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Contents Lesson 11-1Sequences Lesson 11-2Functions Lesson 11-3Graphing Linear Functions Lesson 11-4The Slope Formula Lesson 11-5Slope-Intercept Form Lesson 11-6Scatter Plots Lesson 11-7Graphing Systems of Equations Lesson 11-8Graphing Linear Inequalities

Lesson 1 Contents Example 1Identify Arithmetic Sequences Example 2Identify Geometric Sequences Example 3Identify Geometric Sequences

Example 1-1a State whether the sequence 23, 15, 7, –1, –9 … is arithmetic. If it is, state the common difference. Write the next three terms of the sequence. 23, 15, 7, –1, –9 –8 –8 –8 –8 Answer: The terms have a common difference of –8, so the sequence is arithmetic. –9, –8 Answer: The next three terms are –17, –25, and –33. Continue the pattern to find the next three terms. –17,–25,–33 –8

Example 1-1b State whether the sequence 29, 27, 25, 23, 21, … is arithmetic. If it is, state the common difference. Write the next three terms of the sequence. Answer: arithmetic; –2; 19, 17, 15

Example 1-2a State whether the sequence –1, 3, –9, 27, –81,… is geometric. If it is, state the common ratio. Write the next three terms of the sequence. Answer: The terms have a common ratio of –3, so the sequence is geometric. Continue the pattern to find the next three terms. –1, 3, –9, 27, –81

Example 1-2b Answer: The next three terms are 243, –729, and 2,187. –81,243, –729,2,187

Example 1-2c State whether the sequence 1, –2, 4, –8, 16,… is geometric. If it is, state the common ratio. Write the next three terms of the sequence. Answer: geometric; –2; –32, 64, –128

Example 1-3a State whether the sequence 6, 12, 13, 26, 27, … is geometric. If it is, state the common ratio. Write the next three terms of the sequence. 6, 12, 13, 26, 27 Answer: Since there is no common ratio, the sequence is not geometric. However, the sequence does have a pattern. Multiply the last term by 2, and then add 1 to get the next term. Answer: The next three terms are 54, 55, and , 54, 55,110

Example 1-3c State whether the sequence 1, 3, 7, 15, 31, … is geometric. If it is, state the common ratio. Write the next three terms of the sequence. Answer: not geometric; 63, 127, 255

End of Lesson 1

Lesson 2 Contents Example 1Find a Function Value Example 2Find a Function Value Example 3Make a Function Table Example 4Functions with Two Variables Example 5Functions with Two Variables

Example 2-1a Substitute 4 for x into the function rule. Answer: Find the function value of

Example 2-1b Answer: –5 Find the function value of

Example 2-2a Answer: Find the function value of Substitute –6 for x into the function rule. Simplify.

Example 2-2b Answer: 2 Find the function value of

Example 2-3a Complete the function table for Input x Rule 4x – 1 Output f(x) –3 –2 –1 0 1 Substitute each value of x, or input, into the function rule. Then simplify to find the output.

Example 2-3b Answer: 1 0 Output f(x) Rule 4x – 1 Input x

Example 2-3c Complete the function table for Input x Rule 3x – 2 Output f(x) –3 –2 –1 0 1 Answer:

Example 2-4a PARKING FEES The price for parking at a city lot is $3.00 plus $2.00 per hour. Write a function using two variables to represent the price of parking for h hours. Words Cost of parking equals $3.00 plus $2.00 per hour. Function Answer: The function represents the situation.

Example 2-4b TAXI The price for a taxi ride is $5.00 plus $4.00 per hour. Write a function using two variables to represent the price of riding a taxi for h hours. Answer:

Example 2-5a PARKING FEES The price for parking at a city lot is $3.00 plus $2.00 per hour. How much would it cost to park at the lot for 2 hours? Substitute 2 for h into the function rule. Answer: It will cost $7.00 to park for 2 hours.

