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Representing Relationships

Representing Relationships. Chapter 4, Lesson 1. Real-World Link. To achieve orbit, the space shuttle must travel at a rate of about 5 miles per second. The table shows the total distance d that the craft covers in certain periods of time t.

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Representing Relationships

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  1. Representing Relationships Chapter 4, Lesson 1

  2. Real-World Link To achieve orbit, the space shuttle must travel at a rate of about 5 miles per second. The table shows the total distance d that the craft covers in certain periods of time t. • Write an algebraic expression for the distance in miles for any number of seconds t. • 5t • Describe the relationship in words. • The distance is 5 times the number of seconds.

  3. Real-World Link To achieve orbit, the space shuttle must travel at a rate of about 5 miles per second. The table shows the total distance d that the craft covers in certain periods of time t. c. Graph the ordered pairs. Describe the shape of the graph.

  4. Linear Equation An equation with a graph that makes a straight line. Can have more than one variable.

  5. Example 1 Write an equation to find the number of liters in any number of quarts. Describe the relationship in words. The rate of change is 0.95. EQUATION: = 0.95q IN WORDS: The rate of change between quarts and liters is 0.95.

  6. Example 2 About how many liters are in 8 quarts? EQUATION: = 0.95q = 0.95(8) = 7.6 There are about 7.6 liters in 8 quarts.

  7. Got it? 1 & 2 The total cost of tickets to the school play is shown in the table. a. Write an equation to find the total cost of any number of tickets. Describe the relationship in words. EQUATION: c = 4.5t WORDS: Each ticket cost $4.50. b. Use the equation to find the cost of 15 tickets. c = 4.5t c = 4.5(15) c = 67.50 The cost of 15 tickets is $67.50.

  8. Example 3 The total distance Marlon ran one week is shown by the graph. Write an equation to find the number of miles run y after any number of days x. • Find the rate of change. m = = • Find the y-intercept. y = mx + b y = 3.5x + b 7 = 3.5(2) + b 0 = b The slope is 3.5 and the y-intercept is 0. The equation is y = 3.5x.

  9. Example 4 Using the same equation in Example3, how many miles will Marion run after 2 weeks? y = 3.5x Let x by 14, since x is in days. y = 3.5(14) y = 49 Marion will run 49 miles in 2 weeks.

  10. Got it? • Write an equation to find the total number of trees y that can be saved for any number of tons of paper x. • y = 17x • Use the equation to find how many trees could be saved if 500 tons of paper are recycled. The number of trees saved by recycling paper is shown. • 8,500 trees

  11. Multiple Representations of Linear Equations Words: The number of trees is equal to 17 times the number of tons of paper. Equation: y = 17x Table: Graph:

  12. Example 5 Chloe completes in jump rope competitions. Her average rate is 225 jumps per minute. • Write an equation to find the number of jumps in any of amount of minutes. j = 225m • Make a table with 1, 2, 3, 4, and 5 minutes. Graph the points.

  13. Got it? Paul earns $7.50 an hour working at a grocery store. • Write an equation the find the amount of money Paul earned m for any number of hours h. • Make a table to find the earnings if he works 5, 6, 7, and 8 hours. Graph the coordinate points.

  14. Relations Chapter 4, Lesson 2

  15. Relations: a set of ordered pairs

  16. Example 1 Domain: {-4, -3, 0, 2} Range: {-8, -4, 6) Express the relation {(2, 6), (-4, 8), (-3, 6), (0, -4)} as a table and a graph. State the domain and range.

  17. Got it? Express the relation {(-5, 2), (3, -1), (6, 2), (1, 7)} as a table and a graph. State the domain and range. Domain: {-4, -3, 0, 2} Range: {-8, -4, 6)

  18. Example 2 It cost $3 per hour to park at the Wild Wood Amusement Park. Make a table using x and y coordinates that represent the total cost for 3, 4, 5, and 6 hours. Graph the ordered pairs.

  19. Got it? 2 A movie rental store charges $3.95 per movie rental. Make a table using x and y coordinates for the total cost of 1,2, 3, and 4 movies. Graph the ordered pairs.

  20. Warm-Up Are these relations functions? Why or why not? • {(7, -4), (8, 2), (8, 1), (-4, 5)} • {(9, 3), (2, 3), (4, -3), (7, 2)}

  21. Functions Chapter 4, Lesson 3

  22. Vocabulary Function: Where every domain (input) is matched up with exactly one range (output) Example: m = 14p If m represent the amount of money you earn, and p is the number of pizza’s you deliver. How much money will you make? Depends on how many pizza’s you deliver. m = dependent p = independent

  23. number of mile number of hours number of goals final score

  24. Functions Example 1: Find f(-3) if f(x) = 2x + 1. f(x) = 2x + 1 f(-3) = 2(-3)+ 1 f(-3) = -6 + 1 So, f(-3) = -5.

  25. Function Tables: A way to organize the domain, range and rule on a table. Independent Variable = Domain Dependent Variable = Range Example 2: Choose four values for x and make a function table for f(x) = x + 5. Then state the domain and range. Domain is {-2, -1, 0, 1} Range is {3, 4, 5, 6}

  26. Got it? 1 & 2 Choose four values for x to complete the function table for the function f(x) = x – 7. Then state the domain and range. Domain: {-1, 0, 1, 2} Range: {-8, -7, -6, -5}

  27. Example 3 There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j. Identify the independent and dependent variable. Ask: How many peanuts p(j) are there? It depends… So, p(j) or the number of peanuts are the dependent variable. Logic tells us that the number of jars is the independent variable.

