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0203 ch 2 day 3

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0203 ch 2 day 3

1. 1. 2.2 Graphs of Functions2 Corinthians 4:16-18  So we do not lose heart. Thoughour outer self is wasting away, our inner self is beingrenewed day by day. For this light momentary afﬂiction ispreparing for us an eternal weight of glory beyond allcomparison,  as we look not to the things that are seenbut to the things that are unseen. For the things that areseen are transient, but the things that are unseen areeternal.
2. 2. Basic Functions you need to know the graphs of:
3. 3. Basic Functions you need to know the graphs of: Constant
4. 4. Basic Functions you need to know the graphs of: Constant Linear
5. 5. Basic Functions you need to know the graphs of: Constant Linear Quadratic
6. 6. Basic Functions you need to know the graphs of: Constant Linear Quadratic Cubic
7. 7. Basic Functions you need to know the graphs of: Constant Linear Quadratic Cubic Quartic
8. 8. Basic Functions you need to know the graphs of: Constant Linear Quadratic Cubic Quartic Square Root
9. 9. Basic Functions you need to know the graphs of: Constant Linear Quadratic Cubic Quartic Square Root Absolute Value
10. 10. Basic Functions you need to know the graphs of: Constant Linear Quadratic Cubic Quartic Square Root Absolute Value Inverse
11. 11. Basic Functions you need to know the graphs of: Constant Linear Quadratic There are others Cubic (trig, for example), but this is a very Quartic good start! Square Root Absolute Value Inverse
12. 12. We can use graphs to help us ﬁnd the range of a function
13. 13. We can use graphs to help us ﬁnd the range of a functiony = 16 − x 2 What is the D and R?
14. 14. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0
15. 15. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 2 x ≤ 16 { x : x ∈[ −4, 4 ]}
16. 16. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { x : x ∈[ −4, 4 ]}
17. 17. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]}
18. 18. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]} Did you recognize it?
19. 19. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]} Did you recognize it? 2 2 2 y = 16 − x
20. 20. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]} Did you recognize it? 2 2 2 y = 16 − x y 2 = 16 − x 2
21. 21. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]} Did you recognize it? 2 2 2 y = 16 − x y 2 = 16 − x 2 x 2 + y 2 = 16
22. 22. We can use graphs to help us ﬁnd the range of a function y = 16 − x 2 What is the D and R?D: 2 16 − x ≥ 0 R: Graph it and look! 2 x ≤ 16 { y : y ∈[ 0, 4 ]} { x : x ∈[ −4, 4 ]} Did you recognize it? 2 2 2 y = 16 − x y 2 = 16 − x 2 x 2 + y 2 = 16 Circle!
23. 23. Sketch a graph of this piecewise function: ⎧ x + 4, x ≤ 0 ⎪ 2 y = ⎨ 4 − x , 0 < x ≤ 2 ⎪−x + 4, x > 2 ⎩
24. 24. Sketch a graph of this piecewise function: ⎧ x + 4, x ≤ 0 ⎪ 2 y = ⎨ 4 − x , 0 < x ≤ 2 ⎪−x + 4, x > 2 ⎩so ... can this be done on the calculator?
25. 25. Sketch a graph of this piecewise function: ⎧ x + 4, x ≤ 0 ⎪ 2 y = ⎨ 4 − x , 0 < x ≤ 2 ⎪−x + 4, x > 2 ⎩so ... can this be done on the calculator? Google: syntax for ti-84 piecewise
26. 26. Sketch a graph of this piecewise function: ⎧ x + 4, x ≤ 0 ⎪ 2 y = ⎨ 4 − x , 0 < x ≤ 2 ⎪−x + 4, x > 2 ⎩so ... can this be done on the calculator? Google: syntax for ti-84 piecewise http://www.tc3.edu/instruct/sbrown/ti83/funcpc.htm
27. 27. Sketch a graph of this piecewise function: ⎧ x + 4, x ≤ 0 ⎪ 2 y = ⎨ 4 − x , 0 < x ≤ 2 ⎪−x + 4, x > 2 ⎩ so ... can this be done on the calculator? Google: syntax for ti-84 piecewise http://www.tc3.edu/instruct/sbrown/ti83/funcpc.htmand do it on the calculator and discuss how dot mode is best for this
