Relations and Functions PRE-ALGEBRA LESSON 8 -1 Determine if and where Will and Pedro will meet if they start driving from the same intersection. Pedro travels 2 blocks east, 2 blocks south, 2 blocks east, and 2 blocks north. Will starts traveling 2 blocks north, 4 blocks east, and 2 blocks south. Yes, they will meet 4 blocks east. 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 (For help, go to Lesson 1 -10. ) Graph each point. 1. A(3, 4) 2. B(– 3, 1) 3. F(2, 0) 4. D(2, – 2) 5. C(– 4, – 3) 6. E(0, – 4) Check Skills You’ll Need 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 Solutions 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 Is each relation a function? Explain. a. {(0, 5), (1, 6), (2, 4), (3, 7)} List the domain values and the range values in order. Draw arrows from the domain values to their range values. There is one range value for each domain value. This relation is a function. 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 (continued) b. {(0, 5), (1, 5), (2, 6), (3, 7)} There is one range value for each domain value. This relation is a function. 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 (continued) c. {(0, 5), (0, 6), (1, 6), (2, 7)} There are two range values for the domain value 0. This relation is not a function. Quick Check 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 Is the time needed to mow a lawn a function of the size of the lawn? Explain. No; two lawns of the same size (domain value) can require different lengths of time (range values) for mowing. Quick Check 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 a. Graph the relation shown in the table. Domain Value – 3 – 5 3 3 5 Graph the ordered pairs (– 3, 5), (– 5, 3), (3, 5), and (5, 3). Range Value 5 5 3 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 (continued) b. Use the vertical-line test. Is the relation a function? Explain. Pass a pencil across the graph as shown. Keep the pencil vertical (parallel to the y-axis) to represent a vertical line. The pencil does not pass through two points at any one of its positions, so the relation is a function. Quick Check 8 -1
Relations and Functions PRE-ALGEBRA LESSON 8 -1 Is each relation a function? Explain. 1. {(– 2, – 1), (4, 2), (– 8, – 4), (6, 3)} Yes; there is only one range value for each domain value. 2. {(5, 0), (7, 2), (9, 4), (5, 1)} No; the domain value 5 has two range values, 0 and 1. 3. Graph the relation in the table. x y Is the relation a function? Explain. – 1 0 7 7 Yes; there is one range value for each domain value. Check students’ graphs. 1 2 7 7 8 -1
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Identify the data needed: At a fundraiser, $5 was collected from each parent and $1 was collected from each child. How much money was raised? the number of parents and children at the fundraiser 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 (For help, go to Lesson 1 -3. ) Evaluate each expression for x = 2. 1. 2 + x 2. x – 12 3. 8 x – 13 4. 24 ÷ 2 x Check Skills You’ll Need 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Solutions 1. 2 + x = 2 + 2 = 4 2. x – 12 = 2 – 12 = – 10 3. 8 x – 13 = 8(2) – 13 = 16 – 13 = 3 4. 24 ÷ 2 x = 24 ÷ 2(2) = 24 ÷ 4 = 6 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Find the solution of y = 4 x – 3 for x = 2. y = 4 x – 3 y = 4(2) – 3 Replace x with 2. y = 8 – 3 y = 5 Multiply. Subtract. A solution of the equation is (2, 5). Quick Check 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 The equation a = 5 + 3 p gives the price for admission to a park. In the equation, a is the admission price for one car with p people in it. Find the price of admission for a car with 4 people in it. a = 5 + 3 p a = 5 + 3(4) a = 5 + 12 a = 17 Replace p with 4. Multiply. Add. A solution of the equation is (4, 17). The admission price for one car with 4 people in it is $17. Quick Check 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Graph