Example 2-5b TAXI The price for a taxi ride is $5.00 plus $1.00 per hour. How much would it cost for a 3 hour taxi ride? Answer: $17.00

End of Lesson 2

Lesson 3 Contents Example 1Graph a Function Example 2Use x- and y-intercepts

Example 3-1a (x, y)yx – 3x Step 1 Choose some values for x. Make a function table. Include a column of ordered pairs of the form (x, y). 0 – 3 –3 (0, –3) 1 – 3 –2 (1, –2) 2 – 3 –1(2, –1) 3 – 3 0 (3, 0) Graph

Example 3-1b Step 2 Graph each ordered pair. (0, –3) y = x – 3 Answer: Draw a line that passes through each point. Note that the ordered pair for any point on this line is a solution of The line is the complete graph of the function. (1, –2) (2, –1) (3, 0)

Example 3-1b Check It appears from the graph that (–1, –4) is also a solution. Check this by substitution. Write the function. Simplify.

Example 3-1c Answer: Graph

Example 3-2a MULTIPLE-CHOICE TEST ITEM Which graph represents DC B A

Example 3-2b Solve the Test Item The graph will cross the x-axis when Replace y with 0. Subtract 1. Simplify. Divide by 2. Simplify. Read the Test Item You need to decide which of the four graphs represents

Example 3-2c The graph will cross the y-axis when Replace x with 0. Simplify. The x-intercept is and the y-intercept is1. Graph D is the only graph with both of these intercepts. Answer: D

Example 3-2e MULTIPLE-CHOICE TEST ITEM Which graph represents Answer: C

End of Lesson 3

Lesson 4 Contents Example 1Positive Slope Example 2Negative Slope Example 3Zero Slope Example 4Undefined Slope

Example 4-1a Find the slope of the line that passes through A(3, 3) and B(2, 0). Simplify. Definition of slope

Example 4-1b Check When going from left to right, the graph of the line slants upward. This is consistent with a positive slope. Answer: The slope is 3.

Example 4-1c Find the slope of the line that passes through A(4, 3) and B(1, 0). Answer: 1

Example 4-2a Find the slope of the line that passes through X(–2, 3) and Y(3, 0). Definition of slope Simplify.

Example 4-2b Check When going from left to right, the graph of the line slants downward. This is consistent with a negative slope. Answer: The slope is.

Example 4-2c Find the slope of the line that passes through X(–3, 3) and Y(1, 0). Answer:

Example 4-3a Find the slope of the line that passes through P(6, 5) and Q(2, 5). Definition of slope Simplify.

Example 4-3b Answer: The slope is 0. The slope of any horizontal line is 0.

Example 4-3c Find the slope of the line that passes through P(1, 6) and Q(2, 6). Answer: 0

Example 4-4a Find the slope of the line that passes through G(2, 4) and H(2, 6). Definition of slope Simplify.

Example 4-4b Answer: Division by 0 is not defined. So, the slope is undefined. The slope of any vertical line is undefined.

Example 4-4c Find the slope of the line that passes through G(–2, 1) and H(–2, 0). Answer: undefined

End of Lesson 4

Lesson 5 Contents Example 1Find Slopes and y-intercepts of Graphs Example 2Find Slopes and y-intercepts of Graphs Example 3Graph an Equation Example 4Graph an Equation to Solve Problems Example 5Graph an Equation to Solve Problems Example 6Graph an Equation to Solve Problems

Example 5-1a State the slope and the y-intercept of the graph of the equation of Answer: The slope of the graph is and the y-intercept is –5. Write the equation in the form

Example 5-1b Answer: State the slope and the y-intercept of the graph of the equation of

Example 5-2a Answer: The slope of the graph is –2 and the y-intercept is 8. Write the original equation. State the slope and the y-intercept of the graph of the equation of Subtract 2x from each side. Simplify. Write the equation in the form

Example 5-2b Answer: slope = –3; y-intercept = 5 State the slope and the y-intercept of the graph of the equation of

Example 5-3a Graph using the slope and y-intercept. Step 1 Find the slope and y-intercept. Step 2 Graph the y-intercept (0, 2). (0, 2) y-intercept

Example 5-3b Step 4 Draw a line through the two points. Answer: right 3 up 2 change in y: up 2 units change in x: right 3 units Step 3 Use the slope to locate a second point on the line. (0, 2)

Example 5-3c Graph using the slope and y-intercept. Answer:

Example 5-4a MOVIE RENTAL A movie rental store charges $4 to rent a movie. If a movie is returned late, the charge is $3 extra per day. The total cost is given by the equation where x is the number of days the movie is late. Graph the equation.

Example 5-4b Answer: First find the slope and the y-intercept. Plot the point (0, 4). Then locate another point up 3 and right 1. y-intercept Draw the line.

