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Relations and functions

Relations and functions. Chapter 5. 5.1 – representing relations. Chapter 5. New terms. A set is a collection of distinct objects. An element of a set is one object in the set. A relation associates the elements of one set with the elements of another. example.

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Relations and functions

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  1. Relations and functions Chapter 5

  2. 5.1 – representing relations Chapter 5

  3. New terms A set is a collection of distinct objects. An element of a set is one object in the set. A relation associates the elements of one set with the elements of another.

  4. example • Communities can be associated with the territories they are in. Consider the relation represented in this table. • Describe this relation in words. • Represent this relation: • As a set of ordered pairs • As an arrow diagram. a) The relation shows the association “is located in” from a set of northern communities to a set of territories. For example, Yellowknife is located in the Northwest Territories. b) If we listed them as ordered pairs it would look like this: {(Hay River, NWT), (Iqaluit, Nunavut), (Nanisivik, Nunavut), (Old Crow, Yukon), (Whitehorse, Yukon), (Yellowknife, NWT)}

  5. example Different breeds of dogs can be associated with their mean heights. Consider this relation. Represent it as a table, and as an arrow diagram.

  6. example In this diagram: Describe the relation in words. List 2 ordered pairs that belong in the relation.

  7. Pg. 262-263, #5, 6, 7, 9, 11, 12, 14. Independent Practice

  8. 5.2 – properties of functions Chapter 5

  9. Input/output What is the rule for the input/output table below? We can think of relations as similar to this input/output machine. We put a number in, and do a few things to it, and then get the output. x is the input, and y is the output.

  10. Domain and range The set of first elements of a relation is called the domain. The set of related second elements of a relation is called the range. A function is a special type of relation where each element in the domain is associated with exactly one element in the range.

  11. functions A function is a special type of relation where each element in the domain is associated with exactly one element in the range. Which of these is a function?

  12. functions List a set of ordered pairs: List a set of ordered pairs: Domain and Range: Domain and Range:

  13. example • For each relation below: • Determine whether the relation is a function. • Identify the domain and range of each relation that is a function. • A relation that associates given b)shapes with the number of right angles in the shape: {(right triangle, 1), (acute triangle, 0), (square, 4), (rectangle, 4),(regular hexagon, 0)}

  14. Independent & dependent variables In the workplace, a person’s pay, P dollars, often depends on the number of hours worked, h. So, we say that P is the dependent variable. Since the number of hours work, h, does not depend on the pay, we say that h is the independent variable.

  15. example The table shows the masses, mgrams, of different numbers of identical marbles, n. Why is this relation also a function? Identify the independent variable and the dependent variable. Justify the choices. Write the domain and range.

  16. Pg. 270-273, #4, 5, 8, 9, 11, 12. Independent practice

  17. example The equation V = –0.08d + 50 represents the volume, Vlitres, of gas remaining in a vehicle’s tank after travelling dkilometres. The gas tank is not refilled until it is empty. Describe the function. Write the equation in function notation. Determine the value of V(600) What does this number represent? Determine the value of d when V(d) = 26. What does this number represent? a) So, in this equation, which is the independent variable? Why? Function notation is a different way to write an equation for a relation. It just gives us a way to identify the dependent and independent variables in the equation itself. • The amount of gas remaining depends on how far you’ve travelled—it doesn’t make sense the other way around. • Instead of V, we use V(d), which means V depending on d. V(d) = –0.08d + 50

  18. example The equation V = –0.08d + 50 represents the volume, Vlitres, of gas remaining in a vehicle’s tank after travelling dkilometres. The gas tank is not refilled until it is empty. Describe the function. Write the equation in function notation. Determine the value of V(600) What does this number represent? Determine the value of d when V(d) = 26. What does this number represent? V(d) = –0.08d + 50 c) We let V(d) = 26 26 = –0.08d + 50 –24 = –0.08d 300 = d When you have driven for 300 km there will be 26 L of gas left. b) When it says V(600) that means that we want to evaluate the expression for when d = 6. V(600) = –0.08(600) + 50 V(600) = 2 That means that after travelling 600 km, there will be 2 litres of gas left.

  19. Pg. 271-273, #6, 14, 15, 17, 18, 22. Independent practice

  20. 5.3 – interpreting and sketching graphs Chapter 5

  21. graphs

  22. example Each point on this graph represents a bag of popping corn. Explain the answer to each question below. Which bag is the most expensive? What does it cost? Which bag has the least mass? What is this mass? Which bags have the same mass? What is this mass? Which bags cost the same? What is this cost? Which of bags C or D has the better value for money?

