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FUNCTIONS AND GRAPHS

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Title: FUNCTIONS AND GRAPHS


1
FUNCTIONS AND GRAPHS
2
Aim 1.2 What are the basics of functions and
their graphs?
  • Lets Review
  • What is the Cartesian Plane or Rectangular
    Coordinate Plans?
  • How do we find the x and y-intercepts of any
    function?
  • How do we interpret the viewing rectangle
  • -10,10, 1 by -10, 10,1?

3
IN THIS SECTION WE WILL LEARN
  • How to find the domain and range?
  • Determine whether a relation is a function
  • Determine whether an equation represents a
    function
  • Evaluate a function
  • Graph functions by plotting points
  • Use the vertical line to identify functions

4
WHAT IS A RELATION?
  • A relation is a set of ordered pairs.
  • Example (4,-2), (1, 2), (0, 1), (-2, 2)
  • Domain is the first number in the ordered pair.
    Example(4,-2)
  • Range is the second number in the ordered pair.
  • Example (4,30)

5
EXAMPLE 1
  • Find the domain and range of the relation.
  • (Smith, 1.0006), (Johnson, 0.810), (Williams,
    0.699), (Brown, 0.621)

6
PRACTICE
  • Find the domain and range of the following
    relation.

7
HOW DO WE DETERMINE IF A RELATION IS A FUNCTION?
  • A relation is a function if each domain only has
    ONE range value.
  • There are two ways to visually demonstrate if a
    relation is a function.
  • Mapping
  • Vertical Line Test

8
DETERMINE WHETHER THE RELATION IS A FUNCTION
  • (1, 6), (2, 6), (3, 8), (4, 9)
  • (6, 1), (6, 2), (8, 3), (9, 4)

9
FUNCTIONS AS EQUATIONS
  • Functions are usually given in terms of equations
    instead of ordered pairs.
  • Example y 0.13x2 -0.21x 8.7
  • The variable x is known the independent variable
    and y is the dependent variable.

10
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A
FUNCTION?
  • x2 y 4
  • Steps
  • Solve the equation for y in terms of x.
  • Note
  • If two or more y values are found then the
    equation is not a function.

11
HOW DO WE DETERMINE IF AN EQUATION REPRESENTS A
FUNCTION
  • x2 y2 4
  • Steps
  • Solve the equation for y in terms of x.
  • Note
  • If two or more y values are found then the
    equation is not a function.

12
PRACTICE
  • Solve each equation for y and then determine
    whether the equation defines y as a function of
    x.
  • 2x y 6
  • x2 y2 1

13
WHAT IS FUNCTION NOTATION?
  • We use the special notation f(x) which reads as f
    of x and represents the function at the number x.
  • Example f (x) 0.13x2 -0.21x 8.7
  • If we are interested in finding f (30), we
    substitute in 30 for x to find the function at
    30.
  • f (30) 0.13(30)2 -0.21 (30) 8.7
  • Now lets try to evaluate using our calculators.

14
HOW DO WE EVALUATE A FUNCTION?
  • F (x) x2 3x 5
  • Evaluate each of the following
  • f (2)
  • f (x 3)
  • f (-x)
  • Substitute the 2 for x and evaluate.
  • Then repeat.

15
GRAPHS OF FUNCTIONS
  • The graph of a function is the graph of the
    ordered pairs.
  • Lets graph
  • f (x) 2x
  • g (x) 2x 4

16
USING THE VERTICAL LINE TEST
  • The Vertical Line Test for Functions
  • If any vertical line intersects a graph in more
    than one point, the graph does not define y as a
    function of x.

17
Practice
  • Use the vertical line test to identify graphs in
    which y is a function of x.

18
SUMMARY ANSWER IN COMPLETE SENTENCES.
  • What is a relation?
  • What is a function?
  • How can you determine if a relation is a
    function?
  • How can you determine if an equation in x and y
    defines y as a function of x? Give an example.

