2'1 Relations and Functions

1
2.1 Relations and Functions

• Objective
• To graph a relation, state its domain and range,
  and determine if it is a function.
• To find the value of functions for given elements
  of the domain.

2

• Vocabulary
• Domain- all the x-values of the function
• Range- all the y-values of the function
• Function- is a special type of relation in which
  each element of the domain is paired with exactly
  one element from the range.
• Vertical Line Test- used to determine whether a
  relations is a function or not
• Discrete- when the domain of the function has a
  set number of solutions
• Continuous- when the domain of the function has
  infinite number of solutions

3

• One-to-one function- each element of the domain
  pairs to exactly one unique element of the range
• Domain Range
• 1 D
• 2 B
• 3 C
• A

Onto Function- each element of the range
corresponds to an element of the
domain Domain Range 1 D 2 B 3 C 4
Both one-to-one and onto- each element of the
domain is paired to exactly one element of the
range, and each element of the range corresponds
to a unique element of the domain.
Domain Range 1 D 2 B 3 C 4 A
4

• Find the domain and range. State if each relation
  is a function or not.
• (1) (2) x y
• 10 1 .5 -3
• 20 2 2 .8
• 30 3 .5 .8
• Also, if it is a function state whether it is
  one-to-one, onto, or both one-to-one and onto.

5

• Find the domain and range. State if each
  relation is a function or not. Interval notation.
• (3) (4)

6

• Graph each relation. Find the domain and range
  (interval notation). State if each relation is a
  function or not and state whether it is discrete
  or continuous
• (5) (4, 5), (6, 5), (3, 5) (6)

7

• Find each value if f(x) 4x 10 and
• (7) f(3) (8) g(-1/3) (9) f(a 2)

8

• Pre-calculus screening questions.
• (10) Over what interval is this function
  increasing, decreasing, and
• constant?

y f(x)
(-5, 5)
(0, 5)
(-9, 2)
(7, 0)
(3, 0)
(5, -2)
9

• Pre-calculus screening questions.
• (11) Find the domain and the range of the
  function?

y f(x)
(-5, 5)
(0, 5)
(-9, 2)
(7, 0)
(3, 0)
(5, -2)
10

• Pre-calculus screening questions.
• (12) What is the value of f(7)?
• What is the value of f(-5)?

y f(x)
(-5, 5)
(0, 5)
(-9, 2)
(7, 0)
(3, 0)
(5, -2)
11

• Pre-calculus screening questions.
• (13) The value of f(x) 5, find x?
• The value of f(x) 2, find x?

y f(x)
(-5, 5)
(0, 5)
(-9, 2)
(7, 0)
(3, 0)
(5, -2)
12

• Pre-calculus screening questions.
• For what values of x is f(x) lt 0?
• For what values of x is f(x) gt 0?

y f(x)
(-5, 5)
(0, 5)
(-9, 2)
(7, 0)
(3, 0)
(5, -2)
13

• 2.1 Assignment
• 2.1 Worksheet and
• Page 65, (11 13), (15 20), (21 22), (25
  31 odd), 47
• List domain and range in interval notation.

14
2.2 Linear Equations

• Objective
• To identify equations that are linear and graph
  them.
• To write linear equations in standard form
• To determine the intercepts of a line and use
  them to graph an equation

15

• Vocabulary
• Standard Form- x and y are on the same side of
  the equation with no fractions or decimals. Ax
  By C
• Linear Equations- an equation whose graph is a
  line and it can be written is standard form
• Slope Intercept form- the equation of a line,
• y mxb
• X-intercept- where the graph crosses the x-axis
• Y-intercept- where the graph crosses the y-axis

16

• State whether each equation is linear or not. An
  equation is only linear if it can be written in
  standard form (Ax By C)
• (1) (2) h(x) 1.1 2x (3)
• (4) g(x) 5 (5) 2x 4y 10 (6) yx 4y 10

17

• Write each equation in standard form.
• (7) (8) y 6 0 (9)

18

• Find the x-intercept and the y-intercept of each
  graph.
• (10) 2x y 5 (11) (12) b 2a 7

19

• 2.2 Assignment
• Page 72, (17 23 odd), (26 40 even), 45, 47,
  65, 71

20
2.3A Slope

• Objective
• To determine the slope of a line
• To use slope and a point to graph an equation
• To determine if two lines are parallel,
  perpendicular, or neither
• To solve problems by identifying and using a
  pattern.