  23. example Describe the journey for each segment of the graph.

  24. example Samuel went on a bicycle ride. He accelerated until he reached a speed of 20 km/h, then he cycled for 30 min at approximately 20 km/h. Samuel arrived at the bottom of a hill, and his speed decreased to approximately 5 km/h for 10 min as he cycled up the hill. He stopped at the top of the hill for 10 min. Sketch a graph of speed a function of time. Label each section of the graph, and explain what it represents.

  25. Pg. 281-283, #3, 5, 7, 8, 9, 12, 14 Independent practice

  26. 5.4 – graphing data Chapter 5

  27. graphing To rent a car for less than one week from Ace Car Rentals, the cost is $65 per day for the first three days, then $60 a day for each additional day. • Why are the points on the graph not joined? • Is this relation a function? How can you tell? • What is the domain? What is the range?

  28. You need to complete this on graph paper, and hand it in. Your graphs need to be neat and labeled. Pg. 286, #1, 2 Independent practice

  29. 5.5 – graphs of relations and functions Chapter 5

  30. Relations and functions In an environmental study, Joe collected data on the numbers of different species of bird he heard or saw in a 1-h period every 2 h for 24 h. Alice collected data on the temperature in the area at the end of each 1-h period. They plotted their data. Does each graph represent a relation? How about a function? Which of these graphs should have the data points connected?

  31. Vertical line test A graph represents a function when no two points on the graph lie on the same vertical line.

  32. Try IT! Are these functions?

  33. example Determine the domain and range of the graph of each function. a) b)

  34. Example This graph shows the number of fishing boats, n, anchored in an inlet in the Queen Charlotte Islands as a function time, t. Identify the dependent variable and the independent variable. Justify. Why are the points on the graph not connected? Determine the domain and range of the graph.

  35. Example Here is a graph of the function f(x) = –3x + 7. Determine the range value when the domain value is –2. Determine the domain value when the range value is 4.

  36. Pg. 294-297, #4, 7, 8, 9, 11, 14, 16, 17 Independent practice

  37. 5.6 – properties of linear relations Chapter 5

  38. Linear relations The cost for a car rental is $60, plus $20 for every 100 km driven. The independent variable is the distance driven and the dependent variable is the cost. Table of Values: For a linear relation, a constant change in the independent variable results in a constant change in the dependent variable.

  39. Linear relations As a set of ordered pairs: A graph:

  40. Rate of change You can pick any segment of the graph to look at—if it’s a linear graph, it will always give you the same amount. Rise is the vertical distance between two points, while run is the horizontal distance between them. Rate of change (also known as slope) tells you the steepness of a graph. It is the rate at which the dependent variable is changing at.

  41. Linear equations General form of a linear equation: y = mx + b y-intercept (or initial amount) dependent variable independent variable slope (or rate of change) slope = 0.2 y-intercept = 60 C = 0.2d + 60

  42. example Which table represents a linear equation?

  43. example Graph each equation.i) y = –3x + 25 ii) y = 2x2 + 5iii) y = 5 iv) x = 1 Which equations in part a represent linear relations? How do you know?

  44. example Which relation is linear? Justify. A new car is purchased for $24 000. Every year, the value of the car decreases by 15%. The value is related to time. For a service call, an electrician charges $75 flat rate, plus $50 for each hour he works. The total cost for service is related to time.

  45. example A water tank on a farm near Swift Current, Saskatchewan holds 6000 L. Graph A represents the tank being filled at a constant rate, while Graph B represents the tank being emptied at a constant rate. Identify the independent and dependent variables. Determine the rate of change of each relation, then describe what it represents.

  46. Pg. 308-310, #3, 5, 6, 7, 10, 12, 14, 17, 19. Independent practice

  47. 5.7 – interpreting graphs of linear functions Chapter 5

  48. Linear functions The point where the graph intersects the horizontal axis (or the x-axis) is called either the horizontal incercept or the x-intercept. The point where the graph intersects the vertical axis (or the y-axis) is called either the vertical incercept or the y-intercept. What’s the domain and range for this graph? What is the slope?

  49. example This graph shows the fuel consumption of a scooter with a full tank of gas at the beginning of a journey. Write the coordinates of the points where the graph intersects the axes. Determine the x and y-intercepts. Describe what the points of intersection represent. What are the domain and range of this function?

  50. example Sketch a graph of the linear function f(x) = –2x + 7.

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