19
AIM 1.2B WHAT KIND OF INFORMATION CAN WE OBTAIN
FROM GRAPHS OF FUNCTIONS?
  • Note the closed dot indicates the graph does not
    extend from this point and its part of the graph.
  • Open dot indicates that the point does not extend
    and the point is not part of the graph.

20
HOW DO WE IDENTIFY DOMAIN AND RANGE FROM A
FUNCTIONS GRAPH?
  • Domain set of inputs
  • Found on x axis
  • Range set of outputs
  • Found on y -axis

21
  • Using set builder notation it would look like
    this for the domain
  • Using Interval Notation -4, 2
  • What would it look like for the range using both
    set builder and interval notation?

22
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM
ITS GRAPH
  • Use Set Builder Notation.
  • Domain
  • Range

23
IDENTIFY THE DOMAIN AND RANGE OF A FUNCTION FROM
ITS GRAPH
24
IDENTIFYING INTERCEPTS FROM A FUNCTIONS GRAPH
  • We can say that -2, 3, and 5 are the zeros of the
    function. The zeros of the function are the
  • x- values that make
  • f (x) 0.
  • Therefore, the real zeros are the x-intercepts.
  • A function can have more than one x-intercept,
    but at most one y-intercept.

25
SUMMARY ANSWER IN COMPLETE SENTENCES.
  • Explain how the vertical line test is used to
    determine whether a graph is a function.
  • Explain how to determine the domain and range of
    a function from its graph.
  • Does it make sense? Explain your reasoning.
  • I graphed a function showing how paid vacation
    days depend on the number of years a person works
    for a company. The domain was the number of paid
    vacation days.

26
AIM 1.3 HOW DO WE IDENTIFY INTERVALS ON WHICH A
FUNCTION IS INCREASING OR DECREASING?
  • Increasing, Decreasing and Constant Functions
  • A function is increasing on a open interval, I if
  • f (x1) lt f(x2) whenever x1ltx2 for any x1 and
    x2 in the interval.

27
  • A function is decreasing on an open interval, I,
    if f(x1) gt f (x2) whenever x1 gt x2
  • for any x1 and x2 in the interval.

28
  • A function is constant on an open interval, I,
    f(x1) f (x2) for any x1 and x2 in the interval.

29
  • Note
  • The open intervals describing where function
    increase, decrease or are constant use
  • x-coordinates and not y-coordinates.

30
Example 1 Increases, Decreases or Constant
  • State the interval where the function is
    increasing, decreasing or constant.

31
Practice
  • State the interval where the function is
    increasing, decreasing or constant.

32
WHAT IS A RELATIVE MAXIMA?
  • Definition of a Relative Maximum
  • A function value f (a) is a relative maximum of f
    if there exists an open interval containing a
    such that f (a) gt f (x) for all x ? a in the open
    interval.

33
WHAT IS A RELATIVE MINIMA?
  • Definition of a Relative Minimum
  • A function value f (b) is a relative minimum of f
    if there exists an open interval containing b
    such that f (b) lt f (x) for all x ? b in the open
    interval.

34
HOW DO WE IDENTIFY EVEN AND ODD FUNCTIONS AND
SYMMETRY?
  • Definition of Even and Odd Functions
  • The function f is an even function if
  • f (-x) f (x) all x in the domain of f.
  • The right side of the equation of an even
    function does not change if x is replaced with
    x.
  • The function f is an odd function if f (-x) -f
    (x)
  • for all x in the domain of f.
  • Every term on the right side of the equation of
    an odd function changes its sign if x is replaced
  • with x.

35
DETERMINE IF FUNCTION IS EVEN,ODD OR NEITHER
  • f (x) x3 - 6x
  • Steps
  • Replace x with x and simplify.
  • If the right side of the equation stays the same
    it is an even function.
  • If every term on the right side changes sign,
    then the function is odd.

36
DETERMINE IF FUNCTION IS EVEN, ODD OR NEITHER
  • g (x) x4 - 2x2
  • h(x) x2 2 x 1
  • Steps
  • Replace x with x and simplify.
  • If the right side of the equation stays the same
    it is an even function.
  • If every term on the right side changes sign,
    then the function is odd.