y = 4 x – 2. Make a table of values to show ordered-pair solutions. x 4 x – 2 (x, y) – 2 0 4(– 2) – 2 = – 8 – 2 = – 10 4(0) – 2 = 0 – 2 = – 2 (– 2, – 10) (0, – 2) 2 4(2) – 2 = 8 – 2 = 6 (2, 6) Graph the ordered pairs. Draw a line through the points. Quick Check 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Graph each equation. Is the equation a function? a. y = – 3 b. x = 4 For every value of x, y = – 3. For every value of y, x = 4. This is a horizontal line. The equation y = – 3 is a function. This is a vertical line. The equation y = 4 is not a function. 8 -2 Quick Check
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 1 2 Solve y – x = 3 for y. Then graph the equation. Solve the equation for y. 1 2 y – x = 3 1 1 1 y – x + x = 3 + x 2 2 2 1 y = x + 3 2 1 Add x to each side. 2 Simplify. 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 (continued) Make a table of values. 1 2 1 – 2 2 1 0 2 1 2 2 x Graph. x + 3 (x, y) (– 2) + 3 = – 1 + 3 = 2 (– 2, 2) (0) + 3 = 0 + 3 = 3 (0, 3) (2) + 3 = 1 + 3 = 4 (2, 4) Quick Check 8 -2
Equations With Two Variables PRE-ALGEBRA LESSON 8 -2 Find the solution for each equation for x = 2. 1. y = – 2 x + 5 2. y = 7 x (2, 1) (2, 14) 3. y = 3 x – 9 (2, – 3) Solve each equation for y. Then graph each equation. 4. y – 2 x = 3 5. 2 x + 2 y = 8 y = 2 x + 3 y = –x + 4 8 -2
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Floyd has $35 in the bank. He writes a check for $52 and makes a deposit of $10. What is his new balance? – $7 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 (For help, go to Lesson 1 -6. ) Find each difference. 1. – 4 – 5 2. 3 – (– 2) 3. 6 – 9 4. – 1 – (– 1) Check Skills You’ll Need 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Solutions 1. – 4 – 5 = – 4 + (– 5) = – 9 2. 3 – (– 2) = 3 + 2 = 5 3. 6 – 9 = 6 + (– 9) = – 3 4. – 1 – (– 1) = – 1 + 1 = 0 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Find the slope of each line. a. b. rise 4 rise slope = = = 4 run 1 – 6 slope = = = – 2 run 3 Quick Check 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Find the slope of the line through E(7, 5) and F(– 2, 0). difference in y-coordinates 0 – 5 5 slope = difference in x-coordinates – 2 – 7 – 9 = = 9 Quick Check 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Find the slope of each line. a. b. – 3 – (– 3) 0 – 1 – 3 – 4 slope = 4 – (– 2) = = 0 6 slope = – 2 – (– 2) = 0 Slope is 0 for a horizontal line. Division by zero is undefined. Slope is undefined for a vertical line. Quick Check 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 A ramp slopes from a warehouse door down to a 1 street. The function y = – x + 4 models the ramp, where x is 5 the distance in feet from the bottom of the door and y is the height in feet above the street. Graph the equation. Step 1 Since the y-intercept is 4, graph (0, 4). Step 2 Since the slope is – , move 5 1 1 unit down from (0, 4). Then move 5 units right to graph a second point. Step 3 Draw a line through the points. Quick Check 8 -3
Slope and y-intercept PRE-ALGEBRA LESSON 8 -3 Find the slope of the line through each pair of points. 1. A(2, 4), B(– 2, – 4) 2. F(– 5, 1), G(0, – 9) 2 – 2 4 3. Identify the slope and y-intercept of y = – x + 3. 3 Then graph the line. 4 3 – ; 3 8 -3
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 Find the slope of each line. a. 10 x + 5 y = 28 – 2 b. 3 x +3 y = 5 – 1 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 (For help, go to Lesson 8 -3. ) Find the slope of the line through each pair of points. 1. A(3, 1), B(2, 1) 2. S(3, 4), T(1, 2) 3. P(0, – 2), Q(0, 2) 4. C(– 5, 2), D(4, – 1) Check Skills You’ll Need 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 Solutions 1. 2. 3. 4. 0 1 – 1 2 – 3 – 2 2 – 4 slope = = = 1 – 2 1 – 3 4 2 – (– 2) slope = = ; slope is undefined. 0 0 – 3 – 1 – 2 1 slope = = – 3 9 4 – (– 5) slope = = = 0 – 1 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 A long-distance phone company charges its customers a monthly fee of $4. 95 plus 9¢ for each minute of a long-distance call. a. Write a function rule that relates the total monthly bill to the number of minutes a customer spent on long-distance calls. Words total bill is $4. 95 plus 9¢ times number of minutes Let = the number of minutes. m Let t( m ) = total bill, a function of the number of minutes. Rule t( m ) = 4. 95 + 0. 09 A rule for the function is t(m) = 4. 95 + 0. 09 m. 8 -4 • m
