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Chapter 3: Problem 33

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

Short Answer

Expert verified
Y-intercept: (0, -2); X-intercept: (1, 0); Vertical asymptote: x = -2; Horizontal asymptote: y = 4. Graph verified.

Step by step solution

01

Identify the Y-intercept

To find the y-intercept of the function, set \( x = 0 \) in the function \( r(x) = \frac{4x - 4}{x + 2} \). This gives \( r(0) = \frac{4(0) - 4}{0 + 2} = \frac{-4}{2} = -2 \). Thus, the y-intercept is the point \((0, -2)\).
02

Identify the X-intercept

The x-intercepts occur when the numerator of the rational function equals zero. Set \( 4x - 4 = 0 \) and solve for \( x \):\[ 4x - 4 = 0 \rightarrow 4x = 4 \rightarrow x = 1 \].Hence, the x-intercept is the point \((1, 0)\).
03

Determine the Vertical Asymptote

The vertical asymptote occurs where the denominator of the function equals zero since division by zero is undefined. Set \( x + 2 = 0 \) and solve for \( x \):\[ x + 2 = 0 \rightarrow x = -2 \].Therefore, the vertical asymptote is at \( x = -2 \).
04

Determine the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is the quotient of the leading coefficients. Here, both the numerator and denominator have degree 1, so the horizontal asymptote is \( y = \frac{4}{1} = 4 \).
05

Sketch the Graph

Using the intercepts and asymptotes found, sketch the graph on the coordinate plane. Plot the y-intercept at \((0, -2)\), the x-intercept at \((1, 0)\), draw a vertical asymptote as a dashed line at \( x = -2 \), and a horizontal asymptote as a dashed line at \( y = 4 \). The curve should approach these asymptotes as \( x \) moves towards infinity but never touch them.
06

Confirm with Graphing Device

Use a graphing calculator or software to plot \( r(x) = \frac{4x - 4}{x + 2} \) and check that the sketch matches: y-intercept at \((0, -2)\), x-intercept at \((1, 0)\), vertical asymptote at \( x = -2 \), and horizontal asymptote at \( y = 4 \).

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Key Concepts

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

Intercepts of Rational Functions
Intercepts are key points where a graph intersects the axes of a coordinate plane. For a rational function like \( r(x) = \frac{4x - 4}{x + 2} \), we have two types of intercepts to consider:
  • Y-Intercept: This is where the graph of the function crosses the y-axis. To find it, we set \( x = 0 \) and solve: \( r(0) = \frac{4(0) - 4}{0 + 2} = -2 \). Thus, the y-intercept is at the point \( (0, -2) \), where the graph meets the y-axis.
  • X-Intercept: This occurs where the graph crosses the x-axis. Setting the numerator equal to zero, \( 4x - 4 = 0 \), and solving for \( x \), we find \( x = 1 \). Therefore, the x-intercept is at \( (1, 0) \).
Identifying these intercepts is crucial for accurately sketching the graph of a rational function.
Asymptotes of Rational Functions
Asymptotes are invisible lines that a graph approaches but never quite touches. They provide important information about the behavior of a rational function as \( x \) approaches certain values.
  • Vertical Asymptote: This occurs where the denominator of the function is zero. For \( r(x) = \frac{4x - 4}{x + 2} \), set \( x + 2 = 0 \). Solving, we get \( x = -2 \). This means there鈥檚 a vertical asymptote at \( x = -2 \), indicating the graph will approach this line as \( x \) gets close to -2 but will never cross it.
  • Horizontal Asymptote: To find this, compare the degrees of the numerator and denominator. When they are equal, the horizontal asymptote is the ratio of their leading coefficients. Here, both degrees are 1, leading to a horizontal asymptote at \( y = \frac{4}{1} = 4 \). This line represents the value that the function approaches as \( x \) moves towards infinity.
Understanding asymptotes helps in predicting long-term behavior of the rational function's graph.
Graphing Rational Functions
Graphing rational functions involves combining knowledge of intercepts and asymptotes to sketch the curve accurately.
  • Start by plotting the intercepts: the y-intercept \( (0, -2) \) and the x-intercept \( (1, 0) \). These points provide a starting foundation on the graph.
  • Next, draw the vertical asymptote as a dashed line at \( x = -2 \), since the graph will not cross this line.
  • Similarly, represent the horizontal asymptote as a dashed line at \( y = 4 \), showing that the graph approaches but never reaches this line at infinity.
With these elements in place, sketch the curve: notice how the graph moves away from the intercepts, curves gently towards the asymptotes, and follows their direction without ever touching them. As a final step, use a graphing tool to confirm your plot matches these features.

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Most popular questions from this chapter

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}$$

After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$c(t)=\frac{30 t}{t^{2}+2}$$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right)$$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\). Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Find all horizontal and vertical asymptotes (if any). $$s(x)=\frac{3 x^{2}}{x^{2}+2 x+5}$$

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$
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