37
PRACTICE
  • Determine if function is Even, Odd or Neither
  • f (x) x2 6
  • g(x) 7x3 x
  • h (x) x5 1

38
  • The function on the left is even.
  • What does that mean in terms of the graph of the
    function?
  • The graph is symmetric with respect to the
    y-axis. For every point (x, y) on the graph, the
    point (-x, y) is also on the graph.
  • All even functions have graphs with this kind of
    symmetry.

39
  • The graph of function f (x) x3 is odd.
  • It may not be symmetrical with respect to the
    y-axis. It does have symmetry in another way.
  • Can you identify how?

40
  • For each point (x, y) there is a point (-x, -y)
    is also on the graph.
  • Ex. (2, 8) and (-2, -8) are on the graph.
  • The graph is symmetrical with respect to the
    origin.
  • All ODD functions have graphs with origin
    symmetry.

41
SUMMARYANSWER IN COMPLETE SENTENCES.
  • What does it mean if a function f is increasing
    on an interval?
  • If you are given a functions equation, how do
    you determine if the function is even, odd or
    neither?
  • Determine whether each function is even, odd or
    neither.
  • a. f (x) x2- x4 b. f (x) x(1- x2)1/2

42
AIM 1.3B HOW DO WE UNDERSTAND AND USE PIECEWISE
FUNCTIONS?
  • A piecewise function is a function that is
    defined by two (or more) equations over a
    specified domain.

43
Example ( DO NOT COPY) READ
  • A cellular phone company offers the following
    plan
  • 20 per month buys 60 minutes
  • Additional time costs 0.40 per minute

44
HOW DO WE EVALUATE A PIECEWISE FUNCTION?
45
PRACTICE
  • Find and interpret each of the following
  • a. C (40) b. C (80)

46
HOW DO WE GRAPH A PIECEWISE FUNCTION?
47
  • We can use the graph of a piecewise function to
    find the range of f.
  • What would the range be for the piecewise
    function? ( For previous piecewise function)

48
  • Some piecewise functions are called step
    functions because the graphs form discontinuous
    steps.
  • One such function is called the greatest integer
    function, symbolized by int (x) or
  • int (x) greatest integer that is less than or
    equal to x.
  • For example
  • a. int (1) 1, int (1.3) 1, int (1.5) 1,
    int (1.9) 1
  • b. int (2) 2, , int (2.3) 2 , int (2.5)
    2, int (2.9) 2

49
Graph of a Step Function
50
FUNCTION AND DIFFERENCE QUOTIENTS
  • Definition of the Difference Quotient of a
    Function
  • The expression for
    h?0 is called
  • the difference quotient of the function f.

51
HOW DO WE EVALUATE AND SIMPLIFY A DIFFERENCE
QUOTIENT?
  • If f (x) 2x2 x 3, find and simplify each
    expression
  • f ( x h)
  • Try
  • Steps
  • Replace x with (x h) each time x appears in the
    equation.

52
PRACTICE
  • If f (x) -2x2 x 5, find and simplify each
    expression
  • f (x h)

53
SUMMARY ANSWER IN COMPLETE SENTENCES.
  • What is a piecewise function?
  • Explain how to find the difference quotient of a
    function f,
  • If an equation for f is given.

54
AIM 1.4 HOW DO WE IDENTIFY A LINEAR FUNCTION AND
ITS SLOPE?
Data presented in a visual form as a set of
points is called a scatter plot. A scatter plot
shows the relationship between two types of data.
55
  • Regression line is the line that passes through
    or near the points. This is the line that best
    fits the data points.
  • We can write an equation that models the data and
    allows us to make predictions.

56
WHAT IS THE DEFINITION OF A SLOPE OF A LINE?
57
PRACTICE
  • Find the slope of a line.
  • (-3, 4) and (-4, -2)
  • (4, -2) and (-1, 5)

58
POINT-SLOPE FORM EQUATION
  • The point-slope of the equation of a nonvertical
    line with slope m that passes through the point
    (x1,y1) is
  • y y1m (x x1)

59
HOW DO WE WRITE AN EQUATION IN POINT-SLOPE FORM?
  • Write an equation in point-slope form for the
    line with slope 4 that passes through the point
    (-1, 3).
  • Then solve the equation for y.
  • Steps
  • Write the point-slope form equation.
  • y y1m (x x1)
  • Substitute the given values.
  • Then solve for y.

