Chapter 2 Functions , Special Functions, and Translations

1. Chapter 2Functions, Special Functions, and Translations Unit Essential Questions: How do step functions apply to postage rates? How do you use transformations to help graph absolute value functions? How can you model data with linear equations?

2. Section 2.1 Part 1Relations and Functions Students will be able to graph relations Students will be able to identify functions

3. Warm Up Graph each ordered pair on the coordinate plane. • (-4, -8) • (3, 6) • (0, 0) • (-1, 3)

4. Key Concepts Relation - a set of pairs of input and output values. -3 -1 3 3 4 4

5. Example 1 When skydivers jump out of an airplane, they experience free fall. At 0 seconds, they are at 10,000ft, 8 seconds, they are at 8976ft, 12 seconds, they are at 7696ft, and 16 seconds, they are at 5904ft. How can you represent this relation in four different ways?

6. Example 1 (Continued) • 0 10000 • 8976 • 12 9696 • 16 5904

7. Key Concepts Domain - the set of all inputs (x-coordinates) Range - the set of all outputs (y-coordinates) Function- a relation in which each element in the domain corresponds to exactly one element of the range. Vertical Line Test- if any vertical line passes through more than one point on the graph of a relation, then it is not a function.

8. Example 2 Determine whether each relation is a function. State the domain and range. {(0, 1), (1, 0), (2, 1), (3, 1), (4, 2)} Domain: {0, 1, 2, 3, 4} Yes, it is a function. Range: {0, 1, 2} b) {(1, 4), (3, 2), (5, 2), (1, -8), (6, 7)} Domain: {1, 3, 5, 6} No, it is not a function. Range: {-8, 2, 4, 7} c) {(1, 3), (2, 3), (3, 3), ( 4, 3), ( 5, 3)} Domain: {1, 2, 3, 4, 5} Yes, it is a function. Range: {3} d) {(4, 9), (4, 3), (4, 0), (4, 4), (4, 1)} Domain: {4} No, it is not a function. Range: {0, 1, 3, 4, 9}

9. Example 3 Use the vertical line test. Which graphs represent a function? Not a Function Function Not a Function

10. Section 2.1 Part 2Relations and Functions Students will be able to write and evaluate functions

11. Warm Up • Can you have a relation that is not a function? yes, a relation is any set of pairs of input and output values. • Can you have a function that is not a relation? no, a function is a relation.

12. Key Concepts Function Rule- an equation that represents an output value in terms of an input value. You can write the function rule in function notation. Independent Variable- x, represents the input value. Dependent Variable- y, represents the output value. (Call dependent because it depends on the input value)

13. Output value Input value Output value Input value ƒ(x) =5x+ 3 ƒ(1) =5(1)+ 3 ƒ of 1 equals 5 times 1 plus 3. ƒ of x equals 5 times x plus 3.

14. The function described by ƒ(x) = 5x + 3 is the same as the function described by y = 5x + 3. And both of these functions are the same as the set of ordered pairs (x, 5x+ 3). y = 5x + 3 (x, y) (x, 5x + 3) Notice that y = ƒ(x) for each x. ƒ(x) = 5x + 3 (x, ƒ(x)) (x, 5x + 3) The graph of a function is a picture of the function’s ordered pairs.

15. Example 1 Evaluate each function for the given value of x, and write the input x and output f (x) in an ordered pair. f (x) = -2x + 11 for x = 5, -3, and 0 f(5) = -2(5) + 11 f(5) = 1 f(-3) = -2(-3) + 11 f(-3) = 17 f(0) = -2(0) + 11 f(0) = 11 (5, 1) (-3, 17) (0, 11)

16. Example 2 Write a function rule to model the cost per month of a long-distance cell phone calling plan. Then evaluate the function for given number of minutes. Monthly service fee: $4.99 Rate per minute: $.10 Minutes used: 250 minutes

17. Section 2.2Direct Variation Students will be able to write and interpret direct variation equations

18. Warm Up Solve each equation for y.

19. Key Concepts Direct Variation- a linear function defined by an equation of the form y=kx, where k ≠ 0. Constant of Variation- k, where k = y/x.