  28. Example 4 There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j. What values of the domain and range make sense for this situation? DOMAIN: Ask: positive or negative numbers, whole, decimals? only positive whole numbers RANGE: The range depends on the x-values, and since there are 770 peanuts in each jar, the range will be multiples of 770.

  29. Example 4 There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j. Write a function to represent the total number of peanuts. p(j) = 770j

  30. Example 5 There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j. How many peanuts are there in 7 jars of peanut butter? p(j) = 770j p(7) = 770(7) p(7) = 5,390 There will be 5,390 peanuts in 7 jars of peanut butter.

  31. The domain is the set of numbers for the independent variable. The range is the set of numbers for the dependent variable.

  32. Got it? 3-5 A scrapbooking store is selling rubber stamps for $4.95 each. The total sales f(n) is a function of the number of rubber stamps n sold. • Identify the independent and dependent variable. independent = n dependent = f(n) • What values of the domain and range make sense for this situation? Domain: positive whole numbers Range: multiples of 4.95 • Write a function equation to represent total sales. f(n) = 4.95n • Determine the total cost of 5 stamps. $24.95

  33. Linear Functions Chapter 4, Lesson 4

  34. Sometimes functions are written with two variables, x and y. x represents the domain y represents the range

  35. Example 1 The school stores buys book covers for $2 each and notebooks for $1. Toni has $5 to spend. The function y = 5 – 2x represents this situation. Graph the function and interpret the points graphed. Chose values for x and substitute them to find y. Graph the ordered pairs. Toni has 4 options at the book store. 5 notebooks, 1 cover and 3 notebooks, or 2 covers and 1 notebook.

  36. Got it? 1 The farmer’s market sells apples for $2 per pound and pears for $1 per pound. Mallory has $10 to spend. The function y = 10 – 2x represents this situation. Graph this function and interpret the points. Mallory can purchase 10 pounds of pears, or 8 pounds of pear and 1 pound of apples, or 2 pounds of pears and 2 pounds of apples.

  37. Example 2 Graph y = x + 2. Make a function table. Graph the ordered pairs.

  38. Got it? 2 Graph these functions. • y = x – 5 • y = -2x

  39. Words: the value of y is one less than the corresponding value of x. Representing Functions Equation: y = x – 1 Table: Graph:

  40. Linear Functions: a function where the graph is a line. Example: y = mx + b Continuous vs. discrete data Continuous – no space between data values Discrete – have space between data values

  41. Each person that enters the store receives a coupon for $5 off his or her entire purchase. Example 3 y = 5x • Write a function to represent the total value of coupons given out. • Make a function table for 5, 10, 15, and 20 and graph the points. • Is the function continuous or discrete? Explain. There can only be a whole number of customers, so the graph is discrete.

  42. A store sells trail mix for $5.95 per pound. Got it? 3 y = 5.95x • Write a function to represent the total cost of any number of pounds. • Make a function table for 1, 2, 3, 4, and 5 pounds and graph the points. • Is the function continuous or discrete? Explain. There can be decimals of pounds and cost so the function is continuous.

  43. Make ordered pairs from the x-value and y-value. Then graph the coordinates and draw a line through, IF the function is continuous.

  44. Warm-up Use a map diagram to show if these relations are function. • {(1, -3), (3, 8), (2, 8), (-7, 1)} • {(-4, 0), (7, -2), (-4, 5), (2, 9)}

  45. Compare Properties of Functions Chapter 4, Lesson 5

  46. Real-World Link Make a table to represent the cost of Carlos’s membership. Describe the rate of change for each function. Carlos has a rate of 0, and Stephanie has a rate of 5. Carlos and Stephanie are members to the science museum. Carlos’s members can be represented by the function c = 9.99. The cost of Stephanie’s membership is shown by the table.

  47. Example 1 A zebra’s main predator is a lion. Lions can run at a speed of 53 feet per second over short distances. The graph shows the speed of a zebra. Compare their speeds. Lion’s rate of change = 53 Find the zebra’s rate of change The zebra has a faster rate than the lion.

  48. Got it? 1 A 2013 Ford Acura has a gas mileage of 21 miles per gallon. The gas mileage of a 2013 Audi is represented by this graph. Compare their gas mileage. The Ford Acura has a better gas mileage. The Acura has 21, and the Audi has a rate of 19.

  49. Example 2 Compare the functions’ y-intercepts and rate of change. They both have the same y-intercept: 0 The rate of change for Japan is 140. The function m = 140h, where m is the miles traveled in h hours, represents the speed of the first Japanese high speed train. The speed today’s high speed train in China is shown by the table. The rate of change for China is 217. China’s high speed train is faster than Japan’s train.

  50. Example 2 The function m = 140h, where m is the miles traveled in h hours, represents the speed of the first Japanese high speed train. The speed today’s high speed train in China is shown by the table. If you ride each train for 5 hours, how far will you travel on each? Japan: y = 140h y = 140(5) y = 700 You will travel 700 miles on Japan’s train. You will travel 1,085 miles in 5 hours on the Chinese train.

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