60
PRACTICE
  • Write an equation in point-slope form for the
    line with slope 6 that passes through the point
    (2, -5).
  • Then solve the equation for y.

61
HOW DO WE WRITE AN EQUATION FOR A LINE WHEN WE
ONLY HAVE TWO POINTS?
  • Write an equation in point-slope form for the
    line that passes through the points (4, -3) and
    (-2, 6).
  • Then solve the equation for y.
  • What would you need to do first?
  • Steps
  • Find the slope.
  • Then choose any pair of points and substitute
    into the point-slope form equation.
  • Then solve the equation for y.

62
PRACTICE
  • Write an equation in point-slope form for the
    line that passes through the points (-2,-1) and
    (-1, -6).
  • Then solve the equation for y.

63
SUMMARY ANSWER IN COMPLETE SENTENCES.
  • What is the slope of a line and how is it found?
  • Describe how to write an equation of a line if
    you know at least two points on that line.
  • How do you derive the slope-intercept equation
    from y y1m (x x1)?

64
AIM 1.4B HOW DO WE WRITE AND GRAPH LINEAR
EQUATIONS IN THE FORM OF Y MXB?
  • Slope intercept equation is y mx b where m
    is the slope and b is the y-intercept of the
    equation.
  • Graphing y mx b using the slope and
    y-intercept.
  • Graph the y-intercept first. (0, b)
  • Then use the slope to get to the other points on
    the line.
  • Lets try

65
EQUATIONS OF HORIZONTAL LINES
  • A horizontal line has a m0. Therefore the
    equation is y0x b which can be simplified to
  • y b.
  • Graph y 3 or f (x) 3
  • Note This is a constant function

66
EQUATIONS OF VERTICAL LINES
  • The slope of a vertical line is undefined. The
    equation of a vertical line is x a, where a is
    the x-intercept of the line.
  • Note
  • No vertical line represents a function.

67
WHAT IS THE GENERAL FORM OF THE EQUATION OF A
LINE?
  • Every line has an equation that can be written in
    the general form
  • Ax By C or Ax By- C 0
  • Where A, B, C are real numbers and A and B are
    not both zeros.

68
FINDING THE SLOPE AND THE Y-INTERCEPT
  • Find the slope and the y-intercept of the line
    whose equation is 3x 2y 4 0.
  • Steps
  • The equation is given in general form. Express in
    y mx b by solving for y.
  • Then the slope and y-intercept can be identified.

69
PRACTICE
  • Find the slope and the y-intercept of the line
    whose equation is 3x 6y 12 0.
  • Then use slope and y-intercept to graph the line.

70
HOW DO WE FIND THE INTERCEPTS FROM THE GENERAL
FORM OF THE EQUATION OF A LINE?
  • Graph using the intercepts 4x 3y 60
  • Steps
  • To find the x-intercept. Set y 0 and solve for
    x.
  • To find the y-intercept. Set x 0 and solve for
    y.
  • Graph the points and draw a line connecting these
    points.

71
PRACTICE
  • Graph using the intercepts 3x 2y 60

72
REVIEW OF THE VARIOUS EQUATIONS OF LINES
73
SUMMARYANSWER IN COMPLETE SENTENCES.
  • How would you graph the equation x 2. Can this
    equation be expressed in slope-intercept form?
    Explain.
  • Explain how to use the general form of a lines
    equation to find the lines slope and
    y-intercept.
  • How do you use the intercepts to graph the
    general form of a lines equation?

74
Aim 1.5 How do we find the average rate of
change?
  • Slope as a rate of change
  • Example 1
  • The line on the graph for the number of women
    and men living alone are shown in the graph.
  • Describe what the slope represents.

