Chapter 12: Rational Equations and Functions
Introduction
The final chapter of this text introduces the concept of rational functions; that is, equations in which the variable appears in the denominator of a fraction. A common rational function is the inverse variation model, similar to the direct variation model you studied in a Concept in chapter 4. We finish the chapter with solving rational equations and using graphical representations to display data.
Chapter Outline
- 12.1. Inverse Variation Models
- 12.2. Graphs of Rational Functions
- 12.3. Division of Polynomials
- 12.4. Excluded Values for Rational Expressions
- 12.5. Multiplication of Rational Expressions
- 12.6. Division of Rational Expressions
- 12.7. Addition and Subtraction of Rational Expressions
- 12.8. Rational Equations Using Proportions
- 12.9. Clearing Denominators in Rational Equations
- 12.10. Surveys and Samples
Chapter Summary
Summary
This chapter begins by talking about inverse variation models, the graphs of rational functions, and the division of polynomials. It then moves on to discuss rational expressions in detail, including the multiplication, division, addition, and subtraction of rational expressions. Next, instruction is given on solving rational equations by using proportions and by clearing denominators. Finally, surveys and samples in statistics are highlighted.
Rational Equations and Functions; Statistics Review
Define the following terms used in this chapter.
- Inverse variation
- Asymptotes
- Hyperbola
- Points of discontinuity
- Least common multiple
- Random sampling
- Stratified sampling
- Biased
- Cherry picking
- What quadrants are the branches of a hyperbola located if \begin{align*}k<0\end{align*}?
Are the following examples of direct variation or inverse variation?
- The number of slices \begin{align*}n\end{align*} people get from sharing one pizza
- The thickness of a phone book given \begin{align*}n\end{align*} telephone numbers
- The amount of coffee \begin{align*}n\end{align*} people receive from a single pot
- The total cost of pears given that nectarines cost $0.99 per pound
For each variation equation:
- Translate the sentence into an inverse variation equation.
- Find \begin{align*}k\end{align*}, the constant of variation.
- Find the unknown value.
- \begin{align*}y\end{align*} varies inversely as \begin{align*}x\end{align*}. When \begin{align*}x=5, y=\frac{2}{15}\end{align*}. Find \begin{align*}y\end{align*} when \begin{align*}x=- \frac{1}{2}\end{align*}.
- \begin{align*}y\end{align*} is inversely proportional to the square root of \begin{align*}y\end{align*}. When \begin{align*}x=16, y=0.5625\end{align*}. Find \begin{align*}y\end{align*} when \begin{align*}x=\frac{1}{8}\end{align*}.
- Habitat for Humanity uses volunteers to build houses. The number of days it takes to build a house varies inversely as the number of volunteers. It takes eight days to build a house with twenty volunteers. How many days will it take sixteen volunteers to complete the same job?
- The Law of the Fulcrum states the distance you sit to balance a seesaw varies inversely as your weight. If Gary weighs 20.43 kg and sits 1.8 meters from the fulcrum, how far would Shelley sit, assuming she weighs 36.32 kilograms?
For each function:
- Graph it on a Cartesian plane.
- State its domain and range.
- Determine any horizontal and/or vertical asymptotes the function may have.
- \begin{align*}y=\frac{4}{x}\end{align*}
- \begin{align*}f(x)=\frac{2}{4-x}\end{align*}
- \begin{align*}g(x)=\frac{-1}{x+1}\end{align*}
- \begin{align*}y=\frac{6}{3x+1}-2\end{align*}
- \begin{align*}f(x)=\frac{3}{x}-5\end{align*}
Perform the indicated operation.
