Graphs and Functions

1
Graphs and Functions
Chapter 3
2
3.1

• Graphing Equations

3
Vocabulary

• Ordered pair a sequence of 2 numbers where the
  order of the numbers is important
• Axis horizontal or vertical number line
• Origin point of intersection of two axes
• Quadrants regions created by intersection of 2
  axes
• Location of a point residing in the rectangular
  coordinate system created by a horizontal (x-)
  axis and vertical (y-) axis can be described by
  an ordered pair. Each number in the ordered pair
  is referred to as a coordinate

4
Graphing an Ordered Pair
5
Graphing an Ordered Pair
Note that the order of the coordinates is very
important, since (-4, 2) and (2, -4) are located
in different positions.
6
Vocabulary

• Paired data are data that can be represented as
  an ordered pair
• A scatter diagram is the graph of paired data as
  points in the rectangular coordinate system
• An order pair is a solution of an equation in two
  variables if replacing the variables by the
  appropriate values of the ordered pair results in
  a true statement.

7
Solutions of an Equation
Example
Determine whether (3, 2) is a solution of 2x
5y 4. Let x 3 and y 2 in the equation.
2x 5y 4 2(3) 5( 2) 4
Replace x with 3 and y with 2. 6
( 10) 4 Compute the products.
4 4 True
So (3, -2) is a solution of 2x 5y 4
8
Solutions of an Equation
Example
Determine whether ( 1, 6) is a solution of 3x
y 5. Let x 1 and y 6 in the equation.
3x y 5 3( 1) 6 5 Replace x
with 1 and y with 6. 3 6 5
Compute the product. 9 5
False
So ( 1, 6) is not a solution of 3x y 5
9
Linear Equations

• Linear Equation in Two Variables
• A linear equation in two variables is an equation
  that can be written in the form
• Ax By C
• where A and B are not both 0. This is called
  standard form.

10
Graphing Linear Equations
Example
Graph the linear equation 2x y -4.
Continued.
11
Graphing Linear Equations
Example continued
Graph the linear equation 2x y 4. Let x
1. Then 2x y 4 becomes
2(1) y 4 Replace x with 1. 2
y 4 Simplify the left side. y
4 2 6 Subtract 2 from both sides.
y 6 Multiply both sides by
1. So one solution is (1, 6)
Continued.
12
Graphing Linear Equations
Example continued
Graph the linear equation 2x y 4. For the
second solution, let y 4. Then 2x y 4
becomes
2x 4 4 Replace y with 4.
2x 4 4 Add 4 to both sides.
2x 0 Simplify the right side.
x 0 Divide both sides by 2. So the
second solution is (0, 4)
Continued.
13
Graphing Linear Equations
Example continued
Graph the linear equation 2x y 4. For the
third solution, let x 3. Then 2x y 4
becomes
2( 3) y 4 Replace x with 3.
6 y 4 Simplify the left side.
y 4 6 2 Add 6 to both sides.
y 2 Multiply both sides by
1. So the third solution is ( 3, 2)
Continued.
14
Graphing Linear Equations
Example continued
Now we plot all three of the solutions (1, 6),
(0, 4) and ( 3, 2).
And then we draw the line that contains the three
points.
15
Graphing Linear Equations
Example
Continued.
16
Graphing Linear Equations
Example continued
Let x 4.
y 3 3 6 Simplify the right
side. So one solution is (4, 6)
Continued.
17
Graphing Linear Equations
Example continued
For the second solution, let x 0.
y 0 3 3 Simplify the right
side. So a second solution is (0, 3)
Continued.
18
Graphing Linear Equations
Example continued
For the third solution, let x 4.
y 3 3 0 Simplify the right
side. So the third solution is ( 4, 0)
Continued.
19
Graphing Linear Equations
Example continued
Now we plot all three of the solutions (4, 6),
(0, 3) and ( 4, 0).
And then we draw the line that contains the three
points.
20
Intercepts

• Intercepts of axes (where graph crosses the axes)
• Since all points on the x-axis have a
  y-coordinate of 0, to find x-intercept, let y 0
  and solve for x
• Since all points on the y-axis have an
  x-coordinate of 0, to find y-intercept, let x 0
  and solve for y

21
Intercepts
Example

• Find the y-intercept of 4 x 3y
• Let x 0.
• Then 4 x 3y becomes
• 4 0 3y Replace x with 0.
• 4 3y Simplify the right side.

