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Constant Rate of Change

Constant Rate of Change. Lesson 1. Marcus can download two songs from the Internet each minute. This is shown by the table below. Compare the change in number of songs y to the change in time x. What is the rate of change? The number of songs increases by 2, and the time increases by 1.

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Constant Rate of Change

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  1. Constant Rate of Change Lesson 1

  2. Marcus can download two songs from the Internet each minute. This is shown by the table below. • Compare the change in number of songs y to the change in time x. What is the rate of change? • The number of songs increases by 2, and the time increases by 1. • Graph the ordered pairs. • Describe the pattern. • The points appear to make • a straight line. Real World Link

  3. 1. Relationships that have a straight-line graph 2. The rate of change is the same – “constant rate of change” Linear Relationships

  4. The balance in an account after several transactions is shown. Is the relationship linear? If so, find the constant rate of change. Since the rate is constant, this shows a linear relationship. The constant rate of change is or -10. Each transaction involved a $10 withdrawal. Example 1

  5. a. b. No, this is not a linear relationship. It does not have a constant rate of change. Yes; the points make a line so this is a linear relationship. The constant rate of change is -2.5 minutes per volunteer. Got it? 1

  6. Words: Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change. Symbols: is constant and is also constant. Example: there is a constant rate of change and Proportional Linear Relationships

  7. Use the table to determine if there is a proportional linear relationship between the temperature in degrees and Fahrenheit and a temperature in degrees Celsius. Explain your reasoning. Since the ratios are not equal, degrees Fahrenheit and Celsius are not proportional. Since the rate of change is constant, this is a linear relationship. Example 2

  8. Use the table to determine if there is proportional linear relationship between the mass of an object in kilograms and the weight of the object in pounds. Explain your reasoning. It is a proportional linear relationship. The ratio is a constant , and the rate of change is . Got it? 2

  9. Slope Lesson 2

  10. Slope =

  11. Find the slope of the treadmill. Slope = = = The slope of the treadmill is . Example 1

  12. A hiking trail rises 6 feet for every horizontal change of 100 feet. What is the slope of the hiking trail? Slope = = The slope of the trail is . Got it? 1

  13. The graph shows the cost of muffins at a bake sale. Find the slope of the line. The vertical change is 2 and the horizontal change is 1. The slope is or 2. Example 2

  14. The table shows the number of pages Garrett has left to read after a certain number of minutes. The points lie on a line. Find the slope of the line. Slope = Choose any points: (1, 12) and (3, 9) Slope = The slope is - . Example 3

  15. a. b. The slope is . The slope is . Got it? 2 & 3

  16. Symbols: m = slope m = Slope Formula

  17. Find the slope of the line that passes through R(1, 2), S(-4, 3). m = m = m = The slope of the line is - . Example 4

  18. a. A(2, 2), B(5, 3) b. J(-7, -4), K(-3, -2) Got it? 4

  19. Equations in y = mx Form Lesson 3

  20. Find the slope of this graph. m = m = Solve for y. y = mx

  21. Words: “y varies directly with x” Symbols: m = or y = mx, where m is the constant or slope. Example: y = 3x Graph: Direct Variation

  22. The amount of money Robin earns while babysitting varies directly with the time as shown in the graph. Determine the amount that Robin earns per hour. Find “nice” points on the graph to find the constant of variation. (2, 15) and (4, 30) m = m = = 7.5 m = = 7.5 So Robin earns $7.5 and hour babysitting. Example 1

  23. Two minutes after a skydiver opens his parachute, he has descended 1,900 feet. After 5 minutes, he descends 4,750 feet. If the distance varies directly with the time, at what rate is the skydiver descending? Hint: Find the two points on the line. -950 feet/minute Got it? 1

  24. A cyclist can ride 3 miles in 0.25 hour. Assume that the distance biked in miles varies directly with time in hours x. This situation can be represented by y = 12x. Graph the equation. How far can the cyclist ride per hour? Make a table of values. 12 miles per hour Graph the values. The slope is 12 since the equation is y = 12x. Example 2

  25. A grocery store sells 6 oranges for $2. Assume that the cost of the oranges varies directly with the number of oranges. This situation can be represented by y = x. Graph the equation. What is the cost per orange? about $0.33 per orange Got it? 2

  26. In a proportional relationship, how is the unit rate represented on a graph? It is the slope. When comparing two different direct variation equations, what’s the difference? y = 3x y = 12x The rate or slope. Comparing Direct Variation

  27. The distance d in miles covered by a rabbit in t hours can be represented by d = 35t. The distance covered by a grizzly bear is shown on the graph. Which animal is faster, or has the fastest rate? Rabbit: d = 35t, so the rate is 35. Grizzly Bear: Find the slope of the line. The rate is 30. 35 is greater than 30, so the rabbit is faster. Example 3

  28. Damon’s earnings for four weeks from a part time job are shown in the table. Assume that his earnings vary directly with the number of hours worked. He can take a job that will pay him $7.35 per hour. What job is the better pay? Explain. His current job is a better pay. He currently earns $7.50. Got it? 3