75
  • Solution Note x represents the year and y-
    number of women.
  • Choose two points from the womens graph.
  • Find the slope and then describe the slope.
  • Remember to include the units.

76
Practice
  • Use the ordered pairs and find the rate of change
    for the green line or men graph. Express slope
    two decimal places and describe what it
    represents.

77
Average Rate of Change
  • If the graph of the function is not a straight
    line, the average rate of change between any two
    points is the slope of the line containing the
    two points.
  • This line is called the secant line.

78
Problem
  • Looking at the graph, what is the mans average
    growth rate between the ages 13 and 18.

79
The Average Rate of Change of a Function
80
Example 1
  • Find the average rate of change of f (x) x2
    from
  • Steps
  • Use

81
  • Find the average rate of change of f (x) x2
    from

82
Average Velocity of an Object
  • Suppose that a function expresses an objects
    position, s (t), in terms of time, t. The average
    velocity of the object from t1 to t2 is

83
Example 2
  • The distance, s (t), in feet, traveled by a ball
    rolling down a ramp is given by the function
  • s (t) 5t2,
  • where t is the time, in seconds after the ball
    is released. Find the balls average velocity
    from
  • t1 2 seconds to t2 3 seconds
  • t1 2 seconds to t2 2.5 seconds
  • t1 2 seconds to t2 2.01 seconds

84
Summary Answer in complete sentences.
  • If two lines are parallel, describe the
    relationship between their slopes and
    y-intercept.
  • If two line are perpendicular, describe the
    relationship between their slopes.
  • What is the secant line?
  • What is the average rate of change of a function?

85
Aim 1.6 How do we recognize transformations?
  • Review Algebras Common Graphs (distribute)
  • Vertical Shift

86
Vertical Shifts
  • In general if c is positive, y f (x) c shifts
    upward c units. If c is negative it shifts
    downward c units.

87
Example 1
88
Practice
  • Use the graph of to obtain the
    graph of

89
Horizontal Shifts
  • In general, if c is positive, y f (x c)
    shifts the graph of f to the left c units and y
    f (x c) shifts the graphs of f to the right c
    units.
  • These are called horizontal shifts of the graph
    of f.

90
Example 2
91
Note
92
Example 3
  • Use the graph of f (x ) x2 to obtain the graph
    of h (x) (x 1)2 3.
  • Steps to combining a shift
  • Graph the original function.
  • Then shift horizontally.
  • Then shift vertically.

93
Reflections of Graphs
  • Reflection about the x-axis
  • The graph of y - f (x) is the graph of y f
    (x) reflected about the x- axis.

94
Example 4
  • Use the graph of to obtain the
    graph of

95
Reflections of Graphs
  • Reflection about the y-axis
  • The graph of y f (-x) is the graph of y f (x)
    reflected over the y axis.
  • Example The point (2, 3) reflected over the
    y-axis is (-2, 3).

96
Vertical Stretching
  • Let f be a function and c be a positive real
    number.
  • If c gt 1 the graph of y c f (x) is the graph y
    f (x) vertically stretched.
  • How?
  • By multiplying the y-coordinates by c.

97
Vertical Shrinking
  • Let f be a function and c be a positive real
    number.
  • If 0 lt c lt 1 the graph of y c f (x) is the
    graph y f (x) vertically shrunk.
  • How?
  • By multiplying the y-coordinates by c.

98
Example 6
  • Use the graph of f (x) x3 to obtain the graph
    of
  • h (x)

99
Horizontal Shrinking
  • Let f be a function and c be a positive real
    number.
  • If c gt 1 the graph of y f (cx) is the graph
  • y f (x) horizontal shrink.
  • How?
  • By dividing x- coordinates by c.

100
Horizontal Stretching
  • Let f be a function and c be a positive real
    number.
  • If 0 lt c lt 1 the graph of y f (cx) is the graph
  • y f (x) horizontal stretch.
  • How?
  • By dividing x- coordinates by c.