28. 28. Absolute Value is a piecewise function
29. 29. Absolute Value is a piecewise function ⎧ x, x ≥ 0 x = ⎨ ⎩−x, x < 0
30. 30. Absolute Value is a piecewise function ⎧ x, x ≥ 0 x = ⎨ ⎩−x, x < 0do on your calculator using piecewise method
31. 31. Greatest Integer Function
32. 32. Greatest Integer Function y = [ x]
33. 33. Greatest Integer Function y = [ x]the output is the greatest integer less than or equal to the input
34. 34. Simplify
35. 35. Simplify1. y = [ x ] if x=3.2
36. 36. Simplify1. y = [ x ] if x=3.2 y = [ 3.2 ] y=3
37. 37. Simplify1. y = [ x ] if x=3.2 y = [ 3.2 ] y=32. y = [ x ] if x=3.9
38. 38. Simplify1. y = [ x ] if x=3.2 y = [ 3.2 ] y=32. y = [ x ] if x=3.9 y = [ 3.9 ] y=3
39. 39. Simplify1. y = [ x ] if x=3.2 y = [ 3.2 ] y=32. y = [ x ] if x=3.9 y = [ 3.9 ] y=33. y = [ x ] if x=4
40. 40. Simplify1. y = [ x ] if x=3.2 y = [ 3.2 ] y=32. y = [ x ] if x=3.9 y = [ 3.9 ] y=33. y = [ x ] if x=4 y = [4] y=4
41. 41. Simplify1. y = [ x ] if x=3.2 4. y = [ x ] if x=-3.2 y = [ 3.2 ] y=32. y = [ x ] if x=3.9 y = [ 3.9 ] y=33. y = [ x ] if x=4 y = [4] y=4
42. 42. Simplify1. y = [ x ] if x=3.2 4. y = [ x ] if x=-3.2 y = [ 3.2 ] y = [ −3.2 ] y=3 y = −42. y = [ x ] if x=3.9 y = [ 3.9 ] y=33. y = [ x ] if x=4 y = [4] y=4
43. 43. Simplify1. y = [ x ] if x=3.2 4. y = [ x ] if x=-3.2 y = [ 3.2 ] y = [ −3.2 ] y=3 y = −42. y = [ x ] if x=3.9 5. y = [ x ] if x=-3.9 y = [ 3.9 ] y=33. y = [ x ] if x=4 y = [4] y=4
44. 44. Simplify1. y = [ x ] if x=3.2 4. y = [ x ] if x=-3.2 y = [ 3.2 ] y = [ −3.2 ] y=3 y = −42. y = [ x ] if x=3.9 5. y = [ x ] if x=-3.9 y = [ 3.9 ] y = [ −3.9 ] y=3 y = −43. y = [ x ] if x=4 y = [4] y=4
45. 45. Simplify1. y = [ x ] if x=3.2 4. y = [ x ] if x=-3.2 y = [ 3.2 ] y = [ −3.2 ] y=3 y = −42. y = [ x ] if x=3.9 5. y = [ x ] if x=-3.9 y = [ 3.9 ] y = [ −3.9 ] y=3 y = −43. y = [ x ] if x=4 Graph y = [ x ] on the calculator y = [4] in dot and connected modes; then on board by hand; discuss y=4
46. 46. What test do we use to determine if a graph is afunction?
47. 47. What test do we use to determine if a graph is afunction? Vertical Line TestIt veriﬁes that for a function there is only oney-value for any given x-value.
48. 48. Find the Domain and Range. Discuss. 2 x − 2x1. f (x) = x−2
49. 49. Find the Domain and Range. Discuss. 2 x − 2x1. f (x) = x−2behaves like y=xhole at x=2D: { x : x ≠ 2}R: { y : y ≠ 2}
50. 50. Find the Domain and Range. Discuss. 2 x − 5x + 62. f (x) = 2 x − 2x + 1
51. 51. Find the Domain and Range. Discuss. 2 x − 5x + 6 ( x − 3)( x + 2 )2. f (x) = 2 f (x) = x − 2x + 1 ( x − 1)( x − 1)
52. 52. Find the Domain and Range. Discuss. 2 x − 5x + 6 ( x − 3)( x + 2 )2. f (x) = 2 f (x) = x − 2x + 1 ( x − 1)( x − 1)V.A. at x=1
53. 53. Find the Domain and Range. Discuss. 2 x − 5x + 6 ( x − 3)( x + 2 )2. f (x) = 2 f (x) = x − 2x + 1 ( x − 1)( x − 1)V.A. at x=1but look at the graph for 2<x<3 ... why is that blank?
54. 54. Find the Domain and Range. Discuss. 2 x − 5x + 6 ( x − 3)( x + 2 )2. f (x) = 2 f (x) = x − 2x + 1 ( x − 1)( x − 1)V.A. at x=1but look at the graph for 2<x<3 ... why is that blank?(fraction is negative ... can’t have that with ABS)
55. 55. Find the Domain and Range. Discuss. 2 x − 5x + 6 ( x − 3)( x + 2 )2. f (x) = 2 f (x) = x − 2x + 1 ( x − 1)( x − 1)V.A. at x=1but look at the graph for 2<x<3 ... why is that blank?(fraction is negative ... can’t have that with ABS)D: { x : x ∈( −∞,1) U (1,2 ) U ( 3,∞ )}R: { y : y ∈[ 0,∞ )}
56. 56. HW #3“See things as you would have them instead ofhow they are.” Robert Collier