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Foundations & Pre-calculus 10 Chapter 5—Relations and Functions Outline Date Topic 5.1 Representing Relations • table, set of ordered pairs, arrow diagram, etc.. 5.3 Interpreting and Sketching Graphs Assignment p.262 #3a, 4,6,8 5.2 Properties of Functions Part I + 5.4 Graphing Data • Review—practice—graphing tov • What is a function • Graphing data • Domain and range given data (not graph) • Discrete vs. continuous data 5.2 Properties of Functions Part II: Function Notation • Function notation 5.5 Graphs of Relations and Functions • Vertical line test for functions • Number line Worksheet • Domain and Range o inequalities and interval notation 5.6 Properties of Linear Relations • Detemine if it is linear from table, ordered pairs, graph, equation (graph equation using table of values) • Determine rate of change from graph 5.7 Interpreting Graphs of Linear Functions • Determine intercepts and domain/range • Sketch graph given function notation—find t for g(t)=-3 • Match graph to rate of change and y-intercept Graphing Review worksheet p.270 #1-5, 8,9 p.286 #1 Review Chapter 5 Test p. 281 #1-6,11,13,16,17 p.271 #6,7,14-19, 12 Function notation worksheet Number line worksheet p.293 #1-8,10-11,16,17,19 Domain and Range Worsheet p.307 #1-5, 8,11,12,14,16,17 p. 319 #1-8,10,11, 15, 16, 17 p. 326 #3, 5, 6, 9-11, 13, 16, 18 Group Review Foundations & Pre-calculus 10 5.1 Representing Relations A set is ____________________________________________________________ An element is _______________________________________________________ A relation is ________________________________________________________ Relations can be represented in several ways: As a table: The heading of each column describes each set. Fruit Apple Apple Blueberry Cherry Huckleberry Colour Red Green Blue Red Blue As a set of ordered pairs: (element of first set, element of second set) In words or a phrase: An Arrow Diagram: The two ovals represent the sets. Each arrow associates an element of the first set with an element of the second set. When elements of one or both sets in the relation are numbers, the relation can be represented as a ________________________________ Foundations & Pre-calculus 10 Ex. #1:Different breeds of dogs can be associated with their mean heights. Consider the relation represented by this graph. Represent the relation as: a) A table b) An arrow diagram Sometimes a relation contains so many ordered pairs that it is impossible to list all of them or to represent them in a table. Ex. #2: a) Describe the relation in words. b) List two ordered pairs that belong to the relation. Foundations & Pre-calculus 10 5.3 Interpreting and Sketching Graphs Discuss Make Connections and Construct Understanding. Time Speed Speed Speed 1. Tyler rides up and down a steep hill on his bike. Which graph best represents his ride? Time Time 2. Draw a graph that is a reasonable representation of each of the following situations. (Label your axes) The temperature of water in a pot on the stove Your height over the course of your lifetime Your health over your lifetime if you smoke Define: Dependent Variable: _________________________________________________ __________________________________________________________________ Independent Variable: ________________________________________________ __________________________________________________________________ Foundations & Pre-calculus 10 Ex. #1: Describe the journey for each segment of the graph. Segment Graph OA The graph goes up to the right, so as time increases, the distance from Winnipeg _________________. AB The graph is horizontal, so as time increases, the distance __________________________ BC The graph goes up to the right, so as time increases, the distance __________________ CD The graph is horizontal, so as time increases, the distance _________________________ DE Journey The graph goes down to the right, so as time increases, the distance __________________ What was the total driving time? What are the dependent and the independent variables? Foundations & Pre-calculus 10 Ex. #2: Karys went on a bicycle ride. She accelerated until she reached a speed of 20 km/h, then she cycled for 30 min. at approximately 20 km/h. Karys arrived at the bottom of a hill, and her speed decreased to approximately 5 km/h for 10 min as she cycled up the hill. She stopped at the top of the hill for 10 min. Sketch a graph of speed as a function of time. Label each section of the graph, and explain what it represents. Foundations & Pre-calculus 10 Review—Practice –Graphing using a table of values and algebra 1. Find the value of y when 2 3 4 2. Find the value of x when 12 5 2 3. Find the value of x when 5 2 1 4. Find the value of y when 