Algebra 2 Chapter 2

1. Algebra 2 Chapter 2

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

3. 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 February 70F March 75F April 78F Represent this relation in the 4 ways.

4. 2.1 Relations and Functions • Table Month Temp 1 69 º F 70 º F 2 3 75 º F 4 78 º F

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

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

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

8. 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

9. 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

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

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

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

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

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

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

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

17. 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

18. 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

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

20. 2.1 Relations and Functions Example 5 (continued) 20

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

22. 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

23. 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

24. 2.1 Relations and Functions 15 3.50 • 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.

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

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

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

28. 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

29. 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

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

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

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

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

34. 2.2 Direct Variation output input Constant of variation

35. 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.

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

37. 2.2 Direct Variation • Example 2 x y NO! – 2 3 – 3 2 15 10

38. 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

39. 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

40. 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

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

42. 2.3 Linear Functions & Slope Intercept Form

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

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

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

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

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

48. 2.3 Linear Functions & Slope Intercept Form Slope-intercept Form where m is the slope of the line and (0, b) is the y-intercept.

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

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