21

• Vocabulary
• Slope Formula-
• Parallel Lines- have the same slope
• Perpendicular Lines- have a negative reciprocal
  slope
• Not Parallel or Perpendicular lines- dont have
  the same slope or negative reciprocal slope

22

• Find the slope and determine if the slope rises
  to the right, falls to the right, is horizontal,
  or is vertical.
• (1) (-3, -1), (5, 7) (2) (6, 4), (3, 4) (3) (8,
  2), (8, -1)

23

• Find the slope of each.
• (4) 4x 8 0 (5) 3x 2y 12 (6) 4y 10 15

24

• Find the value of r so the given slope would go
  through the 2 points.
• (7) (5, 0), (r, 9) m 3 (8) (r, 3), (7, 7)
  slope undefined

25

• (9) Graph a line through (-2, 2) that is parallel
  to a line whose slope is 1

26

• (10) Graph a line through (3, 3) that is
  perpendicular to the graph of y 3

27

• (11) Graph a line through (-4, -2) that has an
  undefined slope.

28

• (12) Graph the line perpendicular to the graph of
  3x 2y 24 that intersects it at its
  x-intercept.

29

• 2.3A Assignment
• Page 80 (13 17 odd), 31, 33, 46, 49

30
2.3B The Average Rate of Change

• Objective
• Obtain information from/about the Graph of a
  Function

31

• 1) Suppose you drop a ball from a cliff 1000 feet
  high. You measure the distance (s) the ball has
  fallen after time (t) using a motion detector and
  obtain the data
• Time, t(seconds) Distance, s(feet) Scatter
  Plot
• 0 0
• 1 16
• 2 64
• 3 144
• 4 256
• 5 400
• 6 576
• 7 784
• a) Draw a line from point (0, 0) to (2, 64). Find
  the average rate of change of the ball between 0
  and 2 seconds and interpret the result.

32

• 2) b) Draw a line from point (5, 400) to (7,
  784). Find the average rate of change of the ball
  between 5 and 7 seconds and interpret the result.
• Time, t(seconds) Distance, s(feet) Scatter
  Plot
• 0 0
• 1 16
• 2 64
• 3 144
• 4 256
• 5 400
• 6 576
• 7 784
• c) What is happening to the average rate of
  change as time passes?

33

• 3) d) This function can be represented as s
  f(t). Where s is distance and t is time. Find the
  average rate of change between 4 and t.
• Time, t(seconds) Distance, s(feet)
• 0 0
• 1 16
• 2 64
• 3 144
• 4 256
• 5 400
• 6 576
• 7 784

34

• Here is a graph of the average rate of change.
  Would this find the exact rate of change? What
  kind of functions will the average rate of change
  find the exact rate of change?

35
(No Transcript)
36
(No Transcript)
37

• 2.3B Assignment
• 2.3B Worksheet (1 5)

38
2.4 Writing Linear Equations

• Objective
• To write an equation of a line in slope-intercept
  form given the slope and one or two points.
• To write an equation of a line that is parallel
  or perpendicular to the graph of a given equation

39

• Vocabulary
• Slope Intercept form- y mx b
• Point Slope form-
• Parallel Lines- have the same slope
• Perpendicular Lines- have a negative reciprocal
  slope
• Not Parallel or Perpendicular lines- dont have
  the same slope or negative reciprocal slope

40

• State the slope and y-intercept of the graph of
  each equation.
• (1) 4x 7y 12 (2) 2x -8 (3) 7y -15

41

• Write the equation in slope-intercept form for
  each graph.
• (4)

42

• Write the equation in slope-intercept form for
  each graph.
• (5)

43

• Write the equation in slope-intercept form.
• (6) Slope 4/5 passes through (10, -3)

44

• Write the equation in slope-intercept form.
• (7) Passes through (4, 3) and (7, -2)

45

• Write the equation in slope-intercept form.
• (8) x-intercept ½, y-intercept 4

46

• Write the equation in slope-intercept form.
• (9) Passes through (4, 6), parallel to the graph
  of

47

• Write the equation in slope-intercept form.
• (10) Passes through (3, 2), perpendicular to the
  graph of y 2x 5

48

• Find the value of k.
• (11) 5x ky 8, (3, -1)

49

• 2.4 Assignment
• Page 87, (9 15 odd), 18, 19, 23, 25, 27, 30,
  31, 50, 61

50
2.5 Scatter Plots

• Objective
• To draw scatter plots
• To find and use prediction equations

51
1a. Draw a scatter plot using linear regression
of the data on the graph below to show the
relationship between distance (d) and time (t).
52

• 1b. Find a prediction equation to show how the
  distance and the time are related.
• 1c. Find the time needed to travel 300 km.
• 1d. In a sentence, describe the real-world
  meaning of the slope in the context of this
  problem situation.

