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Algebra 2 Chapter 2

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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

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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: January 69F February 70F March 75F April 78F Represent this relation in the 4 ways.

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**2.1 Relations and Functions**

Table Month Temp 1 69 º F 2 70 º F 3 75 º F 4 78 º F

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**2.1 Relations and Functions**

Ordered Pairs {( ), ( ), ( ), ( )} 1,69 2,70 3,75 4,78

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**2.1 Relations and Functions**

Mapping 1 69º F 2 70º F 3 75º F 4 78º F

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**2.1 Relations and Functions**

Graph 3 68 70 72 74 76 78 1 2 4

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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

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**2.1 Relations and Functions**

Example 2: Write the domain and range from example 1. Domain: { } Range: { } 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.

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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!

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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. 17

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**2.1 Relations and Functions**

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 18

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**2.1 Relations and Functions**

Example 5 (continued) (9,1) 19

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**2.1 Relations and Functions**

Example 5 (continued) 20

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

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.) 22

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**2.1 Relations and Functions**

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. 23

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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) = (2.5) C(2.5) = Jennifer will charge $23.75. 15 3.50

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**2.1 Relations and Functions**

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

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**2.1 Relations and Functions**

Example 7a – Domain: x > 0 Range: ARN 26

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**2.1 Relations and Functions**

Example 7b – Domain: – 4 < x < 4 Range: – 4 < y < 4 27

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**2.1 Relations and Functions**

Example 8 – The relationship between your weekly salary S and the number of hours worked h is described by the following function. 28

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**2.1 Relations and Functions**

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. 29

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**2.1 Relations and Functions**

Example 8 (continued) – a.) 30

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**2.1 Relations and Functions**

Example 8 (continued) – b.) (h, ) 31

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**2.1 Relations and Functions**

Assignment: p (#25, 26, 29 – 33, 39 – 44, 48) 32

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

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2.2 Direct Variation 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.

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**2.2 Direct Variation x y Example 1 1 3 3 9 7 21 YES! k = 3**

YES! 1 3 3 9 7 21 k = 3 So y = kx would mean y = 3x.

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2.2 Direct Variation Example 2 x y NO! – 2 3 2 – 3 10 15

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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? – 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. 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? = d = $112

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

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**2.3 Linear Functions & Slope Intercept Form**

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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?

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**2.3 Linear Functions & Slope Intercept Form**

Example 1a – (‒10, 2) and (4, ‒5)

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**2.3 Linear Functions & Slope Intercept Form**

Example 1b – (6, ‒1) and (5, ‒1) 0 in numerator

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**2.3 Linear Functions & Slope Intercept Form**

Example 1c – (‒2, 5) and (‒2, 1) 0 in denominator 0 in denominator The slope is UNDEFINED!

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**2.3 Linear Functions & Slope Intercept Form**

Assignment: p.78 (#9-15)

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**2.3 Linear Functions & Slope Intercept Form**

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?

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**2.3 Linear Functions & Slope Intercept form**

Example 2a – Slope = – 3 y-intercept is (0,5)

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**2.3 Linear Functions & Slope Intercept Form**

Example 2b – Slope = y-intercept = over 3 up 2

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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?

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**2.3 Linear Functions & Slope Intercept Form**

Example 3a 2x + 3y – 15 = 0 – 2x – 2x 3y – 15 = – 2x 3y = – 2x + 15

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**2.3 Linear Functions & Slope Intercept Form**

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**2.3 Linear Functions & Slope Intercept Form**

Example 3b – 12 = 10y – 3x + 3x x 12 + 3x = 10y Slope = y-intercept =

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**2.3 Linear Functions & Slope Intercept Form**

Example 4 – What is the graph of 24 = 4x + 3y? 24 = 4x + 3y –4x –4x – 4x + 24 = 3y

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**2.3 Linear Functions & Slope Intercept Form**

Example 4 (continued) – 8

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

Assignment: p.78(#17-31 odds)

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**2.3 Linear Functions & Slope Intercept Form**

Example 5 – A horizontal line has slope 0. Graph y = – 5. m = 0 b = – 5

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**2.3 Linear Functions & Slope Intercept Form**

Example 6 – The slope of a vertical line is UNDEFINED. Graph x = 3. Slope = undefined y-intercept = NONE

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

Assignment: p.78(#32 – 52 evens)

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**2.4 More About Linear Equations**

Point-slope form where m is the slope of the line passing through the point ( 𝒙 𝟏 , 𝒚 𝟏 ).

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**2.4 More About Linear Equations**

Example 1 Use the given information to write an equation in point-slope form.

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**2.4 More About Linear Equations**

a. slope = through (– 1 , 3)

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**2.4 More About Linear Equations**

b.) slope = 0 through (22, – 1)

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**2.4 More About Linear Equations**

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)

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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.

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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.

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**2.4 More About Linear Equations**

a.) parallel to y = x + 2 through (– 5, 3)

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**2.4 More About Linear Equations**

b.) perpendicular to y = – 2x + 3

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**2.4 More About Linear Equations**

Assignment: p (#10 – 18, 32, 33, 65, 72, 74)

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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.

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**2.4 More About Linear Equations**

a.) parallel to 3x – 2y = 6 through (– 3, 5)

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**2.4 More About Linear Equations**

b.) perpendicular to 4x + y = 1 through (2,1)

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**2.4 More About Linear Equations**

Example 4 Find the intercepts, and graph the line.

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**2.4 More About Linear Equations**

a.) 4x + 3y = 12

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**2.4 More About Linear Equations**

b.) 4x – 5y = 10

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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

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**2.4 More About Linear Equations**

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?

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**2.4 More About Linear Equations**

Assignment: p.86-87(#26-31,34-41)

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