  29. A 3-year old dog is often considered to be 21 in human years. Assume that the equivalent age in human years y varies directly with its age as a dog x. Write and solve a direct variation equation to find the human-year age of a dog that is 6 years old. We know that when y is 21, x is 3. y = mx 21 = m(3) m = 7 The rate is 7. y = 7x y = 7(6) y = 42 A dog that is 6 years old has an equivalent human age of 42. Example 4

  30. A charter bus travels 210 miles in 3.5 hours. Assume the distance traveled is directly proportional to the time traveled. Write and solve a direct variation equation to find how far the bus will travel in 6 hours. y = 60x 360 miles • A Monarch butterfly can fly 93 miles in 15 hours. Assume the distance traveled is directly proportional to the time traveled. Write and solve a direct variation equation to find how far the Monarch butterfly will travel in 24 hours. y = 9.3x 148.8 miles Got it? 4

  31. Slope-Intercept Form Lesson 4

  32. Slope-Intercept Form

  33. State the slope and the y-intercept of the graph of the equation y = x – 4. y = mx + b, where m is slope and b is y-intercept y = x – 4 m = and b = -4 The slope is and the y-intercept is -4. Example 1

  34. Find the slope and y-intercept for each equation. • y = -5x + 3 -5; 3 • y = x – 6 • y = -x + 5 -1; 5 Got it? 1

  35. Write an equation of a line in slope-intercept form if you know that the slope is -3 and the y-intercept is -4. y = mx + b y = -3x + (-4)or y = -3x – 4 Write an equation of a line in slope-intercept form from the graph below. y = mx + b the slope is 2 and the y-intercept is 4 y = -x + 4 Examples 2 & 3

  36. b. Write an equation in slope-intercept form with a slope of and a y-intercept of -3. a. Write an equation in slope-intercept form for the graph. y = x - 3 y = x - 2 Got it? 2 & 3

  37. Student Council is selling T-shirts during spirit week. It costs $20 for the design and $5 to print each shirt. The cost y to print x shirts is given by y = 5x + 20. Graph this equation using the slope and y-intercept. Step 1: Find the slope and y-intercept. slope = 5 y-intercept = 20 Step 2: Graph the y-intercept (0, 20) Step 3: Go up 5 and over 1 to find another point. Example 4

  38. Student Council is selling T-shirts during spirit week. It costs $20 for the design and $5 to print each shirt. The cost y to print x shirts is given by y = 5x + 20. Interpret the slope and y-intercept. The slope represents the cost of each T-shirt. The y-intercept is the one time fee of $20 for the design. Example 5

  39. A taxi fare y can be determined by the equation y = 0.50x + 3.5, where x is the number of miles traveled. • Graph this equation. • Interpret the slope and y-intercept. The slope is the $0.50 per mile and the y-intercept is the initial charge of $3.50. Got it? 4 & 5

  40. Graph a Line Using Intercepts Lesson 5

  41. x and y intercepts

  42. State the x- and y-intercepts of y = 1.5x – 9. Then use the intercepts to graph the equation. STEP 1: Find the y-intercept. y-intercept is -9. STEP 2: Find the x-intercept, let y = 0. 0 = 1.5x – 9 9 = 1.5x 6 = x STEP 3: Graph the two intercepts and connect to make a line. Example 1

  43. Graph these equations by using the x- and y- intercepts. a. y = x + 5 b. y = x + 3 Got it? 1

  44. Standard form is when the equation is Ax + By = C, and A, B, and C are integers. A, B, and C can NOT be fractions or decimals. Take the equation: 60x + 15y = 4,740 A = 60 B = 15 C = 4,740 Standard Form

  45. Mauldlin Middle School wants to make $4,740 from yearbooks. Print yearbooks x cost $60 and digital yearbooks y cost $15. This can be represented by the equation 60x + 15y = 4,740. Use the x- and y-intercepts to graph the equation. To find the x-intercept, let y = 0. To find the y-intercept, let x = 0 60x + 15y = 4740 60x + 15(0) = 4740 60x = 4740 x = 79 60x + 15y = 4740 60(0) + 15y = 4740 15y = 4740 y = 316 Example 2

  46. Graph the equation, using the intercepts, and interpret the x- and y- intercepts. The x-intercept is at the point (79,0). This means they can sell 79 print yearbooks and still earn $4,740. The y-intercept is at the point (0, 316). This means they can sell 316 digital yearbooks and still earn $4,740. Example 3

  47. Mr. Davis spent $230 on lunch for his class. Sandwiches x cost $6 and drinks y cost $2. This can be represented by the equation 6x + 2y = 230. Graph and interpret the x- and y- intercepts. The x-intercept of 38.33 means that if Mr. Davis bought 38.33 sandwiches, then the total would be $230. The y-intercept of 115 means that if he bought 115 drinks the total would be $230. Got it? 2 & 3

  48. Write Linear Functions Lesson 6

  49. Point-Slope Form

  50. Write an equation in point-slope form for the line that passes through (-2, 3) with a slope of 4. y – y1 = m(x – x1) y – 3 = 4(x – (-2)) y – 3 = 4(x + 2) Example 1

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