3. Day 61. Opener Are these functions? domain range 2 3 7 4 9 6 10 7
4. Day 61. Opener Are these functions? a) domain range b) 2 3 7 4 9 6 10 7
5. Day 61. Opener Are these functions? a) domain range b) 2 3 7 4 9 6 10 7 b) What is the range here: {(-2, 3), (5, 1), (-2, 7), (-3, 8)}
6. Day 61. Opener Are these functions? a) domain range b) 2 3 7 4 9 6 10 7 b) What is the range here: {(-2, 3), (5, 1), (-2, 7), (-3, 8)} c) Give a line perpendicular to: 2y - x = 7
7. 2. Exercise
8. 2. Exercise Find the line perpendicular to 2y - x = 7 that goes through (2, 3). Sketch both lines.
9. 2. Exercise Find the line perpendicular to 2y - x = 7 that goes through (2, 3). Sketch both lines.
10. 2. Exercise What is the vertex of the following parabola y = (x + 3)2 + 4.
11. 2. Exercise What is the vertex of the following parabola y = (x + 3)2 + 4. V(-3, 4)
12. 2. Exercise What is the vertex of the following parabola y = -(x - 3)2 + 4.
13. 2. Exercise What is the vertex of the following parabola y = -(x - 3)2 + 4. V(3, 4)
14. 2. Exercise What is the equation of the following parabola?
15. 2. Exercise What is the equation of the following parabola? y = (x - 1)2 + 1.
16. 2. Exercise What is the equation of the following parabola?
17. 2. Exercise What is the equation of the following parabola? y = -(x - 1)2 + 1.
18. 2. Exercise What is the equation of the following parabola?
19. 2. Exercise What is the equation of the following parabola? y = (x + 2)2 - 3.
20. 2. Exercise Find the vertex, x and y intercepts and sketch the graph ofy = x2 - 6x + 8.
21. 2. Exercise Find the vertex, x and y intercepts and sketch the graph ofy = x2 - 6x + 8.
22. 3. Summarizing1. Slope 5. Parallel lines y2 − y1 m= m1 = m2 x2 − x12. General equation 6. Perpendicular lines Ax + By + C = 0 m1m2 = −13. Slope-intercept form 7. Standard form y = mx + b y = ax 2 + bx + c4. Point-slope equation 8. Vertex form 2 y − y1 = m ( x − x1 ) y = ( x − h) + k
23. Day 7 1. f (x) = −3x + 5 2. y = 2x − 4 2 3. g(x) = x − 9 2 4. y = ( x − 2 ) + 1
24. Day 7Opener 1. f (x) = −3x + 5 2. y = 2x − 4 2 3. g(x) = x − 9 2 4. y = ( x − 2 ) + 1
25. Day 7Opener Sketch the graph of the following functions: 1. f (x) = −3x + 5 2. y = 2x − 4 2 3. g(x) = x − 9 2 4. y = ( x − 2 ) + 1
26. Graphs of Functions.
27. Graphs of Functions.Vertically Shifting the graph of y = f(x)
28. Graphs of Functions.Vertically Shifting the graph of y = f(x) Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph vertically upward a vertically downward a distance c distance c
29. Graphs of Functions.Vertically Shifting the graph of y = f(x) Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph vertically upward a vertically downward a distance c distance c
30. Graphs of Functions.Vertically Shifting the graph of y = f(x) Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph vertically upward a vertically downward a distance c distance c y=x2
31. Graphs of Functions.Vertically Shifting the graph of y = f(x) Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph vertically upward a vertically downward a distance c distance c y=x2+4 y=x2
32. Graphs of Functions.Vertically Shifting the graph of y = f(x) Equation y = f(x) + c with c > 0 y = f(x) - c with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph vertically upward a vertically downward a distance c distance c y=x2+4 y=x2 y=x2-4