22
Intercepts
Example

• Find the x-intercept of 4 x 3y
• Let y 0.
• Then 4 x 3y becomes
• 4 x 3(0) Replace y with 0.
• 4 x Simplify the right side.
• So the x-intercept is (4,0)

23
Graph by Plotting Intercepts
Example

• Graph the linear equation 4 x 3y by plotting
  intercepts.

Plot both of these points and then draw the line
through the 2 points. Note You should still
find a 3rd solution to check your computations.
Continued.
24
Graph by Plotting Intercepts
Example continued
Graph the linear equation 4 x 3y. Along with
the intercepts, for the third solution, let y
1. Then 4 x 3y becomes
4 x 3(1) Replace y with 1. 4
x 3 Simplify the right side. 4 3
x Add 3 to both sides. 7 x
Simplify the left side. So the third
solution is (7, 1)
Continued.
25
Graph by Plotting Intercepts
Example continued
And then we draw the line that contains the three
points.
26
Graphing Nonlinear Equations
Example
x y x
?2 4
?1 1
0 0
1 1
2 4
27
Graphing Nonlinear Equations
Example
x y x
?2 2
?1 1
0 0
1 1
2 2
28
Graphing Nonlinear Equations
Example
x y x
0 0
1 1
4 2
9 3
29
3.2

• Introduction to Functions

30
Relations

• Equations in two variables define relations
  between the two variables.
• There are other ways to describe relations
  between variables.
• Set to set
• Ordered pairs
• A set of ordered pairs is also called a relation.

31
Functions

• Some relations are also functions.
• A function is a set of order pairs that assigns
  to each x-value exactly one y-value.

32
Functions
Example

• Given the relation (4,9), (4,9), (2,3), (10,
  5), is it a function?
• Since each element of the domain is paired with
  only one element of the range, it is a function.
• Note Its okay for a y-value to be assigned to
  more than one x-value, but an x-value cannot be
  assigned to more than one y-value (has to be
  assigned to ONLY one y-value).

33
Functions
Example
Is the relation y x2 2x a function? Since
each element of the domain (the x-values) would
produce only one element of the range (the
y-values), it is a function.
34
Functions
Example
Is the relation x2 y2 9 a function? Since
each element of the domain (the x-values) would
correspond with 2 different values of the range
(both a positive and negative y-value), the
relation is NOT a function.
35
Vertical Line Test

• Relations and functions can also be described by
  graphing their ordered pairs.
• Graphs can be used to determine if a relation is
  a function.
• If an x-coordinate is paired with more than one
  y-coordinate, a vertical line can be drawn that
  will intersect the graph at more than one point.
• If no vertical line can be drawn so that it
  intersects a graph more than once, the graph is
  the graph of a function. This is the vertical
  line test.

36
Vertical Line Test
Example
Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since no vertical line will intersect this graph
more than once, it is the graph of a function.
37
Vertical Line Test
Example
Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since no vertical line will intersect this graph
more than once, it is the graph of a function.
38
Vertical Line Test
Example
Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since vertical lines can be drawn that intersect
the graph in two points, it is NOT the graph of a
function.
39
Vertical Line Test

• Since the graph of a linear equation is a line,
  all linear equations are functions, except those
  whose graph is a vertical line

40
Domain and Range

• Recall that a set of ordered pairs is also called
  a relation.
• The domain is the set of x-coordinates of the
  ordered pairs.
• The range is the set of y-coordinates of the
  ordered pairs.

41
Domain and Range
Example

• Find the domain and range of the relation (4,9),
  (4,9), (2,3), (10, 5)
• Domain is the set of all x-values, 4, 4, 2, 10
• Range is the set of all y-values, 9, 3, 5

42
Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
43
Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
44
Domain and Range
Example
Find the domain and range of the following
relation.

• Input (Animal)
• Polar Bear
• Cow
• Chimpanzee
• Giraffe
• Gorilla
• Kangaroo
• Red Fox

• Output (Life Span)
• 20
• 15
• 10
• 7

45
Domain and Range
Example continued
Domain is Polar Bear, Cow, Chimpanzee, Giraffe,
Gorilla, Kangaroo, Red Fox Range is 20, 15, 10,
7
46
Function Notation

• Specialized notation is often used when we know a
  relation is a function and it has been solved for
  y.
• For example, the graph of the linear equation
  y 3x 2 passes the vertical line
  test, so it represents a function.
• We often use letters such as f, g, and h to name
  functions.
• We can use the function notation f(x) (read f of
  x) and write the equation as f(x) 3x 2.
• Note The symbol f(x) is a specialized notation
  that does NOT mean f x (f times x).

