Circles and Additional Curves

1
Chapter 4

• Circles and Additional Curves

2
Definition

• Circle The set of points on a plane equidistant
  from a central point.

3
Distance formula

• The distance formula gives the distance between
  two points and
• AB

4
CE

• Find the length of the line segment connecting
  A(-2,2)
• And B(6,-4)

5
CE

• Find the length of the line segment connecting
  A(0,0)
• And B(1,1)

6
CE

• Find the length of the line segment connecting
  A(0,0)
• And B(-1,-1)

7

• Standard Form for of a circle with center at the
  origin and radius r

8

• Standard form for the equation of a circle with
  center at (h,k) and radius r

9
CE

• Find the center and the radius of a circle with
  this equation, Graph

10
CE

• Find the center and the radius of a circle with
  this equation

11
CE

• Find the center and the radius of a circle with
  this equation, Graph

12
Page 251

• Test Your Understanding
• 1-10

13
Page 254

• 1- 2

14

• NoteIf an equation of a circle is not in
  standard form You must complete the Square to
  force the equation to be in standard form.

15
CECompleting the Square

• Complete the square on both x and y to write the
  equation in standard form.

16
CEFind the center, radius, and graph
17
Midpoint Formula

• Use the midpoint formula when you have two
  endpoints of a diameter of a circle. This will
  give you the center.

18
.

• Use the distance formula with the center and an
  endpoint. This will give you the radius.

19
Midpoint Formula
20
CEFind the center, radius, and equation of the
circle when

• Points A(2,5) and Q(-4,-3) are endpoints of a
  diameter of a circle.

21
CEFind the center, radius, and equation of the
circle when

• Points A(0,2) and Q(8,2) are endpoints of a
  diameter of a circle.

22
HW

• Page 254 3 13 odd
• Page 255 29,31

23
Section 4.2

• Graphing Radical Functions

24
CE

• Graph

25
CE

• Graph
• Find the domain and range

26
CE

• Graph
• Find the domain and range of

27
ce

• Graph

28
ce

• Graph

29
Graphing Calculators

• Set range to following specifications
• Xmin 5 xmax 5
• Ymin 5 ymax 5

30
Graph on the same screen
31
Graph on the same screen
32
Graph on the same screen
33
Graph on the same screen
34

• Page 263 1- 3
• Page 264 1 7 odd

35
Section 4.4

• Combining Functions

36
Combining functions

• Just as we can add, subtract, multiply, and
  divide numbers, we can do the same with
  functions.

37
Section 4.4

• Combining and Decomposing Functions.

38

• We can
• Add
• Subtract
• Multiply
• Divide functions.

39
ce

• Let f(x) x 4
• Let g(x) x 5
• Find f(x) g(x)

40
ce

• Let f(x) 4x-7
• Let g(x)
• Find a. f g b. f g
• c. fg d. g f

41
Composition

• Plugging a function into another function.
• Noted with a
• I.e f g

42
CE

• Let f(x) 8 3x
• g(x)
• Find a. f g
• b. g f

43
ce

• Let f(x) 3x-1
• Let g(x)
• Find f g
• Find g f

44
ce

• Let f(x)
• Let h(x)
• Find f h
• Find h f

45

• Worksheet

46
Inverse functions

• Section 4.5

47
Definition

• One-to-one Functions
• a function f is one-to-one if and only if for
  each range value there corresponds exactly one
  domain value.

48
Horizontal Line test

• A function f is one-to-one if and only if the
  horizontal lines through the range values
  intersect the graph of f exactly once.

49
ce

• Determine if f is one-to-one
• _f(x) x3

50
ce

• Determine if f is a one-to-one function
• F(x)

51
ce

• Determine if f is a one-to-one function
• F(x)

52
Definition Inverse Function

• Two functions f and g are said to be inverse
  functions if and only if
• (f g)(x) x and
• (g f)(x) x

53
To Find the inverse of a function

• 1. Begin with y f(x)
• 2. Switch the variables y,x
• 3. Solve for y in terms of x to obtain the
  inverse.

54
ce

• Find the inverse g of
• F(x) 2x3, then show
• (f g)(x) x and
• (g f)(x) x
• Graph

55
ce

• Find the inverse g of
• F(x) y 7x-3, then show
• (f g)(x) x and
• (g f)(x) x
• Graph

56
ce

• Find the inverse g of
• F(x) y -x7, then show
• (f g)(x) x and
• (g f)(x) x
• Graph

57
HW Page 295

• 15,16,20,24

58
Chapter 5

• Exponential and Logarithmic Functions

59
Exponential Function

• For b gt 0, the equation
• Defines an exponential function with base b and
  domain all real numbers.

60
CEUse a table to graph
61
Ce Use a table to graph
62
CeUse a table to graph
63
Ce Use a table to graph
64
One to One Property

• If
• Then a b

65
ce

• Solve for x

66
CeSolve for x
67
CESolve for x
68
Hw

• Page 312 test your understanding 2-5
• Exercise 5.1
• 1 5, 19 - 21

69
Section 5.2

• Logarithmic Functions

70
Logarithm

• Use log to solve equations in the form for y.
  

71

• The equation can be manipulated as follows
• That is y equals log x base b.

72
ce

• Convert to
• Logarithmic form

73
ce

• Convert to logarithmic form

74
ce

• Convert to logarithmic

75
ce

• Convert from logarithmic to exponential form.

76
ce

• Convert from logarithmic to exponential form.

77
ce

• Hw Page 321
• 17-33 odd
• 37,39