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Slide 2 Algebra 2 Chapter 2 Algebra 2 Chapter 2 1 Slide 3 2.1 Relations and Functions Relation Any set of inputs and outputs. Maybe represented as a Table Ordered pairs Mapping Graph 2 Slide 4 2.1 Relations and Functions Example 1: The monthly average water temperature of the Gulf of Mexico in Key West, Florida is as follows: January69 F February70 F March75 F April 78 F Represent this relation in the 4 ways. 3 Slide 5 2.1 Relations and Functions Table MonthTemp 1 2 3 4 69 F 70 F 75 F 78 F 4 Slide 6 2.1 Relations and Functions Ordered Pairs {( ), ( ), ( ), ( )} 1,69 2,70 3,75 4,78 5 Slide 7 2.1 Relations and Functions Mapping 1 2 3 4 69 F 70 F 75 F 78 F 6 Slide 8 2.1 Relations and Functions 3 68 70 72 74 76 78 1 2 4 Graph 7 Slide 9 2.1 Relations and Functions Domain the set of inputs of a relation the x-coordinates of the ordered pairs Range the set of outputs of a relation the y-coordinates of the ordered pairs 8 Slide 10 2.1 Relations and Functions Example 2: Write the domain and range from example 1. Domain: { } Range: { } 9 1, 2, 3, 4 69, 70, 75, 78 Slide 11 2.1 Relations and Functions Function a relation where no input (x) repeats. 10 Slide 12 2.1 Relations and Functions Example 3a Is the relation a function? {( 3, 5), (5, 4), (4, 6), (0, 6)} YES! 11 Slide 13 2.1 Relations and Functions Example 3b Is the relation a function? 12 x y 5 4 3 5 99 100 20 10 NO! Slide 14 2.1 Relations and Functions 13 4 5 6 7 8 2 4 6 8 Example 3c Is the relation a function? YES! Slide 15 2.1 Relations and Functions 14 Example 4a Use the vertical line test to determine if the relation is a function. NO! Slide 16 2.1 Relations and Functions 15 Example 4b Use the vertical line test to determine if the relation is a function. NO! Slide 17 2.1 Relations and Functions 16 Example 4c Use the vertical line test to determine if the relation is a function. NO! Slide 18 2.1 Relations and Functions 17 Function Rule An equation that represents an output value in terms of an input value Function Notation f(x) f(x) is read f of x. On a graph, f(x) is y. Slide 19 2.1 Relations and Functions 18 Example 5 Evaluate the function for the given values of x, and write the input x and output as an ordered pair. a. x = 9 b. x = 4 Slide 20 2.1 Relations and Functions 19 Example 5 (continued) (9,1) Slide 21 2.1 Relations and Functions 20 Example 5 (continued) Slide 22 2.1 Relations and Functions 21 Assignment: p.65 (#9 16 all, 18 24 evens) Slide 23 2.1 Relations and Functions 22 Independent Variable Usually x, represents the input value of the function Dependent Variable Usually f(x), represents the output value of the function (The value of this variable depends on the input value.) Slide 24 2.1 Relations and Functions 23 Example 6 To wash her brothers clothes Jennifer charges him a base rate of $15 plus $3.50 per hour. Write a function rule to model the cost of washing her brothers clothes. Slide 25 2.1 Relations and Functions C(x) = ____ + _____ x Then evaluate the function if it takes Jennifer 2 hours to wash his clothes. C(2.5) = 15 + 3.50(2.5) C(2.5) = 23.75 Jennifer will charge $23.75. 24 153.50 Slide 26 2.1 Relations and Functions 25 Example 7 Find the domain and range of each relation. Slide 27 2.1 Relations and Functions 26 Example 7a Domain: x > 0 Range: ARN Slide 28 2.1 Relations and Functions 27 Example 7b Domain: 4 < x < 4 Range: 4 < y < 4 Slide 29 2.1 Relations and Functions 28 Example 8 The relationship between your weekly salary S and the number of hours worked h is described by the following function. Slide 30 2.1 Relations and Functions 29 Example 8 (continued) In the following pairs, the input is the number of hours worked and the output is your weekly salary. Find the unknown measure in each ordered pair. Slide 31 2.1 Relations and Functions 30 Example 8 (continued) a.) Slide 32 2.1 Relations and Functions 31 Example 8 (continued) b.) (h, 135.20) Slide 33 2.1 Relations and Functions 32 Assignment: p.65-66 (#25, 26, 29 33, 39 44, 48) Slide 34 2.2 Direct Variation A function where the ratio of output to input is called direct variation. 33 Slide 35 2.2 Direct Variation 34 output input Constant of variation Slide 36 2.2 Direct Variation For each of the following tables, determine whether y varies directly as x. If so, find the constant of variation and the equation of variation. 35 Slide 37 2.2 Direct Variation Example 1 36 xy 1 3 7 21 9 3 YES! k = 3 So y = kx would mean y = 3x. Slide 38 2.2 Direct Variation Example 2 37 xy 2 2 10 15 3 NO! 3 Slide 39 2.2 Direct Variation Example 3 If y varies directly as x, and y = 4 when x = 25. What is x when y = 10? 