47
Function Notation

• When we want to evaluate a function at a
  particular value of x, we substitute the x-value
  into the notation.
• For example, f(2) means to evaluate the function
  f when x 2. So we replace x with 2 in the
  equation.
• For our previous example when f(x) 3x 2,
  f(2) 3(2) 2 6 2 4.
• When x 2, then f(x) 4, giving us the order
  pair (2, 4).

48
Function Notation
Example

• Given that g(x) x2 2x, find g( 3). Then
  write down the corresponding ordered pair.
• g( 3) ( 3)2 2( 3) 9 ( 6) 15.
• The ordered pair is ( 3, 15).

49
Function Notation
Example
Given the graph of the following function, find
each function value by inspecting the graph.
50
3.3

• Graphing Linear Functions

51
Graphing Linear Functions
Example
Graph the linear function f(x) x 3.
Let x 4.
3 3 6 Simplify the right
side. So one solution is (4, 6)
Continued.
52
Graphing Linear Functions
Example continued
Graph the linear function f(x) x 3.
For the second solution, let x 0.
0 3 3 Simplify the right
side. So a second solution is (0, 3)
Continued.
53
Graphing Linear Functions
Example continued
Graph the linear function f(x) x 3.
For the third solution, let x 4 .
3 3 0 Simplify the right
side. So a third solution is ( 4 , 0)
Continued.
54
Graphing Linear Functions
Example continued
Now we plot all three of the solutions (4, 6),
(0, 3) and ( 4, 0).
And then we draw the line that contains the three
points.
55
Graphing Linear Functions
Example
x f (x) x
?2 ?2
?1 1
0 0
1 1
2 2
56
Intercepts

• Intercepts of axes (where graph crosses the axes)
• Since all points on the x-axis have a
  y-coordinate of 0, to find x-intercept, let y 0
  and solve for x
• Since all points on the y-axis have an
  x-coordinate of 0, to find y-intercept, let x 0
  and solve for y

57
Intercepts
Example

• Find the y-intercept of 4 x 3y
• Let x 0.
• Then 4 x 3y becomes
• 4 0 3y Replace x with 0.
• 4 3y Simplify the right side.

58
Intercepts
Example

• Find the x-intercept of 4 x 3y
• Let y 0.
• Then 4 x 3y becomes
• 4 x 3(0) Replace y with 0.
• 4 x Simplify the right side.
• So the x-intercept is (4,0)

59
Graph by Plotting Intercepts
Example

• Graph the linear equation 4 x 3y by plotting
  intercepts.

Plot both of these points and then draw the line
through the 2 points. Note You should still
find a 3rd solution to check your computations.
Continued.
60
Graph by Plotting Intercepts
Example continued
Graph the linear equation 4 x 3y. Along with
the intercepts, for the third solution, let y
1. Then 4 x 3y becomes
4 x 3(1) Replace y with 1. 4
x 3 Simplify the right side. 4 3
x Add 3 to both sides. 7 x
Simplify the left side. So the third
solution is (7, 1)
Continued.
61
Graph by Plotting Intercepts
Example continued
And then we draw the line that contains the three
points.
62
Graphing Horizontal Lines
Example

• Graph y 3
• Note that this line can be written as y 0x
  3
• The y-intercept is (0, 3), but there is no
  x-intercept!
• (Since an x-intercept would be found by letting
  y 0, and 0 ? 0x 3, there is no x-intercept)
• Every value we substitute for x gives a
  y-coordinate of 3.
• The graph will be a horizontal line through the
  point (0,3) on the y-axis

Continued.
63
Graphing Horizontal Lines
Example continued
64
Graphing Vertical Lines
Example

• Graph x 3
• This equation can be written x 0y 3
• When y 0, x 3, so the x-intercept is (
  3,0), but there is no y-intercept
• Any value we substitute for y gives an
  x-coordinate of 3.
• So the graph will be a vertical line through the
  point ( 3,0) on the x-axis

Continued.
65
Graphing Vertical Lines
Example continued
66
Vertical and Horizontal Lines

• Vertical lines
• Appear in the form of x c, where c is a real
  number
• x-intercept is at (c, 0), no y-intercept unless
  c 0 (y-axis)
• Horizontal lines
• Appear in the form of y c, where c is a real
  number
• y-intercept is at (0, c), no x-intercept unless
  c 0 (x-axis)

67
3.4

• The Slope of a Line

68
Slope

• Slope of a Line

69
Slope
Example

• Find the slope of the line through (4, -3) and
  (2, 2)
• If we let (x1, y1) be (4, -3) and (x2, y2) be (2,
  2), then

Note If we let (x1, y1) be (2, 2) and (x2, y2)
be (4, -3), then we get the same result.
70
Slope-Intercept Form

• Slope-Intercept Form of a line
• y mx b has a slope of m and has a
  y-intercept of (0, b).
• This form is useful for graphing, since you have
  a point and the slope readily visible.

