1 / 72

Relations and Functions

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

hastingsb
Download Presentation

Relations and Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. 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

  3. Relations and Functions PRE-ALGEBRA LESSON 8-1 Solutions 8-1

  4. 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

  5. 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

  6. 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

  7. 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

  8. Graph the ordered pairs (–3, 5), (–5, 3), (3, 5), and (5, 3). Relations and Functions PRE-ALGEBRA LESSON 8-1 a. Graph the relation shown in the table. Domain Value –3 –5 3 5 Range Value 5 3 5 3 8-1

  9. Pass a pencil across the graph as shown. Keep the pencil vertical (parallel to the y-axis) to represent a vertical line. Relations and Functions PRE-ALGEBRA LESSON 8-1 (continued) b. Use the vertical-line test. Is the relation a function? Explain. The pencil does not pass through two points at any one of its positions, so the relation is a function. Quick Check 8-1

  10. xy –1 7 0 7 1 7 2 7 Relations and Functions PRE-ALGEBRA LESSON 8-1 Is each relation a function? Explain. 1. {(–2, –1), (4, 2), (–8, –4), (6, 3)} 2. {(5, 0), (7, 2), (9, 4), (5, 1)} 3. Graph the relation in the table. Is the relation a function? Explain. Yes; there is only one range value for each domain value. No; the domain value 5 has two range values, 0 and 1. Yes; there is one range value for each domain value. Check students’ graphs. 8-1

  11. 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

  12. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 (For help, go to Lesson 1-3.) Evaluate each expression for x = 2. 1. 2 + x2.x – 12 3. 8x – 13 4. 24 ÷ 2x Check Skills You’ll Need 8-2

  13. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Solutions 1. 2 + x = 2 + 2 2.x – 12 = 2 – 12 = 4 = –10 3. 8x – 13 = 8(2) – 13 4. 24 ÷ 2x = 24 ÷ 2(2) = 16 – 13 = 24 ÷ 4 = 3 = 6 8-2

  14. y = 4(2) – 3 Replace x with 2. y = 8 – 3 Multiply. y = 5 Subtract. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Find the solution of y = 4x – 3 for x = 2. y = 4x – 3 A solution of the equation is (2, 5). Quick Check 8-2

  15. a = 5 + 3(4) Replace p with 4. a = 5 + 12 Multiply. a = 17 Add. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 The equation a = 5 + 3p 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 + 3p 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

  16. x 4x – 2 (x, y) –2 4(–2) – 2 = –8 – 2 = –10 (–2, –10) 0 4(0) – 2 = 0 – 2 = –2 (0, –2) 2 4(2) – 2 = 8 – 2 = 6 (2, 6) Graph the ordered pairs. Draw a line through the points. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Graph y = 4x – 2. Make a table of values to show ordered-pair solutions. Quick Check 8-2

  17. For every value of x, y = –3. For every value of y, x = 4. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Graph each equation. Is the equation a function? a.y = –3 b.x = 4 This is a horizontal line. This is a vertical line. The equation y = –3 is a function. The equation y = 4 is not a function. Quick Check 8-2

  18. 1 2 1 2 1 2 1 2 Solve the equation for y. y – x = 3 1 2 1 2 1 2 y – x+ x = 3 + x Add x to each side. y = x + 3 Simplify. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Solve y – x = 3 for y. Then graph the equation. 8-2

  19. Make a table of values. Graph. 1 2 xx + 3 (x, y) –2 (–2) + 3 = –1 + 3 = 2 (–2, 2) 0 (0) + 3 = 0 + 3 = 3 (0, 3) 2 (2) + 3 = 1 + 3 = 4 (2, 4) 1 2 1 2 1 2 Equations With Two Variables PRE-ALGEBRA LESSON 8-2 (continued) Quick Check 8-2

  20. Equations With Two Variables PRE-ALGEBRA LESSON 8-2 Find the solution for each equation for x = 2. 1.y = –2x + 5 2.y = 7x3.y = 3x – 9 Solve each equation for y. Then graph each equation. 4.y – 2x = 3 5. 2x + 2y = 8 (2, 1) (2, 14) (2, –3) y = 2x + 3 y = –x + 4 8-2

  21. 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

  22. 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

  23. Slope and y-intercept PRE-ALGEBRA LESSON 8-3 Solutions 1. –4 – 5 = –4 + (–5) 2. 3 – (–2) = 3 + 2 = –9 = 5 3. 6 – 9 = 6 + (–9) 4. –1 – (–1) = –1 + 1 = – 3 = 0 8-3

  24. rise run 4 1 rise run –6 3 slope = = = 4 slope = = = –2 Slope and y-intercept PRE-ALGEBRA LESSON 8-3 Find the slope of each line. a. b. Quick Check 8-3

  25. difference in y-coordinates difference in x-coordinates 0 – 5 –2 – 7 –5 –9 5 9 slope = = = = Slope and y-intercept PRE-ALGEBRA LESSON 8-3 Find the slope of the line through E(7, 5) and F(–2, 0). Quick Check 8-3