38 4x = 250 x = 62.5 Slide 40 2.2 Direct Variation Example 4 If y varies directly as x, and x = 8 when y = 10, find y when x = 30. 39 300 = 8y 37.5 = y Slide 41 2.2 Direct Variation Example 5 The cost buying sirloin steak is directly proportional with the weight in pounds. If 8.5 lbs of steak cost $47.60, how much does 20 lbs cost? 40 = d = $112 Slide 42 2.2 Direct Variation Assignment: p.71(#7 10, 19 26) 41 Slide 43 2.3 Linear Functions & Slope Intercept Form 42 Slide 44 2.3 Linear Functions & Slope Intercept Form Example 1 What is the slope of the line that passes through the given points? 43 Slide 45 2.3 Linear Functions & Slope Intercept Form Example 1a ( 10, 2) and (4, 5) 44 Slide 46 2.3 Linear Functions & Slope Intercept Form Example 1b (6, 1) and (5, 1) 45 0 in numerator Slide 47 2.3 Linear Functions & Slope Intercept Form Example 1c ( 2, 5) and ( 2, 1) 46 0 in denominator The slope isUNDEFINED! 0 in denominator Slide 48 2.3 Linear Functions & Slope Intercept Form 47 Assignment: p.78 (#9-15) Slide 49 2.3 Linear Functions & Slope Intercept Form Slope-intercept Form 48 where m is the slope of the line and (0, b ) is the y-intercept. Slide 50 2.3 Linear Functions & Slope Intercept Form Example 2 What is an equation of each line in slope-intercept form? 49 Slide 51 2.3 Linear Functions & Slope Intercept form Example 2a 50 Slope = 3 y-intercept is (0,5) Slide 52 2.3 Linear Functions & Slope Intercept Form Example 2b Slope = y-intercept = 51 up 2 over 3 Slide 53 2.3 Linear Functions & Slope Intercept Form Example 3 Write the equation in slope- intercept form. What are the slope and y-intercept? 52 Slide 54 2.3 Linear Functions & Slope Intercept Form Example 3a 2x + 3y 15 = 0 2x 3y 15 = 2x + 15 + 15 3y = 2x + 15 3 3 3 53 Slide 55 2.3 Linear Functions & Slope Intercept Form 54 Slide 56 2.3 Linear Functions & Slope Intercept Form Example 3b 12 = 10y 3x + 3x 12 + 3x = 10y 10 10 10 55 Slope = y-intercept = Slide 57 2.3 Linear Functions & Slope Intercept Form Example 4 What is the graph of 24 = 4x + 3y? 56 24 = 4x + 3y 4x 4x 4x + 24 = 3y 3 3 3 Slide 58 2.3 Linear Functions & Slope Intercept Form Example 4 (continued) 57 8 Slide 59 Assignment: 58 p.78(#17-31 odds) 2.3 Linear Functions & Slope Intercept Form Slide 60 Example 5 A horizontal line has slope 0. Graph y = 5. m = 0 b = 5 59 2.3 Linear Functions & Slope Intercept Form Slide 61 Example 6 The slope of a vertical line is UNDEFINED. Graph x = 3. 60 2.3 Linear Functions & Slope Intercept Form Slope = undefined y-intercept = NONE Slide 62 Assignment: 61 p.78(#32 52 evens) 2.3 Linear Functions & Slope Intercept form Slide 63 2.4 More About Linear Equations Point-slope form 62 Slide 64 2.4 More About Linear Equations Example 1 Use the given information to write an equation in point-slope form. 63 Slide 65 2.4 More About Linear Equations a. slope = through ( 1, 3) 64 Slide 66 2.4 More About Linear Equations b.) slope = 0 through (22, 1) 65 Slide 67 2.4 More About Linear Equations 66 c.) passing through (5, 1) and (7, 1) Slide 68 2.4 More About Linear Equations d.) passing through (4, 1) and ( 6, 5) 67 Slide 69 2.4 More About Linear Equations Slopes of parallel lines are equal / the same. Slopes of perpendicular lines are opposite reciprocals. 68 Slide 70 2.4 More About Linear Equations Example 2 Use the given information to write the equation of the line described in slope-intercept form. 69 Slide 71 2.4 More About Linear Equations a.) parallel to y = x + 2 through ( 5, 3) 70 Slide 72 2.4 More About Linear Equations b.) perpendicular to y = 2x + 3 71 Slide 73 2.4 More About Linear Equations Assignment: p.86-88 (#10 18, 32, 33, 65, 72, 74) 72 Slide 74 2.4 More About Linear Equations Example 3 Use the given information to write the equation of the line described in slope- intercept form. 73 Slide 75 2.4 More About Linear Equations a.) parallel to 3x 2y = 6 through ( 3, 5) 74 Slide 76 2.4 More About Linear Equations b.) perpendicular to 4x + y = 1 through (2,1) 75 Slide 77 2.4 More About Linear Equations Example 4 Find the intercepts, and graph the line. 76 Slide 78 2.4 More About Linear Equations a.) 4x + 3y = 12 77 Slide 79 2.4 More About Linear Equations b.) 4x 5y = 10 78 Slide 80 2.4 More About Linear Equations Example 5 The cost of a taxi ride depends on the distance traveled. You paid $8.50 for a 3-mile ride, and your friend paid $18.50 for an 8-mile ride 79 Slide 81 2.4 More About Linear Equations 80 Example 5 A.) Sketch a graph that models this situation. Slide 82 2.4 More About Linear Equations Example 5 (continued) B.) Write the equation in slope-intercept form for this situation. C.) How much would a 6 mile taxi ride cost? 81 Slide 83 2.4 More About Linear Equations Assignment: p.86-87(#26-31,34-41) 82