71
Slope-Intercept Form
Example

• Find the slope and y-intercept of the line 3x
  y -5.
• First, we need to solve the linear equation for
  y.
• By adding 3x to both sides, y 3x 5.
• Once we have the equation in the form of y mx
  b, we can read the slope and y-intercept.
• slope is 3
• y-intercept is (0, 5)

72
Slope-Intercept Form
Example

• Find the slope and y-intercept of the line 2x
  6y 12.
• First, we need to solve the linear equation for
  y.
• 6y 2x 12 Subtract 2x from both sides.

73
Slope of a Horizontal Line

• For any 2 points, the y values will be equal to
  the same real number.
• The numerator in the slope formula 0 (the
  difference of the y-coordinates), but the
  denominator ? 0 (two different points would have
  two different x-coordinates).
• So the slope 0.

74
Slope of a Vertical Line

• For any 2 points, the x values will be equal to
  the same real number.
• The denominator in the slope formula 0 (the
  difference of the x-coordinates), but the
  numerator ? 0 (two different points would have
  two different y-coordinates),
• So the slope is undefined (since you cant divide
  by 0).

75
Summary of Slope of Lines

• If a line moves up as it moves from left to
  right, the slope is positive.
• If a line moves down as it moves from left to
  right, the slope is negative.
• Horizontal lines have a slope of 0.
• Vertical lines have undefined slope (or no slope).

76
Parallel Lines

• Two lines that never intersect are called
  parallel lines.
• Parallel lines have the same slope
• unless they are vertical lines, which have no
  slope.
• Vertical lines are also parallel.

77
Parallel Lines
Example

• Find the slope of a line parallel to the line
  passing through (0,3) and (6,0)

So the slope of any parallel line is also ½
78
Perpendicular Lines

• Two lines that intersect at right angles are
  called perpendicular lines
• Two nonvertical perpendicular lines have slopes
  that are negative reciprocals of each other
• The product of their slopes will be 1
• Horizontal and vertical lines are perpendicular
  to each other

79
Perpendicular Lines
Example

• Find the slope of a line perpendicular to the
  line passing through (-1,3) and (2,-8)

80
Parallel and Perpendicular Lines
Example

• Determine whether the following lines are
  parallel, perpendicular, or neither.
• 5x y 6 and x 5y 5
• First, we need to solve both equations for y.
• In the first equation,
• y 5x 6 Add 5x to both sides.
• In the second equation,
• 5y x 5 Subtract x from both sides.

81
3.5

• Equations of Lines

82
Slope-Intercept Form

• Slope-Intercept Form of a line
• y mx b has a slope of m and has a
  y-intercept of (0, b).
• This form is useful for graphing, since you have
  a point and the slope readily visible.

83
Slope-Intercept Form
Example

• Find the slope and y-intercept of the line 3x
  y 5.
• First, we need to solve the linear equation for
  y.
• By adding 3x to both sides, y 3x 5.
• Once we have the equation in the form of y mx
  b, we can read the slope and y-intercept.
• slope is 3
• y-intercept is (0, 5)

84
Slope-Intercept Form
Example Find the equation of the line with slope
and y-intercept (0, ?5).
85
Point-Slope Form

• The slope-intercept form uses, specifically, the
  y-intercept in the equation.
• The point-slope form allows you to use ANY point,
  together with the slope, to form the equation of
  the line.

m is the slope (x1, y1) is a point on the line
86
Point-Slope Form
Example

• Find an equation of a line with slope 2,
  through the point ( 11, 12). Write the
  equation in standard form.
• First we substitute the slope and point into the
  point-slope form of an equation.
• y ( 12) 2(x ( 11))
• y 12 2x 22 Use distributive
  property.
• 2x y 12 22 Add 2x to both sides.
• 2x y 34 Subtract 12 from both
  sides.

87
Point-Slope Form
Example

• Find the equation of the line through ( 4, 0)
  and (6, 1). Write the equation in standard
  form.
• First find the slope.

Continued.
88
Point-Slope Form
Example continued
Now substitute the slope and one of the points
into the point-slope form of an equation.
89
Point-Slope Form
Example

• Find the equation of the line passing through
  points (2, 5) and ( 4, 3). Write the equation
  in slope-intercept form.
• First find the slope.