- \begin{align*}\frac{5a}{6}-\frac{5b}{4b}\end{align*}
- \begin{align*}\frac{4}{3m}+\frac{4m}{5}\end{align*}
- \begin{align*}\frac{3x}{2xy}+\frac{4}{3}\end{align*}
- \begin{align*}\frac{2}{5n-2}+\frac{2n}{2}\end{align*}
- \begin{align*}\frac{2x+1}{3x+9}-\frac{x+5}{3x+9}\end{align*}
- \begin{align*}\frac{5m+n}{30n^4}-\frac{4m+n}{30n^4}\end{align*}
- \begin{align*}\frac{r-6}{4r^2-12r+8}-\frac{r+6}{4r^2-12r+8}\end{align*}
- \begin{align*}\frac{2}{16x^3 y^2}+\frac{x-2y}{16x^3 y^2}\end{align*}
- \begin{align*}\frac{n-6}{n+2}+\frac{2n}{5}\end{align*}
- \begin{align*}\frac{8}{4}-\frac{x+5}{x+8}\end{align*}
- \begin{align*}\frac{3x}{2(x+1)}+\frac{6}{7x-6}\end{align*}
- \begin{align*}\frac{11}{8} \cdot \frac{20x^2}{2}\end{align*}
- \begin{align*}\frac{17r}{16} \cdot \frac{7r^4}{16}\end{align*}
- \begin{align*}\frac{15}{18} \cdot \frac{14}{17t}\end{align*}
- \begin{align*}\frac{2(b-11)}{14b} \cdot \frac{b+5}{(b+5)(b-11)}\end{align*}
- \begin{align*}\frac{17w^2}{w+4} \cdot \frac{18(w+4)}{17w^2 (w-9)}\end{align*}
- \begin{align*}\frac{10s^3-30s^2}{30s^2-10s^3} \cdot \frac{s-3}{8}\end{align*}
- \begin{align*}\frac{1}{f-5} \div \frac{f+3}{f^2+6f+9}\end{align*}
- \begin{align*}\frac{(a+8)(a+3)}{4(a+3)} \div \frac{10a^2 (a+10)}{4}\end{align*}
- \begin{align*}\frac{1}{(h-10)(h+7)} \div \frac{(h-4)}{4h(h-10)}\end{align*}
- \begin{align*}\frac{2(5x-8)}{4x^2 (8-5x)} \div \frac{6}{4x^2}\end{align*}
- \begin{align*}\frac{2(q-7)}{40q(q+1)} \div \frac{1}{40q(q+1)}\end{align*}
Solve each equation.
- \begin{align*}\frac{3}{3x^2}=\frac{1}{x}+\frac{1}{3x^2}\end{align*}
- \begin{align*}\frac{2}{5x^2}=-\frac{12}{x-3}\end{align*}
- \begin{align*}\frac{7x}{x-6}=\frac{3}{4x+16}\end{align*}
- \begin{align*}\frac{4}{c-2}=\frac{3}{c+4}\end{align*}
- \begin{align*}\frac{d-4}{4d^2}=\frac{1}{4d^2}+\frac{1}{4d}\end{align*}
- \begin{align*}\frac{1}{2}=\frac{2z-12}{z} - \frac{z+1}{4z}\end{align*}
- \begin{align*}\frac{1}{n}=\frac{1}{n^2} +\frac{6}{n}\end{align*}
- \begin{align*}\frac{1}{2a}=\frac{1}{2a^2}+\frac{1}{a}\end{align*}
- \begin{align*}\frac{k+4}{k^2} =\frac{5k-30}{3k^2}+\frac{1}{3k^2}\end{align*}
- It takes Jayden seven hours to paint a room. Andie can do it in five hours. How long will it take to paint the room if Jayden and Andie work together?
- Kiefer can mow the lawn in 4.5 hours. Brad can do it in two hours. How long will it take if they worked together?
- Melissa can mop the floor in 1.75 hours. With Brad’s help, it took only 50 minutes. How long would it take Brad to mop it alone?
- Working together, it took Frankie and Ricky eight hours to frame a room. It would take Frankie fifteen hours doing it alone. How long would it take Ricky to do it alone?
- A parallel circuit has \begin{align*}R_1=50 \Omega\end{align*} and \begin{align*}R_t=16 \Omega\end{align*}. Find \begin{align*}R_2\end{align*}.
- A parallel circuit has \begin{align*}R_1=6 \Omega\end{align*} and \begin{align*}R_2=9 \Omega\end{align*}. Find \begin{align*}R_T\end{align*}.
- A series circuit has \begin{align*}R_1=200 \Omega\end{align*} and \begin{align*}R_t=300 \Omega\end{align*}. Find \begin{align*}R_2\end{align*}.
- A series circuit has \begin{align*}R_1=11 \Omega\end{align*} and \begin{align*}R_2=25 \Omega\end{align*}. Find \begin{align*}R_T\end{align*}.