Example 5-4c Answer: GAME RENTAL A game rental store charges $5 to rent a movie. If a movie is returned late, the charge is $5 extra per day. The total cost is given by the equation where x is the number of days the game is late. Graph the equation.

Example 5-5a Answer: The total cost is $16. MOVIE RENTAL A movie rental store charges $4 to rent a movie. If a movie is returned late, the charge is $3 extra per day. The total cost is given by the equation where x is the number of days the movie is late. Use the graph to find the cost for returning the movie 4 days late. Locate 4 on the x-axis. Find the y-coordinate on the graph where the x-coordinate is 4.

Example 5-5b Answer: $10 GAME RENTAL A game rental store charges $5 to rent a movie. If a movie is returned late, the charge is $5 extra per day. The total cost is given by the equation where x is the number of days the game is late. Use the graph to find the cost for returning the game 1 day late.

Example 5-6a Answer: The slope 3 represents the rate of change in price each day a movie is late. The y-intercept 4 is the minimum charge for renting a movie. MOVIE RENTAL A movie rental store charges $4 to rent a movie. If a movie is returned late, the charge is $3 extra per day. The total cost is given by the equation where x is the number of days the movie is late. Describe what the slope and y-intercept of the graph represent.

Example 5-6b Answer: The slope 5 represents the rate of change in price each day a game is late. The y-intercept 5 is the minimum charge for renting a game. GAME RENTAL A game rental store charges $5 to rent a movie. If a movie is returned late, the charge is $5 extra per day. The total cost is given by the equation where x is the number of days the game is late. Describe what the slope and y-intercept of the graph represent.

End of Lesson 5

Lesson 6 Contents Example 1Identify a Relationship Example 2Identify a Relationship Example 3Draw a Best-Fit Line Example 4Draw a Best-Fit Line Example 5Draw a Best-Fit Line

Example 6-1a Answer: negative Determine whether a scatter plot of the data for the number of cups of hot chocolate sold at a concession stand and the outside temperature might show a positive, negative, or no relationship. As the temperature decreases, the number of cups of hot chocolate sold increases. Therefore, the scatter plot might show a negative relationship.

Example 6-1b Answer: positive Determine whether a scatter plot of the data for the number of cups of lemonade sold at a concession stand and the outside temperature might show a positive, negative, or no relationship.

Example 6-2a Answer: Therefore, the scatter plot shows no relationship. Determine whether a scatter plot of the data for the birthday and number of sports played might show a positive, negative, or no relationship. The number of sports played does not depend on your birthday.

Example 6-2b Answer: no relationship Determine whether a scatter plot of the data for your age and the color of your hair might show a positive, negative, or no relationship.

Example 6-3a ZOOS The table to the right shows the average and maximum longevity of various animals in captivity. Make a scatter plot using the data. Then draw a line that seems to best represent the data MaximumAverage Longevity (Years) Source: Walker’s Mammals of the World

Example 6-3b Answer: Graph each of the data points. Draw a line that best fits the data.

Example 6-3c PRODUCTION The table to the right shows the average hourly earnings of U.S. production workers since Make a scatter plot using the data. Source: The World Almanac $ $ $ $ $ $ $ Average Hourly Earnings Years Since 1995 U.S. Production Workers Earnings

Example 6-3c Answer:

Example 6-4a ZOOS The table to the right shows the average and maximum longevity of various animals in captivity. Write an equation for the best-fit line MaximumAverage Longevity (Years) Source: Walker’s Mammals of the World The line passes through the points at and. Use these points to find the slope of the line.

Example 6-4b Definition of slope Simplify. The slope is and the y-intercept is 17.5.

Example 6-4b Use the slope and the y-intercept to write the equation. Answer: The equation for the best-fit line is Slope-intercept form

Example 6-4c Answer: PRODUCTION The table to the right shows the average hourly earnings of U.S. production workers since Write an equation for the best- fit line using points (0, 11.43) and (5, 13.76). Source: The World Almanac $ $ $ $ $ $ $ Average Hourly Earnings Years Since 1995 U.S. Production Workers Earnings

Example 6-5a MaximumAverage Longevity (Years) Source: Walker’s Mammals of the World ZOOS The table to the right shows the average and maximum longevity of various animals in captivity. Use the equation to predict the maximum longevity for an animal with an average longevity of 33 years.