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 (continued) b. Find the total monthly bill if the customer made 90 minutes of long-distance calls. t(m) = 4. 95 + 0. 09 m t(90) = 4. 95 + 0. 09(90) Replace m with 90. t(90) = 4. 95 + 8. 10 Multiply. t(90) = 13. 05 Add. The total monthly bill with 90 minutes of long-distance calls is $13. 05. Quick Check 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 Write a rule for the linear function in the table below. x – 2 – 2 f(x) 2 0 3 – 5 – 2 – 13 – 4 – 21 As the x values decrease by 2, the f(x) values decrease by 8. – 8 – 8 So m = = 4. – 2 When x = 0, f(x) = – 5. So b = – 5. A rule for the function is f(x) = 4 x – 5. Quick Check 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 Write a rule for the linear function graphed below. – 2 – 4 slope = 0 – 2 = 2 y-intercept = – 2 A rule for the function is f(x) = 2 x – 2. Quick Check 8 -4
Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8 -4 Write a rule for each function. 1. Sarena earns a salary of $150 a week plus a 10% commission on each sale. f(x) = 0. 10 x + 150 2. x f(x) 0 1 2 3. 2 f(x) = x f(x) = 5 x – 3 8 -4
Scatter Plots PRE-ALGEBRA LESSON 8 -5 Use a graph with 100 squares to draw and shade squares that spell a 3 -letter word. Estimate what fraction of the grid is unshaded. Check students’ answers. 8 -5
Scatter Plots PRE-ALGEBRA LESSON 8 -5 (For help, go to Lesson 1 -10. ) Write the coordinates of each point. 1. A 2. B 3. C 4. D Check Skills You’ll Need 8 -5
Scatter Plots PRE-ALGEBRA LESSON 8 -5 Solutions 1. A(– 2, 2) 3. C(– 3, 0) 2. B(0, 3) 4. D(2, 3) 8 -5
Scatter Plots PRE-ALGEBRA LESSON 8 -5 The scatter plot shows education and income data. a. Describe the person represented by the point with coordinates (10, 30). This person has 10 years of education and earns $30, 000 each year. b. How many people have exactly 14 years of education? What are their incomes? The points (14, 50), (14, 80), and (14, 90) have education coordinate 14. The three people they represent earn $50, 000, $80, 000, and $90, 000, respectively. Quick Check 8 -5
Scatter Plots PRE-ALGEBRA LESSON 8 -5 Use the table to make a scatter plot of the elevation and precipitation data. City Atlanta, GA Boston, MA Chicago, IL Honolulu, HI Miami, FL Phoenix, AZ Portland, ME San Diego, CA Wichita, KS Elevation Above Mean Annual Sea Level (ft) Precipitation (in. ) 1, 050 51 20 42 596 36 18 22 11 1, 072 75 40 1, 305 56 8 44 10 29 8 -5 Quick Check
Scatter Plots PRE-ALGEBRA LESSON 8 -5 Use the scatter plot below. Is there a positive correlation, a negative correlation, or no correlation between temperatures and amounts of precipitation? Explain. The values show no relationship. There is no correlation. Quick Check 8 -5
Scatter Plots PRE-ALGEBRA LESSON 8 -5 Answer the following questions based on the graph. 1. What do you know about the student at point A? read 2 books per month, grade: 80 2. How many students read 3 books per month? 3 3. Is there a positive correlation, a negative correlation, or no correlation between books read and semester grades? positive correlation 8 -5