  26. –3 – (–3) 4 – (–2) 0 6 –1 – 3 –2 – (–2) –4 0 slope = = = 0 slope = = Slope and y-intercept PRE-ALGEBRA LESSON 8-3 Find the slope of each line. a. b. Slope is 0 for a horizontal line. Division by zero is undefined. Slope is undefined for a vertical line. Quick Check 8-3

  27. 1 5 1 5 Step 2 Since the slope is – , move 1 unit down from (0, 4). Then move 5 units right to graph a second point. Slope and y-intercept PRE-ALGEBRA LESSON 8-3 A ramp slopes from a warehouse door down to a street. The function y = – x + 4 models the ramp, where x is 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 3 Draw a line through the points. Quick Check 8-3

  28. 4 3 – ; 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) 3. Identify the slope and y-intercept of y = – x + 3. Then graph the line. –2 2 4 3 8-3

  29. Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8-4 Find the slope of each line. • 10x + 5y = 28 • –2 b. 3x +3y = 5 –1 8-4

  30. 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

  31. Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8-4 Solutions 0 –1 1 – 1 2 – 3 1. slope = = = 0 2. slope = = = 1 3. slope = = ; slope is undefined. 4. slope = = = – –2 –2 2 – 4 1 – 3 4 0 2 – (–2) 0 – 0 –3 9 –1 – 2 4 – (–5) 1 3 8-4

  32. number of minutes Words total bill is $4.95 plus 9¢ times Let = the number of minutes. m Let = total bill, a function of the number of minutes. t( m ) Rule t( m ) = 4.95 + • m 0.09 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. A rule for the function is t(m) = 4.95 + 0.09m. 8-4

  33. t(90) = 4.95 + 0.09(90) Replace m with 90. t(90) = 4.95 + 8.10 Multiply. t(90) = 13.05 Add. 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 + 0.09m The total monthly bill with 90 minutes of long-distance calls is $13.05. Quick Check 8-4

  34. x 2 0 –2 –4 f(x) 3 –5 –13 –21 As the x values decrease by 2, the f(x) values decrease by 8. –2 –2 –2 –8 –8 –8 –8 –2 So m = = 4. Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8-4 Write a rule for the linear function in the table below. When x = 0, f(x) = –5. So b = –5. A rule for the function is f(x) = 4x – 5. Quick Check 8-4

  35. –2 – 2 0 – 2 –4 –2 slope = = = 2 Writing Rules for Linear Functions PRE-ALGEBRA LESSON 8-4 Write a rule for the linear function graphed below. y-intercept = –2 A rule for the function is f(x) = 2x – 2. Quick Check 8-4

  36. 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. 2.3. f(x) = 0.10x + 150 xf(x) 0 0 1 1 2 2 f(x) = x f(x) = 5x – 3 8-4

  37. 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

  38. Scatter Plots PRE-ALGEBRA LESSON 8-5 (For help, go to Lesson 1-10.) Write the coordinates of each point. 1.A2.B3.C4.D Check Skills You’ll Need 8-5

  39. Scatter Plots PRE-ALGEBRA LESSON 8-5 Solutions 1.A(–2, 2) 2.B(0, 3) 3.C(–3, 0) 4.D(2, 3) 8-5

  40. 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

  41. Elevation Above Sea Level (ft) Mean Annual Precipitation (in.) City 1,050 20 596 18 11 1,072 75 40 1,305 Atlanta, GA Boston, MA Chicago, IL Honolulu, HI Miami, FL Phoenix, AZ Portland, ME San Diego, CA Wichita, KS 51 42 36 22 56 8 44 10 29 Scatter Plots PRE-ALGEBRA LESSON 8-5 Use the table to make a scatter plot of the elevation and precipitation data. Quick Check 8-5

  42. 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

  43. 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? 2. How many students read 3 books per month? 3. Is there a positive correlation, a negative correlation, or no correlation between books read and semester grades? read 2 books per month, grade: 80 3 positive correlation 8-5

  44. 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

  45. 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

  46. 4 2 6 – 2 2 – 0 –2 – (– 3) –4 – 0 1 –4 1 4 1 4 Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8-6 Solutions 1. slope = = = 2 y-intercept = 2 A rule for the function is f(x) = 2x + 2. 2. slope = = = – y-intercept = –3 A rule for the function is f(x) = – x – 3. 8-6

  47. Year Wolf Moose Year Wolf Moose Year Wolf Moose 1982 1983 1984 1985 1986 1987 14 23 24 22 20 16 700 900 811 1,062 1,025 1,380 1988 1989 1990 1991 1992 1993 12 11 15 12 12 13 1,653 1,397 1,216 1,313 1,600 1,880 1994 1995 1996 1997 1998 1999 15 16 22 24 14 25 1,800 2,400 1,200 500 700 750 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 8-6

  48. 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. Problem Solving Strategy: Solve by Graphing PRE-ALGEBRA LESSON 8-6 (continued) 8-6

  49. 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. 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. There are about 1,300 moose on the island. Quick Check 8-6

  50. 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? 2. What type of correlation will the data have for a scatter plot of ages and heights of all students in your school? 3. Why is it important to say “about” when making a prediction? when there is no correlation positive The prediction is based on a trend line that approximates the locations of the related points. 8-6

More Related