53

• 2a. Draw a scatter plot using linear regression
  of the data on the graph below to show the
  relationship between studying (x) and grade (y).

54

• 2b. Find a prediction equation to show how the
  study time and the grades are related.
• 2c. Find out how much studying is needed for
  97.
• 2d. In a sentence, describe the real-world
  meaning of the slope in the context of this
  problem situation.

55

• 2.5 Assignment
• Page 97, questions 9, 10a, 22, 25, 30

56
2.6 Special Functions

• Objective
• To identify and graph special functions

57

• Vocabulary
• Direct Variation- a linear function in the form
  of
• y mx b has b 0 and m 0.

58

• Vocabulary
• Constant Function Identity Function
• Which one of these functions is also direct
  variation?

F(x)
F(x)
F(x) x
F(x) 2
(0, 2)
x
x
59

• Vocabulary
• Step Function- Greatest Integer Function

F(x)
x
60

• Graphing absolute value functions and compare the
  differences.
• (1) (2)
• (graph on next slide)

61

• Transformations Compare each equation with the
  parent function.
• (3)
• (graph on next slide)

62

• Graphing absolute value functions and compare
  the differences to the original.
• (1)

63

• Graphing absolute value functions and compare
  the differences to the original.
• (2)

64

• Graphing absolute value functions and compare
  the differences to the original.
• (3)

65

• Evaluate.
• (4) (5) (6) (7) (8)

66

• Graph each function.
• (9)

67

• Graph each function.
• (10)

68

• (11) h(1/4)

69

• 2.6 Assignment
• Page 105 questions 20, 21, 25, 27, 33, 49 a c,
  50, 51, 54

70
2.7 Parent Functions and Transformations

• Objectives
• Identify and use the parent functions.
• Describe transformations of functions.

71

• Parent Functions
• Constant Function Identity Function
• Absolute Value Function Quadratic Function

72

• Identify the type of function represented by each
  graph.
• 1) 2)
• 3) 4)

73

• Transformations of Functions
• Translation
• F(x h) Translates graph h units left.
• F(x h) Translates graph h units right.
• F(x) k Translates graph h units up.
• F(x) k Translates graph h units down
• Reflection
• -F(x) Reflects graph in the x-axis
• F(-x) Reflects graph in the y-axis

74

• Transformations of Functions
• Dilation
• Af(x), A gt 1 Stretches graph vertically
• Af(x), 0 lt A lt 1 Compresses graph vertically
• F(Ax), A gt 1 Compresses graph horizontally
• F(Bx), 0 lt A lt 1 Stretches graph horizontally

75

76

77

• Describe the transformation in each function. The
  graph the function.
• 7) Y (x 2) 5 8) 9)

78

• Write an equation for each function.
• 10)

79

• 2.7 Assignment
• Page 114 (15 37 odd), 41, 52, 55a c, 57

80
2.8 Linear Inequalities

• Objective
• To draw graphs of inequalities in two variables.

81

• Graph each function.
• (1) 2y 3x lt 6

82

• Graph each function.
• (2)

83

• Graph each function.
• (3)

84

• (4) Graph all the points on the coordinate plane
  to the left of x -2. Write an inequality to
  describe these points.

85

• (5) Graph all 2nd quadrant points bounded by the
  lines x -3, x -6, and y 4

86

• 2.8 Assignment
• Page (8 12 even), (15 19 odd), 21, 23, 24,
  27, 43, 50

87
2.9 Piecewise Functions

• Objective
• To solve piecewise functions

88
(No Transcript)
89
(No Transcript)
90
(No Transcript)
91
(No Transcript)
92

• 2.9 Assignment
• 2.9 Worksheet