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Algebra 2 Chapter 2 Algebra 2 Chapter 2 1

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2.1 Relations and Functions Relation – Any set of inputs and outputs. Maybe represented as a Table Ordered pairs Mapping Graph 2

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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

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2.1 Relations and Functions Table MonthTemp 1 2 3 4 69 º F 70 º F 75 º F 78 º F 4

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2.1 Relations and Functions Ordered Pairs {( ), ( ), ( ), ( )} 1,69 2,70 3,75 4,78 5

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2.1 Relations and Functions Mapping 1 2 3 4 69º F 70º F 75º F 78º F 6

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2.1 Relations and Functions 3 68 70 72 74 76 78 1 2 4 Graph 7

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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

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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

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2.1 Relations and Functions Function ‒ a relation where no input (x) repeats. 10

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2.1 Relations and Functions Example 3a Is the relation a function? {( ‒ 3, 5), (5, 4), (4, ‒ 6), (0, ‒ 6)} YES! 11

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2.1 Relations and Functions Example 3b Is the relation a function? 12 x y 5 4 3 5 ‒9‒9 100 20 ‒ 10 NO!

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2.1 Relations and Functions 13 4 5 6 7 8 2 4 6 8 Example 3c Is the relation a function? YES!

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2.1 Relations and Functions 14 Example 4a – Use the vertical line test to determine if the relation is a function. NO!

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2.1 Relations and Functions 15 Example 4b – Use the vertical line test to determine if the relation is a function. NO!

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2.1 Relations and Functions 16 Example 4c – Use the vertical line test to determine if the relation is a function. NO!

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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.

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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

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2.1 Relations and Functions 19 Example 5 (continued) (9,1)

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2.1 Relations and Functions 20 Example 5 (continued)

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2.1 Relations and Functions 21 Assignment: p.65 (#9 – 16 all, 18 – 24 evens)

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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.)

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2.1 Relations and Functions 23 Example 6 To wash her brother’s 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 brother’s clothes.

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2.1 Relations and Functions C(x) = ____ + _____ x Then evaluate the function if it takes Jennifer 2½ 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

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2.1 Relations and Functions 25 Example 7 – Find the domain and range of each relation.

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2.1 Relations and Functions 26 Example 7a – Domain: x > 0 Range: ARN

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2.1 Relations and Functions 27 Example 7b – Domain: – 4 < x < 4 Range: – 4 < y < 4

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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.

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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.

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2.1 Relations and Functions 30 Example 8 (continued) – a.)

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2.1 Relations and Functions 31 Example 8 (continued) – b.) (h, 135.20)

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2.1 Relations and Functions 32 Assignment: p.65-66 (#25, 26, 29 – 33, 39 – 44, 48)

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2.2 Direct Variation A function where the ratio of output to input is called direct variation. 33

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2.2 Direct Variation 34 output input Constant of variation

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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

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2.2 Direct Variation Example 1 36 xy 1 3 7 21 9 3 YES! k = 3 So y = kx would mean y = 3x.

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2.2 Direct Variation Example 2 37 xy – 2 2 10 15 3 NO! – 3

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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

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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

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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

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2.2 Direct Variation Assignment: p.71(#7 – 10, 19 – 26) 41

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2.3 Linear Functions & Slope Intercept Form 42

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2.3 Linear Functions & Slope Intercept Form Example 1 – What is the slope of the line that passes through the given points? 43

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2.3 Linear Functions & Slope Intercept Form Example 1a – ( ‒ 10, 2) and (4, ‒ 5) 44

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2.3 Linear Functions & Slope Intercept Form Example 1b – (6, ‒ 1) and (5, ‒ 1) 45 0 in numerator

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2.3 Linear Functions & Slope Intercept Form Example 1c – ( ‒ 2, 5) and ( ‒ 2, 1) 46 0 in denominator The slope isUNDEFINED! 0 in denominator

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2.3 Linear Functions & Slope Intercept Form 47 Assignment: p.78 (#9-15)

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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.

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2.3 Linear Functions & Slope Intercept Form Example 2 – What is an equation of each line in slope-intercept form? 49

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2.3 Linear Functions & Slope Intercept form Example 2a – 50 Slope = – 3 y-intercept is (0,5)

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2.3 Linear Functions & Slope Intercept Form Example 2b – Slope = y-intercept = 51 up 2 over 3

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2.3 Linear Functions & Slope Intercept Form Example 3 – Write the equation in slope- intercept form. What are the slope and y-intercept? 52

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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

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2.3 Linear Functions & Slope Intercept Form 54

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2.3 Linear Functions & Slope Intercept Form Example 3b – 12 = 10y – 3x + 3x 12 + 3x = 10y 10 10 10 55 Slope = y-intercept =

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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

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2.3 Linear Functions & Slope Intercept Form Example 4 (continued) – 57 8

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Assignment: 58 p.78(#17-31 odds) 2.3 Linear Functions & Slope Intercept Form

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Example 5 – A horizontal line has slope 0. Graph y = – 5. m = 0 b = – 5 59 2.3 Linear Functions & Slope Intercept Form

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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

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Assignment: 61 p.78(#32 – 52 evens) 2.3 Linear Functions & Slope Intercept form

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2.4 More About Linear Equations Point-slope form 62

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2.4 More About Linear Equations Example 1 Use the given information to write an equation in point-slope form. 63

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2.4 More About Linear Equations a. slope = through (– 1, 3) 64

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2.4 More About Linear Equations b.) slope = 0 through (22, – 1) 65

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2.4 More About Linear Equations 66 c.) passing through (5, 1) and (7, 1)

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2.4 More About Linear Equations d.) passing through (4, – 1) and (– 6, 5) 67

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2.4 More About Linear Equations Slopes of parallel lines are equal / the same. Slopes of perpendicular lines are opposite reciprocals. 68

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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

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2.4 More About Linear Equations a.) parallel to y = x + 2 through (– 5, 3) 70

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2.4 More About Linear Equations b.) perpendicular to y = – 2x + 3 71

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2.4 More About Linear Equations Assignment: p.86-88 (#10 – 18, 32, 33, 65, 72, 74) 72

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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

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2.4 More About Linear Equations a.) parallel to 3x – 2y = 6 through (– 3, 5) 74

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2.4 More About Linear Equations b.) perpendicular to 4x + y = 1 through (2,1) 75

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2.4 More About Linear Equations Example 4 Find the intercepts, and graph the line. 76

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2.4 More About Linear Equations a.) 4x + 3y = 12 77

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2.4 More About Linear Equations b.) 4x – 5y = 10 78

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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

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2.4 More About Linear Equations 80 Example 5 A.) Sketch a graph that models this situation.

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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

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2.4 More About Linear Equations Assignment: p.86-87(#26-31,34-41) 82

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