3 4 5 5. Graph the equation 6. Graph the equation y = 2x + 1 x -2 -1 0 1 2 y 6x – 2y = 8 x y Foundations & Pre-calculus 10 5.2 Properties of Functions Part I + 5.4 Graphing Data A __________________ is a special type of relation where each element in the first set is associated with exactly one element in the second set. The set of first elements of a function is called the ______________________. The set of related second elements of a function is called the _________________. Ex. #1: Here are two different ways to relate vehicles and the number of wheels each has. This relation associates a number with a vehicle with that number of wheels This relation associates a vehicle with the number of wheels it has This diagram _____________________ a function because This diagram _____________________ a function because Display as a set of ordered Pairs: Display as a set of ordered Pairs: The set of ordered pairs _____________ The set of ordered pairs _____________ _________________ a function because a function because Foundations & Pre-calculus 10 Ex. #2: Create a table showing the cost of movie tickets. Variable Number of Tickets Purchased 1 2 3 4 5 6 Total Cost ($) Variable a) Is this relation a function? Explain. b) Identify the independent variable and the dependent variable. Explain. c) Write the domain and range. d) Graph the function. Do you connect the points? Explain. Foundations & Pre-calculus 10 Function Notation Function Notation: y is a function of x means y depends on x for its value. This can be expressed as f(x), said f of x or f at x. f is the name of the function and x is the variable used in the function. When we give functions “names”, this is called function notation. When you see f(x) it means the same as the y value. Example: The function y = 2 x − 1 can be expressed using function notation as f(x) = 2x-1. Complete the chart below for each function. Equation Function Notation y = 2x – 1 y = 4x + 1 y = 5x – 7 y = x2 – 5x + 2 y = 2x y = 3x2 y = 4 – 2x Substitution: If f(x) = 3x + 1, then this is the same as y = 3x + 1. f(2), said f at 2, means replace x with 2 and find the value of y. f(-3), said f at -3, means replace x with -3 and find the value of y. Example: f(x) = 3x + 1 f(2) = 3(2)+1 f(2) = 6+1 f(2) = 7 This means when x=2, y=7. f(x) = 3x + 1 f(-3) = 3(-3) + 1 f(-3) = -9 + 1 f(-3) = -8 This means when x=-3, y=-8 Example: Exercises: Using the function, f(x) = 3x + 1 and g(x) = -4x – 4, find the following values: f(7) = g(-2) = f(-4) = g(0) = Using the function, f(x) = x2 + 3x – 5, find the following values: f(1) = f(0) = f(-2) = f(4) = Foundations & Pre-calculus 10 5.2 Properties of Functions Part II—Function Notation Function Machine: What is the rule for the function machine? Which numbers would complete this table for the machine? Input Output 1 5 2 7 9 4 13 Let x be the input value and y be the output value. Create an equation for this function in terms of x and y. Consider the following function machine—it calculates the value of quarters When the input is q quarters, the output or value, V, in dollars is: The equation that describes this function is: Since V is a function of q, we can write this equation using function notation: This notation shows that _____ is the dependent variable and that _____ depends on _____ 3 represents the value of the function when ___________ Foundations & Pre-calculus 10 Ex. #1: The equation 0.08 50 represents the volume, V litres, of gas remaining in a vehicle’s tank after travelling d kilometres. The gas tank is not refilled until it is empty. a) Describe the function. Write the equation in function notation. b) Determine the value of 600. What does this number represent? c) Determine the value of d when 26. What does this number represent? Foundations & Pre-calculus 10 5.5 Graphs of Relations and Functions The relation 2 is graphed: We know that the relation 2 is a function because each value of x associates with _______________________ value of y, and each ordered pair has a different _____________________. The __________________ of a function is the set of values of the independent variable, or the set of ____________________ The __________________ of a function is the set of values of the dependent variable, or the set of _______________________. Vertical Line Test for a Function A graph represents a function when no two points on the graph lie on the same ______________________________ Ex. #1: Which of these graphs represents a function? Foundations & Pre-calculus 10 Ex. #2: Use the graph of the function 3 7 a) Determine the range value when the domain value is 3. b) Determine 1. c) Determine the value of x when 1 d) Circle the coordinate that is represented by 1 10. What are the coordinates at this point? Ex. #3: Use