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Algebra 1 Unit 4 GRAPHING STORIES: Watch the videos on the next two slides and graph the story. GRAPHING STORIES: Watch the video and graph the story. Clock: www.graphingstories.com/2kj GRAPHING STORIES: Watch the video and graph the story. Height of waist off ground: www.graphingstories.com/4my How would you use your calculator to solve 52? Input Press: 5 Output x2 25 The number you entered is the input number (or x-value on a graph). The result is the output number (or yvalue on a graph). The x2 key illustrates the idea of a function. A function is a relation that gives a single output number for every valid input number. A relation is a rule that produces one or more output numbers for every valid input number. There are many ways to represent relations: Graph Equation Table of values A set of ordered pairs Mapping These are all ways of showing a relationship between two variables. A function is a rule that gives a single output number for every valid input number. To help remember & understand the definition: Think of your input number, usually your x-coordinate, as a letter. Think of your output number, usually your y-coordinate, as a mailbox. A function is a rule that gives a single output number for every valid input number. Input number Output number Can you have one letter going to two different mail boxes? Not a FUNCTION A function is a rule that gives a single output number for every valid input number. Input number Output number Can you have two different letters going to one mail box? Are these relations or functions? x 1 2 3 4 y 5 6 7 Function & Relation x 1 2 3 4 y 5 6 7 6 Are these relations or functions? x 1 y 5 6 2 7 Not a Function but a Relation x 1 2 1 1 y 5 6 7 6 Are these relations or functions? x y 1 5 2 6 3 8 11 Not a function But a relation x 1 2 2 3 y 5 6 11 8 In words: Double the number and add 3 As an equation: y = 2x + 3 As a table of values: x y -2 -1 -1 1 0 3 1 5 These all represent the SAME function! As a set of ordered pairs: (-2, -1) (-1,1) (0,3) (1, 5) (2, 7) (3, 9) X Y 1 5 3 4 2 3 5 4 6 5 We know it is a function because non of the “x” values in the table repeat. We can also hold a vertical line up to the points and see that no 2 points are touched by the same vertical line This method of determining a function from a graph is know as the vertical line test #1 Is this a function? X Y -2 -6 -1 -5 0 -4 1 -3 2 -2 Run vertical line across the graph and see if it touches two places at the same time This is a function because it passes the vertical line test. #2 X Y 1 5 3 4 2 3 1 4 4 5 Notice the two points that are touched by the green line. This means that the relation is not a function Functional Notation An equation that is a function may be expressed using functional notation. The notation f(x) (read “f of (x)”) represents the variable y. Functional Notation Con’t 1. For the function f(x) = 2x + 6, the notation f(3) means that the variable x is replaced with the value of 3. f(x) = 2x + 6 f(3) = 2(3) + 6 f(3) = 12 (3, 12) 2. Evaluating Functions Given f(x) = 4x + 8, find each: f(2) = 4(2) + 8 = 16 (2, 16) Evaluating More Functions If f(x) = 3x 1, and g(x) = 5x + 3, find each: 1. f(x) + g(x) = [3x - 1] + [5x + 3] = 3x – 1 + 5x + 3 = 8x + 2 2. f(x) - g(x) = [3x - 1] - [5x + 3] = 3x – 1 – 5x - 3 = -2x - 4