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Unit 5 Relations & Functions Vocabulary: Relation: Ex: Domain: Ex: Name: ______________________________ Period: ________ Date Started: __________ Date Finished: ___________ any set of ordered pairs. (2, 4), (8, 7), (6, 5), (4, 5) all the input values or (x) values. Range: in the relation above, the domain = {2, 4, 6, 8} * always list from least to greatest * ** don’t repeat #s ** all output values or (y) values. Ex: in the relation above, the range = {4, 5, 7} * always list from least to greatest * ** don’t repeat #s ** Practice: Identify the domain & range of the relation. 1) (0, 1), (2, 4), (3, 7), (5, 4) 2) (-1, 2), (-3, -1), (6, 0), (-1, 4) Domain = Domain = Range = Range = Representing a relation: There are several ways to represent a relation… 1) 3) A set of ordered pairs: (-1, 1), (2, 0), (3, 1), (3, 2), (4, 5) 2) A Graph: 4) A Table: Input x Output y -1 1 2 0 3 1 3 2 A Mapping Diagram: -1 2 3 4 0 1 2 5 4 5 Practice: Represent the relation in each way possible. Table (1, 2), (1, 5), (2, 4), (3, 3), (4, 1) Graph Mapping Diagram Functions! What is a function? A function is a special type of relation. For every input (x) there is EXACTLY ONE output (y). The x-values CANNOT repeat. How to tell? 1) Look at all the ordered pairs & make sure that no x-value ever repeats. 2) Graph the relation and do a “vertical line test”. Vertical Line Test 1) Place your pencil vertically on the left side of your paper. 2) Roll your pencil across toward the right. 3) Check to see if the pencil ever touches two points on your graph at the same time. * If it doesn’t, then the relation is a function - It passes the vertical line test. * If it does, then the relation is NOT a function - It fails the vertical line test. Practice: Determine if each relation is a function using both methods. 1) (0, 3), (1, 2), (2, -1), (4, 4), (5, 4) 2) (-2, -1), (0, 2), (2, 3), (-2, -4) Using the following, determine if each relation is a function. * If it is NOT, explain how you know. A list of ordered pairs: (0, 3), (1, 1), (3, 1), (4, 2) (-1, 3), (0, 2), (-1, -1), (1, 4) An input-output table: Input x Output y 0 1 3 4 3 1 1 2 A mapping diagram: Input Output 0 1 1 3 2 4 3 A Graph: Input Output x y -1 3 0 2 -1 -1 1 4 Input -1 0 1 Vertical Line Test Output -1 2 3 4 Other Ways? We’ve seen functions represented as a set of ordered pairs, an input-output table, a mapping diagram, a graph and now we’ll see it as a mathematical rule or equation. Ex: y = 2x + 1 Make an input/output table for the function. Then graph the function. Input y = 2x + 1 Output (x, y) x y Use 0, 1, 2, & 3 for the domain. 0 1 2 3 Does it pass the vertical line test? So, is it a function? Equations in Two Variables: Equations that contain two variables are called equations in two variables. Ex: y = 2x – 5 A solution to an equation in two variables is an ordered pair (x, y). When you substitute the given values for x and y into the equation, you get a true math statement. So, the values for x and y follow the function rule. Ex: Tell whether the ordered pair is a solution of y = 2x – 5 A) (1, -3) B) (4, 7) Function Notation Is a different way to write an equation. Instead of writing y = 2x + 5 we write f(x) = 2x + 5 We can write f(x), g(x), h(x), etc. Ex: For f(x) = 2x + 5, what is f(3)? f(3) = 2(3) + 5 Substitute 3 for x f(3) = 6 + 5 Solve using order of operations f(3) = 11 Finish solving Practice: 1) For f(x) = 3x - 1, what is f(-2)? 2) Find the value of g(6) if g(x) = 2(x + 1) – 1 3) Evaluate the function for the given value of x: h(x) = 2x2 – x + 8; x = 2 Graphs: The graph of an equation in two variables is the set of points in a coordinate plan that represents all the solutions of the equation. An equation whose graph is a line is called a linear equation. Ex: Graph y = 2x – 5 1. Make a table of values. 2. 3. Plot each ordered pair. Connect the points with a line. Slope-Intercept Form: y = mx + b m is slope & b is y-intercept Ex: y = 4x – 3 4 is the slope of the line -3 is where the line intercepts (crosses) the y-axis An equation that is solved for y and the x term comes first on the other side is in slope-intercept form. It is set up to plug values in for x to find ordered pairs that are solutions to the equation. Another name for slope-intercept form is Function Form. Ex: Put the equation in Slope-Intercept Form A) x+y=3 B) x + 3y = -6 Practice: Put each equation in slope-intercept form and then graph it using a table of values. 1) 2x + y = 3 2) x–y=2 3) 2x – 3y = 3 Graphing Horizontal & Vertical Lines: Graph y = 3 Graph x = -2 Is it a function? Is it a function? Practice: Graph y = -3 Graph x = 4 Graph y = 10 Graph x = -9 The Real-World: 1) The table below shows the ages and lengths of five alligators. Age in yrs. Length - in. (x) (y) 2 32 4 59 5 65 5 58 7 96 A) Represent the relation between the alligator’s age and length using ordered pairs. B) Identify the domain and range of the relation. C) Is the relation a function? Explain. 2) The table below shows on alligator’s length at different ages in its life. Age in yrs. Length - in. (x) (y) 1 23 2 36 3 47 4 61 5 73 A) Represent the relation between this alligator’s age and length using ordered pairs. B) Identify the domain and range of the relation. C) Is this relation a function? Explain. 3) The Hawaiian volcano Mauna Loa has erupted many times. In 1859, lava from the volcano traveled 32 miles to the Pacific Ocean at an average speed of 4 miles per hour. The lava’s distance d (in miles) from the ocean t hours after it left the volcano can be approximated by the equation d = 32 – 4t. A) Which variable represents the input? B) Which variable represents the output? C) Make a table of values for the equation. D) How long did it take the lava to reach the ocean? 4) You are at an altitude of 250 feet in a hot-air balloon. You turn on the burner and rise at a rate of 20 feet per minutes for 5 minutes. Write an equation relating your altitude h after you have risen for t minutes. A) Make an input-output table for the domain 0 ≤ t ≤ 5. Input Output t h B) Graph the function: Altitude (ft) Altitude of Hot-Air Balloon Time (min)