Rational Functions Mathematical Practices
Mathematically proficient students are careful about specifying units of measure and clarifying the relationship between quantities in a problem.
Monitoring Progress
Question 1.You drive a car at a speed of 60 miles per hour. What is the speed in meters per second?Answer:60 miles per hour to Meters per second is 26,8224.
Question 2.A hose carries a pressure of 200 pounds per square inch. What is the pressure in kilograms per square centimeter?Answer:200 pounds per square inch to kilograms per square centimeter = 14.06139 kilograms per square centimeter
Question 3.A concrete truck pours concrete at the rate of 1 cubic yard per minute. What is the rate in cubic feet per hour?Answer:
Question 4.Water in a pipe flows at a rate of 10 gallons per minute. What is the rate in liters per second?Answer:
Essential Question How can you recognize when two quantities vary directly or inversely?
EXPLORATION 1
Recognizing Direct VariationWork with a partner. You hang different weights from the same spring.a. Describe the relationship between the weight x and the distance d the spring stretches from equilibrium. Explain why the distance is said to vary directly with the weight.b. Estimate the values of d from the figure. Then draw a scatter plot of the data. What are the characteristics of the graph?c. Write an equation that represents d as a function of x.d. In physics, the relationship between d and x is described by Hookes Law. How would you describe Hookes Law?
EXPLORATION 2
Graphing Rational Functions 72 Exercises
Vocabulary and Core Concept CheckQuestion 1.COMPLETE THE SENTENCEThe function y = \ has a __________ of all real numbers except 3 and a __________ of all real numbers except 4.Answer:
y = \ 2Answer:
ANALYZING RELATIONSHIPS In Exercises 2124, match the function with its graph. Explain your reasoning.Question 21.g = \ + 1Answer:
h = \ + 1Answer:The graph is A because the asymptotes are x=-3 and y=1
Question 23.f = \ 1Answer:
y = \ 1Answer:The graph is D because the asymptotes are x=-3 and y=1
In Exercises 2532, graph the function. State the domain and range.Question 25.f = \Answer:
y = \Answer:
g = \Answer:
g = \Answer:
g = \Answer:
g = \Answer:
Question 41.PROBLEM SOLVINGYour school purchases a math software program. The program has an initial cost of $500 plus $20 for each student that uses the program.a. Estimate how many students must use the program for the average cost per student to fall to $30.b. What happens to the average cost as more students use the program?Answer:
Question 42.PROBLEM SOLVINGTo join a rock climbing gym, you must pay an initial fee of $100 and a monthly fee of $59.a. Estimate how many months you must purchase a membership for the average cost per month to fall to $69.b. What happens to the average cost as the number of months that you are a member increases?Answer:
Question 43.USING STRUCTUREWhat is the vertical asymptote of the graph of the function y = \ + 7?A. x =7
f = \Answer:
y = \Answer:
f = \Answer:
Rational Functions Cumulative Assessment
Question 1.Which of the following functions are shown in the graph? Select all that apply. Justify your answers.Answer:
Question 2.You step onto an escalator and begin descending. After riding for 12 feet, you realize that you dropped your keys on the upper floor and walk back up the escalator to retrieve them. The total time T of your trip down and up the escalator is given byT = \where s is the speed of the escalator and w is your walking speed. The trip took 9 seconds, and you walk at a speed of 6 feet per second. Find two possible speeds of the escalator.Answer:
Question 3.The graph of a rational function has asymptotes that intersect at the point . Choose the correct values to complete the equation of the function.Answer:
Question 4.The tables below give the amounts A of money in two different bank accounts over time t .a. Determine the type of function represented by the data in each table.b. Provide an explanation for the type of growth of each function.c. Which account has a greater value after 10 years? after 15 years? Justify your answers.Answer:
Order the expressions from least to greatest. Justify your answer.Answer:
Question 6.A movie grosses $37 million after the first week of release. The weekly gross sales y decreases by 30% each week. Write an exponential decay function that represents the weekly gross sales in week x. What is a reasonable domain and range in this situation? Explain your reasoning.Answer:
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Big Ideas Math Algebra 2 Answer Key
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Lesson 75 Solving Rational Equations
Essential Question How can you solve a rational equation?