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 Draw 6 nonlinear points. How many different segments can you draw connecting two points? 15 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 (For help, go to Lesson 8 -4. ) Write a rule for each linear function. 1. 2. Check Skills You’ll Need 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 Solutions 6 – 2 2 – 0 4 2 1. slope = = = 2 y-intercept = 2 A rule for the function is f(x) = 2 x + 2. – 2 – (– 3) – 4 – 0 1 – 4 2. slope = = – 1 4 y-intercept = – 3 1 A rule for the function is f(x) = – x – 3. 4 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 Use the data in the table below. Suppose this year there are 16 wolves on the island. Predict how many moose are on the island. Isle Royale Populations Year Wolf 1982 14 1983 1984 Moose Year Wolf Moose 700 1988 12 1, 653 1994 15 1, 800 23 24 900 811 1989 1990 11 15 1, 397 1, 216 1995 1996 16 22 2, 400 1, 200 1985 22 1, 062 1991 12 1, 313 1997 24 500 1986 1987 20 16 1, 025 1, 380 1992 1993 12 13 1, 600 1, 880 1998 1999 14 25 700 750 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 (continued) Step 1 Make a scatter plot by graphing the (wolf, moose) ordered pairs. Use the x-axis for wolves and the y-axis for moose. Step 2 Sketch a trend line. The line should be as close as possible to each data point. There should be about as many points above the trend line as below it. 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 (continued) Step 3 To predict the number of moose when there are 16 wolves, find 16 along the horizontal axis. Look up to find the point on the trend line that corresponds to 16 wolves. Then look across to the value on the vertical axis, which is about 1, 300. There about 1, 300 moose on the island. Quick Check 8 -6
Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8 -6 Answer each question. 1. When is there no trend line for a scatter plot? when there is no correlation 2. What type of correlation will the data have for a scatter plot of ages and heights of all students in your school? positive 3. Why is it important to say “about” when making a prediction? The prediction is based on a trend line that approximates the locations of the related points. 8 -6
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Find the slope and y-intercept for the equation 1. 3 x – y + 5 = 0. Slope is 1. 3; y-intercept is 5. 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 (For help, go to Lesson 8 -2. ) Graph each equation. 1. y = –x – 4 2. y = 2 x – 1 3. – 4 x = 6 y 4. 3 x – 2 y = 5 Check Skills You’ll Need 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Solutions 1. y = –x + 4 2. y = 2 x – 1 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Solutions (continued) 3. – 4 x = 6 y 4 – x = y 6 2 y = – x 4. 3 x – 2 y = 5 3 x – 2 y – 3 x = 5 – 3 x – 2 y = – 3 x + 5 3 5 y = x – 3 2 8 -7 2
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Solve the system y = x – 7 and y = 4 x + 2 by graphing. Step 1 Graph each line. Step 2 Find the point of intersection. The lines intersect at one point, (– 3, – 10). The solution is (– 3, – 10). Check See whether (– 3, – 10) makes both equations true. y = x – 7 – 10 – 3 – 7 – 10 = – 10 Replace x with – 3 and y with – 10. The solution checks. y = 4 x + 2 – 10 4(– 3) + 2 – 10 = – 10 Quick Check 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Quick Check Solve each system of equations by graphing. a. 27 x + 9 y = 36; y = 4 – 3 x b. 8 = 4 x + 2 y; 2 x + y = 5 The lines are the same line. There are infinitely many solutions. The lines are parallel. They do not intersect. There is no solution. 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Find two numbers with a sum of 10 and a difference of 2. Step 1 Write equations. Let x = the greater number. Let y = the lesser number. Equation 1 Sum x + y is = 10. 10 Equation 2 Difference is x – y = Step 2 Graph the equations. The lines intersect at (6, 4). The numbers are 6 and 4. 8 -7 2. 2
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 (continued) Check Since the sum of 6 and 4 is 10 and the difference of 6 and 4 is 2, the answer is correct. Quick Check 8 -7