the graph given of 4 3 a) Determine the domain value when the range value is 5. b) Determine 0). c) Determine the value of x when 7 d) Circle the coordinate that is represented by 3 9. What are the coordinates at this point? Foundations & Pre-calculus 10 Domain and Range of a Graph Visualize: Domain—represented by the shadow of its graph on the _______________ Range—represented by the shadow of its graph on the _____________ You could also imagine ___________________ the function towards the axes. Ex. #4: Determine the domain and range of the graphs of each function Foundations & Pre-calculus 10 Interval Notation A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, 3, 8 is the interval of real numbers between 3 and 8, including 3 and excluding 8. Using inequalities, this interval is described as 3 8 Ex. #5: Determine the domain and range of the following relations, using words, inequalities, and interval notation. y y a) b) 4 4 3 3 2 2 1 1 x x −4 −3 −2 −1 1 −1 2 3 4 5 −4 −3 −2 −1 1 −1 −2 −2 −3 −3 −4 −4 2 3 4 5 Foundations & Pre-calculus 10 5.6 Properties of Linear Relations Discuss Make Connections on p.300 The cost for a car rental is $60, plus $20 for every 100km driven. So, the independent variable is the _____________________________ and the dependent variable is the _______________________. We can identify that this is a linear relation in different ways. • A table of values Distance (km) 0 100 200 300 400 Cost($) 60 80 100 120 140 For a linear relation, a constant change in the independent variable results in a ____________________________ in the dependent variable. • A set of ordered pairs 0,60, 100,80, 200,100, 300,120, 400,140 • A graph Foundations & Pre-calculus 10 We can use each representation to calculate the rate of change—the fraction: The rate of change is _____________; that is, for each additional 1 km driven, the rental cost increases by ___________. The rate of change is _________________ for a linear relation. We can determine the rate of change from the equation that represents the linear function. Let the cost be C dollars and the distance driven by d kilometres. An equation for this linear function is: Ex. #1: Which relation is linear? Justify the answer. a) A new car is purchased for $24000. Every year, the value of the car decreases by 15%. The value is related to time. Every year, the value decreases by 15%. The value of the car is: _______________________________ of its previous value. Time Value ($) (years) 0 1 2 3 Foundations & Pre-calculus 10 b) For a service call, an electrician charges $75 flat rate, plus $50 for each hour he works. The total cost for service is related to time. After the first hour, the cost increases by ______ per hour. Time (h) Cost ($) 0 1 2 3 4 Examples of Linear Relations Examples of Non-linear Relations How can you tell from an equation whether the relation is linear? Foundations & Pre-calculus 10 Ex. #3: A water tank on a farm near Swift Current, Saskatchewan, holds 6000L. Graph A represents the tank being filled at a constant rate. Graph B represents the tank being emptied at a constant rate. a) Identify the independent and dependent variables. Independent variable is ______________ Dependent variable is _______________ b) Determine the rate of change of each relation, then describe what it represents. Foundations & Pre-calculus 10 5.7 Interpreting Graphs of Linear Functions Discuss Make Connections on p.311 Any graph of a line that is not vertical represents a function. We call these functions ______________________________ Each graph below shows the temperature, T degrees Celsius, as a function of time, t hours, for two locations The point where the graph intersects the horizontal axis is _____________ The horizontal intercept is ______ This point of intersection represents: The point where the graph intersects the vertical axis is _____________ The vertical intercept is ______ This point of intersection represents: Domain: Range: The Rate of Change: The point where the graph intersects the horizontal axis is _____________ The horizontal intercept is ______ This point of intersection represents: The point where the graph intersects the vertical axis is _____________ The vertical intercept is ______ This point of intersection represents: Domain: Range: The Rate of Change: Foundations & Pre-calculus 10 We can use the intercepts to graph a linear function. To determine the y-intercept, find the value of y when x=_______ That is evaluate when _________________________________ To determine the x-intercept, find the value of x when y=_______ That is determine the ______________________________________ Ex. #2: Sketch a graph of the linear function 2 7 Ex. #3: Which graph has a rate of change of ½ anda vertical intercept of 6?