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Graphs, Linear Equations,
and Functions
3
3.1 The Rectangular
R.1 Coordinate
Fractions System
Objectives
1.
2.
3.
4.
5.
6.
Interpret a line graph.
Plot ordered pairs.
Find ordered pairs that satisfy a given equation.
Graph lines.
Find x- and y-intercepts.
Recognize equations of horizontal and vertical
lines and lines passing through the origin.
7. Use the midpoint formula.
3.1, 1
Slide 1Section
of 104
3.1, 2
Slide 2Section
of 104
Rectangular (or Cartesian, for Descartes) Coordinate System
Rectangular (or Cartesian, for Descartes) Coordinate System
y
y
8
Quadrant II
Ordered Pair
(x, y)
y-axis
x-axis
6
Origin
4Quadrant I
-8
-6
-4
0 0
-2 0 -2 2
Quadrant III
A
D
2
x
x
4
6
B
8
C
-4
Quadrant IV
-6
Quadrant
A (5, 3)
Quadrant I
B (2, –1)
Quadrant IV
C (–2, –3)
Quadrant III
D (–4, 2)
Quadrant II
-8
3.1, 3
Slide 3Section
of 104
Caution
3.1, 4
Slide 4Section
of 104
EXAMPLE 1
Completing Ordered Pairs
Complete each ordered pair for 3x + 4y = 7.
CAUTION
The parentheses used to represent an ordered pair are also used to
represent an open interval (introduced in Section 2.1). The context of
the discussion tells whether ordered pairs or open intervals are being
represented.
(a) (5, ? )
We are given x = 5. We substitute into the equation to find y.
3x + 4y = 7
3(5) + 4y = 7
Let x = 5.
15 + 4y = 7
4y = –8
y = –2
The ordered pair is (5, –2).
3.1, 5
Slide 5Section
of 104
3.1, 6
Slide 6Section
of 104
1
Completing Ordered Pairs
A Linear Equation in Two Variables
Complete each ordered pair for 3x + 4y = 7.
A linear equation in two variables can be written in the form
Ax + By = C,
(b)( ? , –5)
Replace y with –5 in the equation to find x.
3x + 4y = 7
where A, B, and C are real numbers (A and B not both 0). This form is
called standard form.
Let y = –5.
3x + 4(–5) = 7
3x – 20 = 7
3x = 27
x=9
The ordered pair is (9, –5).
3.1, 7
Slide 7Section
of 104
3.1, 8
Slide 8Section
of 104
Intercepts
Finding Intercepts
When graphing the equation of a line, find the
intercepts as follows.
y
y-intercept (where the line intersects
the y-axis)
let y = 0 to find the x-intercept;
let x = 0 to find the y-intercept.
x-intercept (where the
line intersects
the x-axis)
x
3.1, 9
Slide 9Section
of 104
EXAMPLE 2
Finding Intercepts
Finding Intercepts
Find the x- and y-intercepts of 2x – y = 6, and graph the equation.
We find the x-intercept
by letting y = 0.
2x – y = 6
2x – 0 = 6
Section
Slide 10
of 3.1,
10410
We find the y-intercept
by letting x = 0.
2x – y = 6
Let y = 0.
2x = 6
x = 3 x-intercept is (3, 0).
2(0) – y = 6
Continued.
Find the x- and y-intercepts of 2x – y = 6, and graph the equation.
The intercepts are the two points (3,0) and (0, –6). We show these ordered
pairs in the table next to the figure below and use these points to draw the
graph.
y
Let x = 0.
–y = 6
y = –6 y-intercept is (0, –6).
x
y
3
0
0
–6
x
The intercepts are the two points (3,0) and (0, –6).
Section
Slide 11
of 3.1,
10411
Section
Slide 12
of 3.1,
10412
2
EXAMPLE 3
Graphing a Horizontal Line
Graphing a Vertical Line
Graph y = –3.
Since y is always –3, there is no value of x corresponding to
y = 0, so the graph has no x-intercept. The y-intercept is
(0, –3). The graph in the figure below, shown with a table of
y
ordered pairs, is a horizontal line.
x
y
2
–3
0
–2
Graph x + 2 = 5.
The x-intercept is (3, 0). The standard form 1x + 0y = 3 shows
that every value of y leads to x = 3, so no value of y makes
x = 0. The only way a straight line can have no y-intercept is if it
y
is vertical, as in the figure below.
x
y
3
2
–3
3
0
–3
3
–2
x
x
Section
Slide 13
of 3.1,
10413
Section
Slide 14
of 3.1,
10414
EXAMPLE 4
Graphing a Line That Passes
through the Origin
Horizontal and Vertical Lines
Graph 3x + y = 0.
CAUTION
To avoid confusing equations of horizontal and vertical lines, keep the
following in mind.
1.
2.
An equation with only the variable x will always intersect the
x-axis and thus will be vertical. It has the form x = a.