Example 6-5a Answer: The maximum longevity is about 67 years. Equation for the best-fit line

Example 6-5b Answer: $15.62 PRODUCTION The table to the right shows the average hourly earnings of U.S. production workers since Use the equation to predict the average hourly earnings of U.S. production workers in Source: The World Almanac $ $ $ $ $ $ $ Average Hourly Earnings Years Since 1995 U.S. Production Workers Earnings

End of Lesson 6

Lesson 7 Contents Example 1One Solution Example 2Infinitely Many Solutions Example 3No Solution Example 4Solve by Substitution

Example 7-1a The graphs of the equations appear to intersect at (1, 0). Solve the system and by graphing.

Example 7-1b Answer: The solution of the system is (1, 0). Check this estimate. Check

Example 7-1c Answer: (3, 6) Solve the system and by graphing.

Example 7-2a Write the equation. Add to each side. Both equations are the same. Write in slope-intercept form. Solve the system and by graphing.

Example 7-2b Answer: The solution of the system is all the coordinates of points on the graph of

Example 7-2c Answer: The solution of the system is all the coordinates of the points on the graph of Solve the system and by graphing.

Example 7-3a PHONE RATES One phone company charges $3 a month plus 10 cents a minute for long-distance calls. Another company charges $5 a month plus 10 cents a minute for long-distance calls. For how many minutes will the total monthly charges of the two companies be the same? Let x equal the number of minutes. Let y equal the total monthly cost. Write an equation to represent each company’s charge for long-distance phone calls. 2nd company 1st company

Example 7-3b Graph the system of equations. The graphs appear to be parallel lines.

Example 7-3b Answer: For any amount of minutes, the first company will charge less than the second company. Since there is no coordinate pair that is a solution of both equations, there is no solution of this system of equations.

Example 7-3c Answer: For any amount of minutes, the first company will charge less than the second company. PHONE RATES One phone company charges $20 a month plus 15 cents a minute for long-distance calls. Another company charges $25 a month plus 15 cents a minute for long-distance calls. For how many minutes will the total monthly charges of the two companies be the same?

Example 7-4a Since y must have the same value in both equations, you can replace y with 5 in the first equation. Write the first equation. Replace y with 5. Add 4 to each side. Simplify. Divide each side by 3. Simplify. Solve the system and by substitution.

Example 7-4b Answer: The solution of this system of equations is (3, 5). You can check the solution by graphing. The graphs appear to intersect at (3, 5), so the solution is correct.

Example 7-4c Answer: (3, 3) Solve the system and by substitution.

End of Lesson 7

Lesson 8 Contents Example 1Graph an Inequality Example 2Graph an Inequality to Solve a Problem

Example 8-1a Step 1 Graph the boundary line Since is used in the inequality, make the boundary line a solid line. Graph

Example 8-1b Replace x with 0 and y with 0. Simplify. Step 2 Test a point not on the boundary line, such as (0, 0). Write the inequality.

Step 3 Since (0, 0) is a solution of shade the region that contains (0, 0). Example 8-1b Answer:

Example 8-1c Answer: Graph

Example 8-2a CRAFTS Sylvia is making items to sell at the school craft sale. Each flower basket takes 15 minutes, and each flower pin takes 1 minute to make. She can spend, at most, 60 minutes making crafts one afternoon. Graph the inequality showing the possible numbers of each item she can make that afternoon. Let x represent the number of flower baskets and y represent the number of flower pins. Write an inequality. Words Inequality Time for flower baskets plus time for flower pins is at most

Example 8-2b The related equation is Write the equation. Subtract 15x from each side. Write in slope-intercept form. Test (0, 0) in the original inequality. Write the inequality. Replace x with 0 and y with 0. Simplify.

Example 8-2c Answer: Since Sylvia cannot make a negative number or a fractional number of flower baskets or flower pins, the answer is any pair of integers represented in the shaded region. For example, she could make 1 basket and 30 pins.

Example 8-2d BAKING Josef is baking items to sell at the school bake sale. Each cake takes 40 minutes, and each pre- made brownie takes 1 minute to unwrap. He can spend, at most, 120 minutes baking one afternoon. Graph the inequality showing the possible numbers of each item he can make that afternoon. Answer:

End of Lesson 8

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