EXPLORATION 1
Solving Rational EquationsWork with a partner. Match each equation with the graph of its related system of equations. Explain your reasoning. Then use the graph to solve the equation.
EXPLORATION 2
Solving Rational EquationsWork with a partner. Look back at the equations in Explorations 1 and 1. Suppose you want a more accurate way to solve the equations than using a graphical approach.a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the equations.b. Show how you could use an analytical approach. For instance, you might use the method you used to solve proportions.
Communicate Your Answer
How can you solve a rational equation?Answer:
Use the method in either Exploration 1 or 2 to solve each equation.a. \b. \c. \Answer:
Solve the equation by cross multiplying. Check your solution.Question 1.\(\frac=\fracAnswer:
\frac=\frac\)Answer:
\Answer:
Solve the equation by using the LCD. Check your solution.Question 4.\Answer:
\Answer:
\Answer:
Solve the equation. Check your solution.Question 7.\Answer:
\Answer:
Question 9.Consider the function f = \ 2. Determine whether the inverse of f is a function. Then find the inverse.Answer:
How do the answers in Example 6 change when c = \?Answer:
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Lesson 74 Adding And Subtracting Rational Expressions
Essential QuestionHow can you determine the domain of the sum or difference of two rational expressions?You can add and subtract rational expressions in much the same way that you add and subtract fractions.\ Sum of rational expressions\ Difference of rational expressions
EXPLORATION 1
Adding and Subtracting Rational ExpressionsWork with a partner. Find the sum or difference of the two rational expressions. Then match the sum or difference with its domain. Explain your reasoning.
EXPLORATION 2
Writing a Sum or DifferenceWork with a partner. Write a sum or difference of rational expressions that has the given domain. Justify your answer.a. all real numbers except 1b. all real numbers except 1 and 3c. all real numbers except 1, 0, and 3
Communicate Your Answer
How can you determine the domain of the sum or difference of two rational expressions?Answer:
Your friend found a sum as follows. Describe and correct the error.\Answer:
Find the sum or difference.Question 1.\Answer:
\Answer:
\Answer:
\Answer:
Find the least common multiple of 5×3 and 10×2 15x.Answer:
Find the sum or difference.Question 6.\Answer:
\Answer:
\Answer:
Question 9.Rewrite g = \ in the form g = \ + k. Graph the function. Describe the graph of g as a transformation of the graph of f = \.Answer:
Explain how adding and subtracting rational expressions is similar to adding and subtracting numerical fractions.Answer:Both expressions require a common denominator before adding and subtracting the numerators.
Chapter 7 Test A Geometry Answers Big Ideas Math
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Big Ideas Math Algebra 2 Answers Chapter 7 Rational Functions
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Preparation Tips For High School Students
Go through the below-listed tips to enhance your math learning skills easily. They are as follows
- Choose the best study material to prepare.
- Plan the best time to prepare.
- Listen to music for relaxation.
- Connect the questions with the real-time and learn the problems.
- You have to prepare the timetable.
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Rational Functions Performance Task: Circuit Design
7.37.5 What Did You Learn?
Core Vocabulary
Simplifying Rational Expressions, p. 376Multiplying Rational Expressions, p. 377Dividing Rational Expressions, p. 378
Section 7.4Adding or Subtracting with Like Denominators, p. 384Adding or Subtracting with Unlike Denominators, p. 384Simplifying Complex Fractions, p. 387
Section 7.5Solving Rational Equations by Cross Multiplying, p. 392Solving Rational Equations by Using the Least Common Denominator, p. 393Using Inverses of Functions, p. 395
Mathematical PracticesQuestion 1.In Exercise 37 on page 381, what type of equation did you expect to get as your solution? Explain why this type of equation is appropriate in the context of this situation.Answer:
Question 2.Write a simpler problem that is similar to Exercise 44 on page 382. Describe how to use the simpler problem to gain insight into the solution of the more complicated problem in Exercise 44.Answer:
Question 3.In Exercise 57 on page 390, what conjecture did you make about the value the given expressions were approaching? What logical progression led you to determine whether your conjecture was correct?Answer:
Question 4.Compare the methods for solving Exercise 45 on page 397. Be sure to discuss the similarities and differences between the methods as precisely as possible.Answer:
Performance Task: Circuit Design
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Rational Functions 7172 Quiz
Tell whether x and y show direct variation, inverse variation, or neither. Explain your reasoning.Question 1.