20. a) b) x –1 2 5 y 3 –6 15 x 7 9 –4 y 14 18 –8 Example 1 For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. Does not vary directly k = 2 y = 2x

21. Example 2 For each function, tell whether y varies directly with x. If so, find the constant of variation. a. 3y = 7x + 7 b. 5x = –2y

22. Example 3 Suppose y varies directly with x, and y = 15 when x = 27. Write the function that models the variation. Find y when x = 18. Write the function that models the variation. Find y when x = 18. Find k.

23. 2.6 Special Functions Part 1 Step functions Greatest integer functions Piecewise functions

24. The symbol [| x |] means The greatest integer less than or equal to x. Therefore, [|7.5|]= 7 [|-1.5|]= -2 because -1 > -1.5 [|1.5|]= 1

25. Step functions: A range of values give a certain outcome. Example: Your grades are based on a step function. Grade Scale Letter grades have the following percentage equivalents: A+ 99-100 B+ 91-92 C+ 83-84 D+ 75-76 F 0- 69 A 96-98 B 88-90 C 80-82 D 72-74 A- 93-95 B- 85-87 C- 77-79 D- 70-71

26. Greatest Integer Function is a step function The function is written as It is not an absolute value. The function rounds down to the last integer.

27. Example :Find the value of a number in the Greatest Integer function f(x) =[| x |] f(2.7) = 2 f(0.8) = 0 f(- 3.4) = - 4 It rounds down to the last integer Find the value f( 5.8) = f(⅛) = f(- ⅜) =

28. A step function graph

29. Each step has a closed circle on one side of the step and an open circle on the other.

30. How to graph a step function;f(x)= [| x |] Find the values of x = .., -2, -1, 0, 1, 2, …… f(-2) = -2 f(-1) = - 1 f(0) = 0 f(1) = 1 f(2) = 2

31. Now lets look at 0.5,1.5, -0.5, -1.5 f(-1.5) = -2 It is the same as f( - 2) = -2 f(-0.5) = - 1 f( - 1) = -1 f(0.5) = 0 f(0) = 0 f(1.5) = 1 f(1) = 1 So between 0 and almost 1 it equal 0 f(0.999999999999999999999) = 0

32. How to show all those number equal 0 A close circle at (0, 0) and an open circle at (1, 0). (1, 0) What happens when x = 1?

33. How to show all those number equal 0 A close circle at (0, 0) and an open circle at (1, 0). (1,1) (2,1) (1, 0) What happens when x = 1? It jumps to (1,1)

34. Is the step only one unit long? It will be in f(x) = [| x |]. Here is how I graph them. Find the fill in circles. Draw line segments ending in a open circle.

35. Piecewise Function A piecewise function is any function that is in, well, pieces! Piecewise functions indicate intervals for each part of the function Graph f(x) =

36. y f(x) = f(x) = 1 x

37. y f(x) = f(x) = {1 x < 3 x 3

38. y f(x) = x + 1 f(x) = x

39. y f(x) = f(x) = {x+1 x > 3 x 3

40. Summary of Steps for our example f(x) =

41. Piecewise Functions What to graph Where to graph

42. Graph the piecewise function

43. Section 2.6 Part 2Families of Functions Students will be able to analyze transformations of functions

44. Warm Up Graph each pair of functions on the same coordinate plane. • y = -x 2) y = x + 1 3) y = -1/4x y = x y = 2x - 1 y = -1/4x + 2

45. Key Concepts Parent Function- the simplest form in a set of functions that form a family

46. The Identity Function

47. Constant Function • m = 0 • Horizontal line • f(x) = b

48. The Absolute Value Function

49. The Squaring Function

50. The Cubing Function