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Chapter 13: Problem 17

Find the vertical asymptotes.$$ g(r)=\frac{r-6}{r^{2}-3 r-4} $$

Short Answer

Expert verified
Answer: The vertical asymptotes are at \(r=4\) and \(r=-1\).

Step by step solution

01

Factor the denominator

First, factor the quadratic expression in the denominator to find its roots. $$ r^2 - 3r - 4 = (r-4)(r+1) $$
02

Identify the vertical asymptotes

Next, identify the values of r for which the denominator becomes zero. These are the vertical asymptotes of the function. For \((r-4)(r+1) = 0\), we have two possibilities: 1. \((r-4)=0 \Rightarrow r=4\) 2. \((r+1)=0 \Rightarrow r=-1\) So, there are two vertical asymptotes at \(r=4\) and \(r=-1\).

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Key Concepts

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

Rational Functions and Their Behavior
Rational functions are expressions formed by dividing one polynomial by another. The general form is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions are interesting due to their unique characteristics, such as the presence of vertical asymptotes and horizontal asymptotes, which are special lines the graph approaches but never touches.

Understanding
  • **Numerator** (e.g., \( P(x) \)): Determines where the graph intersects the x-axis. Solving \( P(x) = 0 \) reveals the x-intercepts.
  • **Denominator** (e.g., \( Q(x) \)): Determines where the function is undefined. Solving \( Q(x) = 0 \) identifies x-values where the graph has asymptotic behavior.
Rational functions can have holes (removable discontinuities) when a factor cancels with a corresponding factor in the numerator, which is different from asymptotes that occur where the denominator is zero, but the numerator is not.
Factoring Quadratics for Simplification
Factoring quadratics is a critical skill for understanding rational functions, as it helps identify key features like roots and vertical asymptotes. A quadratic is an expression of the form \( ax^2 + bx + c \). To factor it, you aim to write it as a product of two binomials, \((x - p)(x - q)\), where \(p\) and \(q\) are roots of the equation.

For the given function, the quadratic \( r^2 - 3r - 4 \) factors into \((r - 4)(r + 1)\). This was achieved by finding two numbers that multiply to \(-4\) (constant term) and add to \(-3\) (linear coefficient). These numbers are \(-4\) and \(+1\).

  • Factoring reveals important information:
  • It simplifies expressions and helps in solving equations.
  • Identifies roots that relate to x-intercepts of functions.
  • Allows for the identification of vertical asymptotes by setting each factor in the denominator equal to zero.
Asymptote Identification in Rational Functions
Asymptotes are lines that the graph of a function approaches infinitely close but never actually reaches. They indicate boundaries or constraints on the graph.

For vertical asymptotes, these occur where the denominator of a rational function equals zero, and the numerator is not zero at those points. In simpler terms, it鈥檚 where the function is undefined due to division by zero.

In this problem, the factored denominator \((r-4)(r+1)\) equals zero at two points: \( r = 4 \) and \( r = -1 \). Thus, the vertical asymptotes are at these r-values. These x-values divide the graph into separate regions with distinct behaviors as the inputs move closer to these points.

  • Vertical asymptotes help determine the limits to the graph's behavior and provide insight into how functions behave near specific values.
  • The continuity breaks at these points, illustrating an infinite increase or decrease in the function鈥檚 values.

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Most popular questions from this chapter

Use the division algorithm to find the quotient \(q(x)\) and the remainder \(r(x)\) so that \(a(x)=\) \(q(x) b(x)+r(x)\). $$ \frac{x^{4}-2}{x^{2}-1} $$

Find the domain. $$ f(x)=\frac{x^{3}-8 x+1}{x^{4}+1} $$

For the rational functions in Problems \(32-35,\) find all zeros and vertical asymptotes and describe the long-run behavior, then graph the function. $$ y=\frac{x+3}{x+5} $$

In Exercises \(4-12,\) use the method of Example 2 on page 415 to find the quotient \(q(x)\) and the remainder \(r(x)\) so that \(a(x)=q(x) b(x)+r(x)\). $$ \frac{8 x^{5}+2 x^{4}-3 x^{2}+2 x+2}{8 x^{5}-3 x^{2}+2 x+2} $$

$$ \begin{array}{l} \qquad p(x)=(x+1)\left(x^{4}-2 x^{3}+4 x^{2}-5 x+4\right)+r \\ \text { where } p(x)=x^{5}-x^{4}+2 x^{3}-x^{2}-x-2 \end{array} $$
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