101
Example 7
  • Use the graph y f (x) to obtain each of the
    following graphs.
  • a. g (x) f (2x) b. h( x) f( 1/2x)

102
Sequences of Transformations
  • Transformations involving more than one
    transformation can be graphed performing the
    transformations in the following order
  • Horizontal shifting
  • Stretching or shrinking
  • Reflecting
  • Vertical shifting

103
Summary Answer in complete sentences.
  • What must be done to a functions equation so
    that its graph is shifted vertically upward?
  • What must be done to a functions equation so
    that its graph is shifted horizontally to the
    right?
  • What must be done to a functions equation so
    that its graph is reflected about the x-axis?
  • What must be done to a functions equation so
    that its graph is stretched vertically?

104
Aim 1.7 What are composite functions?
  • Finding the domain of a function (w/ no graph).
  • Example 1 Find the domain of each function.

105
The Algebra of Functions
  • We can combine functions by addition,
    subtraction, multiplication and division by
    performing operations with the algebraic
    expressions that appear on the right side of the
    equations.
  • Example 2 Let f (x)2x 1 and g (x) x2 x
    2
  • Find the following function.
  • Sum (f g ) (x) f (x ) g (x)
  • ( 2x 1) (x2 x 2) 1. Substitute
    given functions.
  • 2. Simplify.

106
The Algebra of Functions
107
Example 3 Adding Functions and Determining the
Domain
  • Find each of the following
  • (f g) (x)
  • The domain of (f g)

108
Composite Functions
  • This is another way to combine functions.
  • Example f (g (x)) can be read as f of g of x
    and it is written as

109
Example 4 Forming Composite Functions
  • Given f (x) 3x 4 and g (x) x2 2x 6, find
    each of the following

110
Example 5 Forming a Composite Function
and Finding Its Domain
111
Practice
  • Given and g(x)
  • Find each of the following
  • 1.
  • 2. the domain of

112
Summary Answer in complete sentences.
  • Explain how to find the domain of a function of a
    radical and a rational equation.
  • If equations for f and g are given, explain how
    find f- g.
  • If equations for two functions are given, explain
    how to obtain the quotient of the two functions
    and its domain. ( ex. f/g(x))
  • Describe a procedure for finding .
  • What is the name of this function?

113
Aim 1. 8 What is an inverse function?
114
Example 1 Verifying Inverse Functions
  • Show that each function is the inverse of the
    other
  • Steps
  • To show that f and g are inverses of each other,
    we must show that f (g (x)) x and f (g (x))
    x.
  • Begin with f (g (x))
  • Then with g (f (x))
  • Do you get x?

115
Practice
  • Show that each function is the inverse of the
    other

116
How do we find the inverse?
  • Find the inverse of
  • f (x) 7x 5
  • Steps
  • Replace f (x) with y.
  • Switch x and y.
  • Solve for y.
  • Then replace y with f-1 (x)

117
How do we find the inverse?
  • Find the inverse of
  • f (x ) x3 1
  • Steps
  • Replace f (x) with y.
  • Switch x and y.
  • Solve for y.
  • Then replace y with f-1 (x)

118
How do we find the inverse?
  • Find the inverse of
  • Steps
  • Replace f (x) with y.
  • Switch x and y.
  • Solve for y.
  • Then replace y with f-1 (x)

119
Practice
  • Find the inverse for the following
  • Steps
  • Replace f (x) with y.
  • Switch x and y.
  • Solve for y.
  • Then replace y with f-1 (x)

120
Properties of Inverse Functions
  • 1. A function f has an inverse that is a
    function, f-1, if there is no horizontal line
    that intersects the graph of the function f at
    more than one point.

121
Properties of Inverse Functions
  • 2. The graph of a functions inverse is a
    reflection of the graph of f about the line y x.

122
Summary Answer in complete sentences.
  • Explain how to determine if two functions are
    inverse of each other.
  • Describe how to find the inverse of a function.
  • What are some properties of a function and its
    inverse? Draw an illustration to support your
    written statement.

123
Aim 1.9 How do we find the distance and
midpoint of a segment?
  • Formulas

124
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