An equation with only the variable y will always intersect the
y-axis and thus will be horizontal. It has the form y = b.
We find the x-intercept
by letting y = 0.
3x + y = 0
3x + 0 = 0
3x + y = 0
Let y = 0.
3(0) + y = 0
Let x = 0.
0+y=0
3x = 0
x=0
We find the y-intercept
by letting x = 0.
y = 0 y-intercept is (0, 0).
x-intercept is (0, 0).
Both intercepts are the same ordered pair, (0, 0). (This means
the graph goes through the origin.)
Section
Slide 15
of 3.1,
10415
Continued.
Graph 3x + y = 0.
Graphing a Line That Passes
through the Origin
To find another point to graph the line, choose any nonzero
number for x, say x = 2, and solve for y.
Section
Slide 16
of 3.1,
10416
Graphing a Line That Passes
through the Origin
Continued.
Graph 3x + y = 0.
These points, (0, 0) and (2, –6), lead to the graph shown below.
As a check, verify that (1, –3) also lies on the line.
y
Let x = 2.
3x + y = 0
3(2) + y = 0
Let x = 2.
x
y
0
0
x-intercept
and
y-intercept
x
6+y=0
y = –6
This gives the ordered pair (2, –6).
Section
Slide 17
of 3.1,
10417
2
–6
1
–3
Section
Slide 18
of 3.1,
10418
3
Finding the Coordinates of a Midpoint
EXAMPLE 5
Use the midpoint formula
If the endpoints of a line segment PQ are (x1, y1) and
(x2, y2), its midpoint M is
x1
2
x2 y1
,
y2
2
Find the coordinates of the midpoint of line segment PQ with
endpoints P(6, −1) and Q(4, −2).
Use the midpoint formula with x1 = 6, x2 = 4, y1 = −1, y2 = −2:
.
6 4 1 ( 2)
,
2
2
10 3
,
2 2
5,
3
2
Midpoint
Section
Slide 19
of 3.1,
10419
Section
Slide 20
of 3.1,
10420
3.2 The Slope of a
Line
R.1
Fractions
Find the Slope of a Line Given Two Points on the Line
Objectives
1. Find the slope of a line, given two points on the
line.
2. Find the slope of a line, given an equation of the
line.
3. Graph a line, given its slope and a point on the
line.
4. Use slopes to determine whether two lines are
parallel, perpendicular, or neither.
5. Solve problems involving average rate of change.
One of the important properties of a line is the rate at
which it is increasing or decreasing. The slope is the
ratio of vertical change, or rise, to horizontal change, or
run.
As we move from P to P :
1
P1
4 ft
P2
12 ft
Section
Slide 21
of 3.1,
10421
Find the Slope of a Line Given Two Points on the Line
2
Section
Slide 22
of 3.1,
10422
Example 1
Finding the Slope of a Line
Find the slope of the line containing the points (–3, 1)
and (3, 3).
( 3,1)
x1, y1
(3,3)
x2 , y 2
m
y2
x2
y1
x1
3 1
3 ( 3)
2
6
1
3
Rise = 3 – 1 = 2
Run = 3 – (–3) = 6
There is a rise of 1 unit
for a run of 3 units.
Section
Slide 23
of 3.1,
10423
Section
Slide 24
of 3.1,
10424
4
Find the Slope of a Line Given the Equation of the Line
Example 2
Find the slope of the line 4x – y = –8.
The intercepts can be used as the two points needed to find the slope.
Let y = 0 to find that the x-intercept is (–2, 0).
Let x = 0 to find that the y-intercept is (0, 8).
m
y2
x2
Finding the Slope of Horizontal and Vertical Lines
Example 3
y1
x1
8 0
0 ( 2)
Find the slope of each line.
a. y = 2
The graph of y = 2 is a horizontal line.
To find the slope, select two different
points on the line, such as (3, 2) and
(0, 2) and use the slope formula.
m
8
or 4
2
y2
x2
y1
x1
2 2
3 0
0
3
0
The rise is 0, so the slope is 0.
Section
Slide 25
of 3.1,
10425
Finding the Slope of Horizontal and Vertical Lines
Example 3
y2
x2
y1
x1
Finding the Slope from an Equation
Example 4
Find the slope of the graph 5x – 6y = 18.
Solve the equation for y.
The graph of x = 2 is a vertical line.
5 x 6 y 18
Find the slope of each line.
b. x = 2
The graph of x = 2 is a vertical line.
To find the slope, select two different
points on the line, such as (2, 2) and
(2, 0) and use the slope formula.
m
Section
Slide 26
of 3.1,
10426
2 0
2 2
2
0
6y
5 x 18
6y
6
5x
6
18
6
5
x 3
6
The slope is the coefficient of x, so the slope is 5/6.
y
Since division by 0 is undefined,
the slope is undefined.