- Write the formula for the total resistance for a parallel circuit with three individual resistors.
- What would be the bias in this situation? To determine the popularity of a new snack chip, a survey is conducted by asking 75 people walking down the chip aisle in a supermarket which chip they prefer.
- Describe the steps necessary to design and conduct a survey.
- You need to survey potential voters for an upcoming school board election. Design a survey with at least three questions you could ask. How will you plan to conduct the survey?
- What is a stratified sample? Name one case where a stratified sample would be more beneficial.
Rational Equations and Functions; Statistics Test
- True or false? A horizontal asymptote has the equation \begin{align*}y=c\end{align*} and represents where the denominator of the rational function is equal to zero.
- A group of SADD members wants to find out about teenage drinking. The members conduct face-to-face interviews, wearing their SADD club shirts. What is a potential bias? How can this be modified to provide accurate results?
- Name the four types of ways questions can be biased.
- Which is the best way to show data comparing two categories?
- Consider \begin{align*}f(x)= -\frac{4}{x}\end{align*}. State its domain, range, asymptotes, and the locations of its branches.
- \begin{align*}h\end{align*} varies inversly as \begin{align*}r\end{align*}. When \begin{align*}h=-2.25, r=0.125\end{align*}. Find \begin{align*}h\end{align*} when \begin{align*}r=12.\end{align*}
- Name two types of visual displays that could be used with a frequency distribution.
- Tyler conducted a survey asking the number of pets his classmates owned and received the following results: 0, 2, 1, 4, 3, 2, 1, 0, 0, 0, 0, 1, 4, 3, 2, 3, 4, 3, 2, 1, 1, 1, 5, 7, 0, 1, 2, 3, 2, 1, 4, 3, 2, 1, 1, 0
- Display this data with a frequency distribution chart.
- Use it to make a histogram.
- Find its five-number summary.
- Draw a box-and-whisker plot.
- Make at least two conclusions regarding Tyler’s survey.
- Find the excluded values, the domain, the range, and the asymptotes of: \begin{align*}f(x)=-\frac{9}{x^2-16}+4.\end{align*}
Perform the indicated operation.
- \begin{align*}\frac{4}{21r^4}+\frac{4r+5t}{21r^4}\end{align*}
- \begin{align*}\frac{a-v}{12a^3}-\frac{a+5v}{12a^3}\end{align*}
- \begin{align*}\frac{8}{g+8}+\frac{g-3}{g-5}\end{align*}
- \begin{align*}\frac{4t}{5t-8}+\frac{24}{12}\end{align*}
- \begin{align*}\frac{4}{5} \cdot \frac{80}{48m}\end{align*}
- \begin{align*}\frac{1}{d-8} \div \frac{d+7}{2d+14}\end{align*}
- \begin{align*}\frac{1}{u-3} \div \frac{u-4}{2u-6}\end{align*}
Solve.
- \begin{align*}\frac{7w}{w-7}=\frac{7w}{w+5}\end{align*}
- \begin{align*}\frac{p-6}{3p^2-6p}=\frac{7}{3}\end{align*}
- \begin{align*}\frac{2}{x^2} =\frac{1}{2x^2}-\frac{x+1}{2x^2}\end{align*}
- \begin{align*}\frac{1}{2}-\frac{1}{4r}=\frac{3}{4}\end{align*}
- \begin{align*}\frac{y-5}{3y^2}=-\frac{1}{3y}+\frac{1}{y^2}\end{align*}
- Working together, Ashton and Matt can tile a floor in 25 minutes. Working alone, it would take Ashton two hours. How long would it take Matt to tile the floor alone?
- Bethany can paint the deck in twelve hours. Melissa can paint the deck in five hours. How long would it take the girls to paint the deck, working together?
- A parallel circuit has \begin{align*}R_t=115 \ \Omega\end{align*} and \begin{align*}R_2=75 \ \Omega\end{align*}. Find \begin{align*}R_1\end{align*}.
- A series circuit has \begin{align*}R_1=13 \ \Omega\end{align*} and \begin{align*}R_t=21 \ \Omega\end{align*}. Find \begin{align*}R_2\end{align*}.
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9622.