Solving Systems of Linear Equations PRE-ALGEBRA LESSON 8 -7 Solve each system by graphing. 1. y = –x – 4 and y = 4 x + 1 (– 1, – 3) 2. y = – 3 x + 12 and 18 x + 6 y = 42 no solution 3. Find two numbers with a sum of 15 and a difference of 1. Show your work. (8, 7) 8 -7
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 Use two of the given digits to make the equation true. 2, 3, 4 ? ? 8 ÷ = 10 2 3 3 4 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (For help, go to Lesson 2 -9. ) Is the given value of x a solution of the inequality? Explain. < 1. x + 3 – 2; x = – 5 2. 5 – x < 4; x = 1 – < 3. – 2 x 6; x = – 4 4. 4 x + 1 > – 7; x = – 2 – Check Skills You’ll Need 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 Solutions > 1. yes; x + 3 – 2; x = – 5 – > – 5 + 3 – 2 – > – 2 true – < 3. yes; – 2 x 6; x = – 4 – < – 2(– 4) 6 – < – 2 + 8 6 – < 6 6 true – 2. no; 5 – x < 4; x = 1 5 – 1 < 4 4 < 4 false 4. no; 4 x + 1 > – 7; x = – 2 4(– 2) + 1 > – 7 – 8 + 1 > – 7 > – 7 false 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 Graph each inequality on a coordinate plane. a. y > 2 x + 1 Step 1 Graph the boundary line. Points on the boundary line do not make y > 2 x + 1 true. Use a dashed line. 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) Step 2 Test a point not on the boundary line. Test (0, 0) in the inequality. y > 2 x + 1 ? 0 2(0) + 1 Substitute. > ? > 0 0 + 1 0 > 1 false Since the inequality is false for (0, 0), shade the region that does not contain (0, 0). 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) b. y 3 x – 2 < – Step 1 Graph the boundary line. Points on the boundary line make y 3 x – 2 true. Use a solid line. 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) Step 2 Test a point not on the boundary line. Test (3, 0) in the inequality. < y 3 x – 2 – ? < 0 3(3) – 2 Substitute. – ? 0 9 – 2 < – 0 7 < true – Since the inequality is true for (3, 0), shade the region containing (3, 0). Quick Check 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 Cashews cost $2/lb. Pecans cost $4/lb. You plan to spend no more than $20. How many pounds of each can you buy? Step 1 Write an inequality. Words cost of cashews plus cost of pecans is at most twenty dollars Let = number of pounds of cashews. y Let = number of pounds of pecans. x Inequality 2 y + 8 -8 4 x < – 20
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) Step 2 Write the equation of the boundary line in slopeintercept form. < 2 y + 4 x 20 – < y – 2 x + 10 – y = – 2 x + 10 Step 3 Graph y = – 2 x + 10 in Quadrant I since weight is not negative. 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) Step 4 Test (1, 1). < y – 2 x + 10 – ? < 1 – 2(1) + 10 – < 1 8 – The inequality is true. (1, 1) is a solution. Step 5 Shade the region containing (1, 1). The graph shows the possible solutions. For example, you could buy 1 pound of pecans and 5 pounds of cashews. Quick Check 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 > Solve the system y x + 1 and y < 2 x + 3 by graphing. – Step 1 > Graph y x + 1 on a coordinate – plane. Shade in red. Step 2 Graph y < 2 x + 3 on the same coordinate plane. Shade in blue. 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 (continued) The solutions are the coordinates of all the points in the region that is shaded in both colors. Check See whether the solution (2, 5) makes both of the inequalities true. > y x + 1 – ? 5 2 + 1 Replace x with 2 and y with 5. > – > 5 3 y The solution checks. – y < 2 x + 3 ? 5 2(2) + 3 Replace x with 2 and y with 5. < 5 < 7 y The solution checks. Quick Check 8 -8
Graphing Linear Inequalities PRE-ALGEBRA LESSON 8 -8 Tell whether the ordered pair is a solution of the inequality. Justify your response. 1. y < 4 x; (3, 0) 2. y > –x – 1; (2, – 3) Yes; 0 < 12. No; – 3 > – 3. < 4. Solve the system y 3 x + 1 and – y > – 2 x + 2 by graphing. 3. y 4 x – 2; (– 1, 1) < – < No; 1 6. – 8 -8
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