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MF10 5 6-Properties Linear Relations

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MF10 5 6-Properties Linear Relations

  1. 1. Properties of Linear Relations Slide 1 In a linear relation the graph is a straight line and both the dependent and independent variables have constant change. Distance (km) Cost ($) 0 60 100 80 200 100 300 120 400 140 Independent Variable Dependent Variable +20 +20 +20 +20 +100 +100 +100 +100
  2. 2. Properties of Linear Relations Slide 2 In a linear relation the graph is a straight line and both the dependent and independent variables have constant change. 0 20 40 60 80 100 120 140 160 0 50 100 150 200 250 300 350 400 450 Cost($) Distance (km) Car Rental Cost +20 Change in Dependent Variable +100 Change in Independent Variable
  3. 3. Slide 3 Example 1: Which table of values represents a linear relation? Justify the answer. Celsius (C) Fahrenheit (F) 0 31 5 41 10 50 15 59 20 68 The relationship between temperature in degrees Celsius, C, and temperature in degrees Fahrenheit, F. Amps (I) Watts (P) 0 0 5 75 10 300 15 675 20 1200 The relationship between the current, I amps, and power, P watts, in an electrical current.
  4. 4. Slide 4 Example 1: Which table of values represents a linear relation? Justify the answer. Celsius (C) Fahrenheit (F) 0 31 5 41 10 50 15 59 20 68 Amps (I) Watts (P) 0 0 5 75 10 300 15 675 20 1200 +5 +5 +5 +5 +9 +9 +9 +9 This is a linear relation. The changes in both variables are constant.
  5. 5. Slide 5 Example 1: Which table of values represents a linear relation? Justify the answer. Celsius (C) Fahrenheit (F) 0 31 5 41 10 50 15 59 20 68 Amps (I) Watts (P) 0 0 5 75 10 300 15 675 20 1200 +5 +5 +5 +5 +9 +9 +9 +9 This is a linear relation. The changes in both variables are constant. +5 +5 +5 +5 +75 +225 +375 +525 This is NOT a linear relation. The changes in P are not constant.
  6. 6. Slide 6 Example 1: Graph the equation and state if it represents a linear relation. 3 25y x   x y -3 -1 0 1 3
  7. 7. Slide 7 Example 1: Graph the equation and state if it represents a linear relation. 3 25y x   x y -3 34 -1 28 0 25 1 22 3 16 Answer: The above equation is a linear relation.
  8. 8. Slide 8 Example 2: Graph the equation and state if it represents a linear relation. 2 2 5y x  x y -3 -1 0 1 3
  9. 9. Slide 9 Example 2: Graph the equation and state if it represents a linear relation. 2 2 5y x  x y -3 23 -1 7 0 5 1 7 3 23 Answer: The above equation is NOT a linear relation (not a straight line).
  10. 10. Slide 10 Example 3: Graph the equation and state if it represents a linear relation. 5y  x y -3 -1 0 1 3
  11. 11. Slide 11 Example 3: Graph the equation and state if it represents a linear relation. 5y  x y -3 5 -1 5 0 5 1 5 3 5 Answer: The above equation is a linear relation.
  12. 12. Slide 12 Example 3: Graph the equation and state if it represents a linear relation. 4x   x y -3 -1 0 1 3
  13. 13. Slide 13 Example 3: Graph the equation and state if it represents a linear relation. 4x   x y -4 -3 -4 -1 -4 0 -4 1 -4 3 Answer: The above equation is a linear relation.
  14. 14. Change of Rate Slide 14 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐷𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐼𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒
  15. 15. Slide 15 Example 1: This graph shows the altitude of a helicopter as it lifts off. a) Identify the dependent and independent variables. b) Find the rate of change. What does it represent?
  16. 16. Slide 16 Example 1: This graph shows the altitude of a helicopter as it lifts off. a) Identify the dependent and independent variables. b) Find the rate of change. What does it represent? The dependent variable is the Altitude. The independent variable is Time.
  17. 17. Slide 17 Example 1: This graph shows the altitude of a helicopter as it lifts off. a) Identify the dependent and independent variables. b) Find the rate of change. What does it represent? The dependent variable is the Altitude. The independent variable is Time. 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. = 240 𝑚 − 120 𝑚 = 120 𝑚 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. = 120 𝑠 − 60 𝑠 = 60 𝑠 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 = 120 𝑚 60 𝑠 = 2 𝑚/𝑠 Answer: The rate of change is 2 m/s. It represents the helicopter climbing 2 metres every second.
  18. 18. Slide 18 Example 2: This graph shows the altitude of a helicopter as it descents to land. a) Identify the dependent and independent variables. b) Find the rate of change. What does it represent?
  19. 19. Slide 19 Example 2: This graph shows the altitude of a helicopter as it descents to land. a) Identify the dependent and independent variables. b) Find the rate of change. What does it represent? The dependent variable is the Altitude. The independent variable is Time. 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑. 𝑣𝑎𝑟. 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 = 180 𝑚 − 0 𝑚 60 𝑠 − 120 𝑠 = 180 𝑚 −60 𝑠 = −3 𝑚/𝑠 Answer: The rate of change is -3 m/s. It represents the helicopter descending 3 metres every second.

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