Answer:
Question 7.The variables x and y vary inversely, and y= 10 when x= 5. Write an equation that relates x and y. Then find y when x = 2.Answer:
Match the equation with the correct graph. Explain your reasoning.Question 8.f = \ + 2Answer:
y = \ 2Answer:
h = \Answer:
Question 11.Rewrite g = \ in the form g = \. Graph the function. Describe the graph of g as a transformation of the graph of f = \.Answer:
Question 12.The time t required to empty a tank varies inversely with the pumping rate r . The rate of a certain pump is 70 gallons per minute. It takes the pump 20 minutes to empty the tank. Complete the table for the times it takes the pump to empty a tank for the given pumping rates.Answer:
Question 13.A pitcher throws 16 strikes in the first 38 pitches. The table shows how a pitchers strike percentage changes when the pitcher throws x consecutive strikes after the first 38 pitches. Write a rational function for the strike percentage in terms of x. Graph the function. How many consecutive strikes must the pitcher throw to reach a strike percentage of 0.60?Answer:
EXPLORATION 1
Multiplying and Dividing Rational ExpressionsWork with a partner. Find the product or quotient of the two rational expressions. Then match the product or quotient with its excluded values. Explain your reasoning.
EXPLORATION 2
Communicate Your Answer
Monitoring Progress
\Answer:
Rational Functions Chapter Review
7.1 Inverse Variation
Tell whether x and y show direct variation, inverse variation, or neither.Question 1.
y = \ + 2Answer:
f = \Answer:
7.3 Multiplying and Dividing Rational Expressions
Find the product or quotient.Question 14.\Answer:
\Answer:
Question 16.\Answer:
\ ÷ Answer:
7.4 Adding and Subtracting Rational Expressions
Find the sum or difference.Question 18.\Answer:
\Answer:
\Answer:
Rewrite the function in the form g = \ h + k. Graph the function. Describe the graph of gas a transformation of the graph of f = \.Question 21.g = \Answer:
g = \Answer:
g = \Answer:
Question 24.Let f be the focal length of a thin camera lens, p be the distance between the lens and an object being photographed, and q be the distance between the lens and the film. For the photograph to be in focus, the variables should satisfy the lens equation to the right. Simplify the complex fraction.Answer:
7.5 Solving Rational Equations
Solve the equation. Check your solution.Question 25.\Answer:
\Answer:
\ 2 = \Answer:
Determine whether the inverse of f is a function. Then find the inverse.Question 28.f = \Answer:
Perform the indicated operation.Question 7.\Answer:
\Answer:
\Answer:
\Answer:
Question 11.Let g = \. Simplify g. Determine whether the graph of f =x 2 and the graph of g are different. Explain your reasoning.Answer:
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Big Ideas Alg 2 Exam Review And Key For Chapter 7
Solving Rational Equations 75 Exercises
Vocabulary and Core Concept CheckQuestion 1.When can you solve a rational equation by cross multiplying? Explain.Answer:
Question 2.WRITINGA student solves the equation \ and obtains the solutions 3 and 4. Are either of these extraneous solutions? Explain.Answer
Monitoring Progress and Modeling with Mathematics
In Exercises 310, solve the equation by cross multiplying. Check your solution.Question 3.\Answer:
\Answer:
\Answer:
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Question 11.USING EQUATIONSSo far in your volleyball practice, you have put into play 37 of the 44 serves you have attempted. Solve the equation \ to find the number of consecutive serves you need to put into play in order to raise your serve percentage to 90%.Answer:
Question 12.USING EQUATIONSSo far this baseball season, you have 12 hits out of 60 times at-bat. Solve the equation 0.360 = \ to find the number of consecutive hits you need to raise your batting average to 0.360.Answer:
Question 13.MODELING WITH MATHEMATICSBrass is an alloy composed of 55% copper and 45% zinc by weight. You have 25 ounces of copper. How many ounces of zinc do you need to make brass?Answer:
Question 14.MODELING WITH MATHEMATICSYou have 0.2 liter of an acid solution whose acid concentration is 16 moles per liter. You want to dilute the solution with water so that its acid concentration is only 12 moles per liter. Use the given model to determine how many liters of water you should add to the solution.Answer:
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