Continued.
90
Point-Slope Form
Example continued
91
Horizontal and Vertical Lines

• Remember that
• nonvertical parallel lines have equal slopes.
• nonvertical perpendicular lines have slopes that
  are negative reciprocals of each other.
• if you rewrite linear equations into
  slope-intercept form, you can easily determine
  slope to compare lines.

92
Horizontal and Vertical Lines
Example

• Find an equation of a line that contains the
  point ( 2, 4) and is parallel to the line x 3y
  6. Write the equation in standard form.
• First, we need to find the slope of the given
  line.
• 3y ? x 6 Subtract x from both sides.

Continued.
93
Horizontal and Vertical Lines
Example continued
94
Horizontal and Vertical Lines
Example

• Find an equation of a line that contains the
  point (3,? 5) and is perpendicular to the line 3x
  2y 7. Write the equation in standard form.
• First, we need to find the slope of the given
  line.
• 2y ? 3x 7 Subtract 3x from both sides.

Continued.
95
Horizontal and Vertical Lines
Example continued
96
3.6

• Graphing Piecewise-Defined Functions and Shifting
  and Reflecting Graphs of Functions

97
Graphing Piecewise-Defined Functions
Example
Graph each piece separately.
x f (x) 3x 1
0 1(closed circle)
1 4
2 7
x f (x) x 3
1 4
2 5
3 6
Values ? 0.
Values gt 0.
Continued.
98
Graphing Piecewise-Defined Functions
Example continued
x f (x) 3x 1
0 1(closed circle)
1 4
2 7
x f (x) x 3
1 4
2 5
3 6
99
Vertical and Horizontal Shifting
Vertical Shifts (Upward or Downward) Let k be a
Positive Number
Graph of Same As Moved
g(x) f(x) k f(x) k units upward
g(x) f(x) ? k f(x) k units downward
Horizontal Shifts (To the Left or Right) Let h be
a Positive Number
Graph of Same As Moved
g(x) f(x ? h) f(x) h units to the right
g(x) f(x h) f(x) h units to the left
100
Vertical and Horizontal Shifting
Example
Begin with the graph of f(x) x2.
Shift the original graph downward 3 units.
101
Vertical and Horizontal Shifting
Example
Begin with the graph of f(x) x.
Shift the original graph to the left 2 units.
102
Reflections of Graphs
Reflection about the x-axis The graph of g(x)
f(x) is the graph of f(x) reflected about the
x-axis.
103
Reflections of Graphs
Example
Reflect the original graph about the x-axis.
104
3.7

• Graphing Linear Inequalities

105
Linear Inequalities in Two Variables

• Linear Inequality in Two Variables
• Written in the form Ax By lt C
• A, B, C are real s, A B are not both 0
• Could use (gt, ?, ?) in place of lt
• An ordered pair is a solution of the linear
  inequality if it makes the inequality a true
  statement.

106
Linear Inequalities in Two Variables

• Graphing a Linear Inequality in Two Variables
• Graph the boundary line found by replacing the
  inequality sign with an equal sign. If the sign
  is lt or gt, graph a dashed line. If the sign is ?
  or ?, graph a solid line.
• Choose a test point not on the boundary line and
  substitute it into original inequality.
• If a true statement is obtained in Step 2, shade
  the half-plane containing the point. If a false
  statement is obtained, shade the half-plane that
  does NOT contain the point.

107
Linear Inequalities in Two Variables
Example

• Graph 7x y gt 14

• Graph 7x y 14 as a dashed line.

• Pick a point not on the graph
  (0,0)

• Test it in the original inequality.
• 7(0) 0 gt 14, 0 gt 14
• True, so shade the side containing (0,0).

108
Linear Inequalities in Two Variables
Example

• Graph 3x 5y ? 2

• Graph 3x 5y 2 as a solid line.

• Pick a point not on the graph
  (0,0), but just barely

• Test it in the original inequality.
• 3(0) 5(0) gt 2, 0 gt 2
• False, so shade the side that does not contain
  (0,0).

109
Linear Inequalities in Two Variables
Example

• Graph 3x lt 15

• Graph 3x 15 as a dashed line.

• Pick a point not on the graph
  (0,0)

• Test it in the original inequality.
• 3(0) lt 15, 0 lt 15
• True, so shade the side containing (0,0).

110
Linear Inequalities in Two Variables
Warning!

• Note that although all of our examples allowed us
  to select (0, 0) as our test point, that will not
  always be true.
• If the boundary line contains (0,0), you must
  select another point that is not contained on the
  line as your test point.