33. Graphs of Functions.
34. Graphs of Functions.Horizontally Shifting the graph of y = f(x)
35. Graphs of Functions.Horizontally Shifting the graph of y = f(x) Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph horizontally to the right horizontally to the left a distance c distance c
36. Graphs of Functions.Horizontally Shifting the graph of y = f(x) Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph horizontally to the right horizontally to the left a distance c distance c
37. Graphs of Functions.Horizontally Shifting the graph of y = f(x) Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph horizontally to the right horizontally to the left a distance c distance c y=x2
38. Graphs of Functions.Horizontally Shifting the graph of y = f(x) Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph horizontally to the right horizontally to the left a distance c distance c y=(x-4)2 y=x2
39. Graphs of Functions.Horizontally Shifting the graph of y = f(x) Equation y = f(x - c) with c > 0 y = f(x + c) with c > 0 The graph of f is shifted The graph of f is shifted Effect on Graph horizontally to the right horizontally to the left a distance c distance c y=(x-4)2 y=x2 y=(x+2)2
40. 1. f (x) = x + 4 32. y = ( x − 2 ) 13. g(x) = x−3
41. Examples. 1. f (x) = x + 4 3 2. y = ( x − 2 ) 1 3. g(x) = x−3
42. Examples. Sketch the graph of the following functions: 1. f (x) = x + 4 3 2. y = ( x − 2 ) 1 3. g(x) = x−3
43. Day 8
44. Day 8
45. Inverse Functions
46. Inverse FunctionsDefinition of One-to-One Function.
47. Inverse FunctionsDefinition of One-to-One Function.Horizontal Line Test.
48. Inverse FunctionsDefinition of One-to-One Function.Horizontal Line Test.A function f is one-to-one if and only if every horizontal line
49. Inverse FunctionsDefinition of One-to-One Function.Horizontal Line Test.A function f is one-to-one if and only if every horizontal lineintersects the graph of f in at most one point.
50. Example.Use the horizontal line test to determine if the function is one-to-one. 1. f (x) = 3x + 2 2 2. g(x) = x − 3
51. Definition of Inverse Function.Let f be a one-to-one function with domain D and range R. Afunction g with domain R and range D is the inverse function off, provided the following condition is true for every x in D andevery y in R: y = f (x) if and only if x = g(y)
52. Example.Let f(x) = x2 - 3 for x ≥ 0. Find the inverse function of f.
53. Example.Let f(x) = x2 - 3 for x ≥ 0. Find the inverse function of f.
54. Theorem on Inverse Functions.Let f be a one-to-one function with domain D and range R. If g isa function with domain R and range D, then g is the inversefunction of f if and only if both of the following conditions are true:(1) g(f(x)) = x for every x in D(2) f(g(y)) = y for every y in R
55. Homework 2.Find the inverse function of f 2 1. f (x) = 3+ x 2 2. f (x) = ( x + 2 ) , x ≥ −2 2x 3. f (x) = x −1 3x + 4 4. f (x) = 2x − 3 2x + 3 5. f (x) = x+2 6. f (x) = 2 3 x
56. Day 9Domain and Range of f and f-1domain of f-1 = range of frange of f-1 = domain of f
57. Guidelines for Finding f-1 in Simple Cases.1. Verify that f is a one-to-one function throughout its domain.2. Solve the equation y = f(x) for x in terms of y, obtaining anequation of the form x = f-1(y).3. Verify the following conditions:(a) f-1(f(x)) = x for every x in the domain of f(b) f(f-1(x)) = x for every x in the domain of f-1
58. Example. 4xFind the inverse function of f (x) = x−2
59. Example. 4xFind the inverse function of f (x) = x−2
60. Day 10Exercises.Find the inverse function of f. 1 1. f (x) = x+3 3x + 2 2. f (x) = 2x − 5
61. Quiz 2.Sketch the following functions: 2 1. y = ( x + 2 ) 2. y = x 3 − 1 3. State the domain and range of the following function
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