.
For the **y-intercept**, we substitute x with zero in the function:
  • \(r(0) = \frac{3(0)^2 + 6}{(0)^2 - 2(0) - 3} = \frac{6}{-3} = -2\).
  • This gives us the point \((0, -2)\) on the y-axis, where the graph crosses the y-axis.
Understanding intercepts allows you to quickly identify key points on a graph.
Asymptotes
Asymptotes describe the behavior of a function as it moves towards infinity or negative infinity. For rational functions, we generally discuss vertical and horizontal asymptotes.

**Vertical asymptotes** occur where the denominator is zero, as the function approaches infinity, meaning the graph climbs up towards or falls down from these lines dramatically:
  • Set the denominator \(x^2 - 2x - 3\) equal to zero: \((x - 3)(x + 1) = 0\).
  • Solving this gives us \(x = 3\) and \(x = -1\), indicating vertical asymptotes.
**Horizontal asymptotes** occur as the x-value approaches infinity and describe the function's leveling behavior:
  • If the degree of the numerator is equal to the degree of the denominator, as it is with \(3x^2 + 6\) and \(x^2 - 2x - 3\), the horizontal asymptote is calculated as the ratio of the leading coefficients \(\frac{3}{1} = 3\).
  • This tells us the horizontal asymptote is \(y = 3\).
Knowing the asymptotes helps you predict the function's behavior at the extremes.
Graphing Rational Functions
Graphing rational functions involves understanding both intercepts and asymptotes. For \( r(x) = \frac{3x^2 + 6}{x^2 - 2x - 3} \), you've identified the intercepts and asymptotes, which guide you in sketching the graph.

Here's a step-by-step process:
  • Mark the **y-intercept** at \((0, -2)\).
  • Draw **vertical asymptotes** as dashed lines at \(x = 3\) and \(x = -1\). The graph will steeply approach these lines but never cross or touch them.
  • Add a **horizontal asymptote** at \(y = 3\) as a dashed line, indicating that as \(x\) becomes very large or very small, the graph will approach this line.
When drawing the graph, keep these guidelines in mind:- The graph does **not** have x-intercepts to plot as cross points on the x-axis.- As the function extends towards positive and negative infinity, the effect of the horizontal asymptote becomes pronounced.By considering these elements, you can accurately represent the shape and path of the rational function on a graph.

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Most popular questions from this chapter

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}-2 x+1}{x^{2}+2 x+1}$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+4}{2 x^{2}+x-1}$$

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

Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{6 x-2}{x^{2}+5 x-6}$$

Show that the polynomial does not have any rational zeros. $$P(x)=x^{50}-5 x^{25}+x^{2}-1$$
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