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Chapter 3: Problem 55

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x^{2}+2 x-4}{x^{2}+x}$$

Short Answer

Expert verified
Intercepts: x-intercepts at \(-2, 1\); no y-intercept. Asymptotes: horizontal at \(y=2\), vertical at \(x=0\) and \(x=-1\). Domain: \((-\infty, -1) \cup (-1, 0) \cup (0, \infty)\). Range: \((-\infty, 2) \cup (2, \infty)\).

Step by step solution

01

Finding the horizontal asymptote

First, observe the degrees of the numerator and the denominator. Both are quadratic polynomials (degree 2). When degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.\[\text{Horizontal Asymptote: } y = \frac{2}{1} = 2\]
02

Finding the vertical asymptotes

To find the vertical asymptotes, set the denominator equal to zero and solve for \(x\):\[\begin{align*}x^2 + x &= 0 \x(x+1) &= 0 \x &= 0 \quad \text{or} \quad x = -1\end{align*}\]Thus, there are vertical asymptotes at \(x = 0\) and \(x = -1\).
03

Finding the x-intercepts

Set the numerator equal to zero to find the x-intercepts:\[2x^2 + 2x - 4 = 0\]This can be simplified to:\[x^2 + x - 2 = 0\]Factoring gives:\[(x+2)(x-1) = 0\]Thus, the x-intercepts are at \(x = -2\) and \(x = 1\).
04

Finding the y-intercept

The y-intercept occurs where \(x = 0\):\[r(0) = \frac{2(0)^2 + 2(0) - 4}{0^2 + 0} = \text{undefined}\]Since the expression is undefined, there is no y-intercept.
05

Sketching the graph

Plot the horizontal asymptote at \(y = 2\), and vertical asymptotes at \(x = 0\) and \(x = -1\). Plot x-intercepts at \(x = -2\) and \(x = 1\). Note the undefined point at \(x = 0\). Draw the function approaching the asymptotes without crossing them.
06

Determining the domain

The domain is all real numbers except where the denominator is zero: \(x eq 0\) and \(x eq -1\). Thus, the domain is \(( -\infty, -1) \cup (-1, 0) \cup (0, \infty)\).
07

Determining the range

Since there are no restrictions on the y-values except for the horizontal asymptote at \(y = 2\), the range is all real numbers except \(y eq 2\). Therefore, the range is \(( -\infty, 2) \cup (2, \infty)\).

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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 intersects the axes. To find intercepts of a rational function, one can identify both the x-intercepts and the y-intercept.

**X-Intercepts**
  • X-intercepts occur where the graph crosses the x-axis, which is when the value of the function is zero. This is found by setting the numerator of the rational function equal to zero, as at these points the output (or y-value) is zero.
  • For the function \( r(x) = \frac{2x^2 + 2x - 4}{x^2 + x} \), the numerator becomes zero at \( x = -2 \) and \( x = 1 \). These are the x-intercepts.

**Y-Intercept**
  • The y-intercept is found by evaluating the function at \( x = 0 \). For the current rational function, substitution gives a division by zero, meaning there is no y-intercept.
  • Sometimes, a function may not pass through the y-axis if it is undefined at \( x = 0 \).
Asymptotes
In discussing rational functions, asymptotes are fundamental in understanding their behavior. Asymptotes can be either horizontal or vertical.
**Horizontal Asymptote**
  • A horizontal asymptote reflects the end behavior of the graph as \( x \) approaches infinity or negative infinity.
  • For the function \( r(x) = \frac{2x^2+2x-4}{x^2+x} \), the degrees of the numerator and denominator are equal. Therefore, the horizontal asymptote is determined by the ratio of the leading coefficients, giving \( y = 2 \).

**Vertical Asymptote**
  • Vertical asymptotes occur at the values of \( x \) for which the denominator is zero, providing these values don鈥檛 also cancel out the numerator.
  • For \( r(x) \), setting \( x^2 + x = 0 \) gives the solutions \( x = 0 \) and \( x = -1 \). These are the vertical asymptotes of the function.
Domain and Range
The domain and range of rational functions are key to understanding the limits in which the function operates.

**Domain**
  • The domain encompasses all possible input values \( x \) where the function is defined. For rational functions, this means excluding values that make the denominator zero.
  • Given the function \( r(x) = \frac{2x^2+2x-4}{x^2+x} \), \( x = 0 \) and \( x = -1 \) create zero in the denominator, so these are excluded from the domain. Thus, the domain is \(( -\infty, -1) \cup (-1, 0) \cup (0, \infty)\).

**Range**
  • The range includes all possible output values (\( y \)-values) of the function.
  • In rational functions with a horizontal asymptote, such as \( y = 2 \), the function cannot actually reach this value in terms of y, hence \( y = 2 \) is not included in the range.
  • Thus, the range of \( r(x) \) is \(( -\infty, 2) \cup (2, \infty)\).
Graphing Rational Functions
Graphing rational functions can help visualize their behavior and characteristics such as intercepts and asymptotes.

**Sketching the Graph**
  • Plot the horizontal asymptote \( y = 2 \) and vertical asymptotes at \( x = 0 \) and \( x = -1 \).
  • Mark the x-intercepts at points \( x = -2 \) and \( x = 1 \).
  • The graph approaches these asymptotes but never crosses them. It "bends" toward these lines as it extends infinitely along the axes.

**Using Technology**
  • A graphing calculator or software can be used to check the accuracy of the drawn graph. This ensures all asymptotic behavior and intercepts are correctly plotted.
By using these steps, one can gain a comprehensive understanding of how the graph of a rational function behaves relative to its mathematical properties.

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Most popular questions from this chapter

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