Section
Slide 27
of 3.1,
10427
Using the Slope and a Point to Graph a Line
Example 5
Section
Slide 28
of 3.1,
10428
Orientation of a Line in the Plane
Graph the line with slope –2/3 and through the point (–5, 5).
Locate the point P(–5, 5).
From the slope formula:
m
change in y
change in x
2
3
P
Down 2
R
Right 3
So, move down 2 units
and then 3 units to the
right to the point R(–2,
3).
Section
Slide 29
of 3.1,
10429
Section
Slide 30
of 3.1,
10430
5
Slopes of Parallel and Perpendicular Lines
Determining Whether Two Lines are Parallel
Example 6
Determine whether the lines passing through (–2, 1)
and (4, 5) and through (3, 0) and (0, –2) are parallel.
Find the slope of each line.
m
m
y2
x2
y2
x2
y1
x1
y1
x1
5 1
4 ( 2)
2 0
0 3
4
6
2
3
2
3
2
3
Because the slopes are equal, the two lines are
parallel.
Section
Slide 31
of 3.1,
10431
Slopes of Parallel and Perpendicular Lines
Section
Slide 32
of 3.1,
10432
Determining Whether Two Lines are Perpendicular
Example 7
Are the lines with equations 2y = 3x – 6 and 2x + 3y = –6
perpendicular?
Find the slope of each line by solving each equation for y.
2y 3 x 6
2x 3y
6
A line with slope 0 is perpendicular to a line with undefined
slope.
y
3
x 3
2
3y
y
2x 6
2
x
3
2
The slopes are negative reciprocals because their product
is –1. The lines are perpendicular.
Section
Slide 33
of 3.1,
10433
Determining Whether Two Lines Are Parallel,
Perpendicular or Neither
Example 8
Determining Whether Two Lines Are Parallel,
Perpendicular or Neither
Example 8 (continued)
Decide whether each pair of lines is parallel,
perpendicular, or neither.
8x – 2y = 4 and 5x + y = –3
Find the slope of each line by first solving each equation
for y.
8x – 2y = 4
–8x
–8x
–2y = –8x + 4
2 2
2
y = 4x – 2
Section
Slide 34
of 3.1,
10434
5x + y = –3
–5x
–5x
y = –5x – 3
Slope is –5.
Decide whether each pair of lines is parallel,
perpendicular, or neither.
Because the slopes are not equal, the lines are not parallel.
To see if the lines are perpendicular, find the product of the
slopes.
5
4
4 · (–5) = –20
3
The lines are not perpendicular
2
1
because the product of their
-5 -4 -3 -2 -1
1 2 3 4 5
slopes is not –1.
-1
-2
The lines are neither
parallel nor perpendicular.
Slope is 4.
Section
Slide 35
of 3.1,
10435
-3
-4
-5
Section
Slide 36
of 3.1,
10436
6
Interpreting Slope as Average Rate of Change
Example 10
Cindy purchased a new car in 2006 for $18,000. In
2011, the car had a value of $7500. At what rate is the
car’s value changing with respect to time?
To determine the average rate of change, we need two
pairs of data. If x = 2006, then y = 18,000 and if x =
2011, then y = 7500.
y 2 y1
average rate of change
x2 x1
7500 18,000
2011 2006
10,500
5
2100
This means the car decreased in value by $2100 each
year from 2006 to 2011.
3.3 Linear Equations
in Two Variables
R.1 Fractions
Objectives
1.
2.
3.
4.
5.
6.
7.
Write an equation of a line, given its slope and yintercept.
Graph a line, using its slope and y-intercept.
Write an equation of a line, given its slope and a point on
the line.
Write an equation of a line, given two points on the line.
Write equations of horizontal or vertical lines.
Write an equation of a line parallel or perpendicular to a
given line.
Write an equation of a line that models real data.
Section
Slide 37
of 3.1,
10437
Write an equation of a line given its slope and y-intercept.
Section
Slide 38
of 3.1,
10438
Write an equation of a line given its slope and y-intercept.
Given the slope m of a line and the y-intercept b of the
line, we can determine its equation.
If we know the slope of a line and its y-intercept, we can
write its equation by substituting these values into the
above equation.
Section
Slide 39
of 3.1,
10439
Graph Lines Using Slope and y-Intercept
Writing an Equation of a Line
Example 1
Find an equation of a line with slope –¾ and y-intercept (0, –3).
m = –3/4 and b = –3. Substitute into the slope-intercept form.
y
y
Section
Slide 40
of 3.1,
10440
mx b
Example 2
Graph the line having slope 3/2 and y-intercept (0, 3).
rise change in y 3
m
run change in x 2
Plot the y-intercept (0, 3).
Move up 3 units and to
the right 2 units.
3
x 3
4
Join the points with a
straight line.
Section
Slide 41
of 3.1,
10441
Section
Slide 42
of 3.1,
10442
7
Write an equation of a line, given its slope and a point on the line.
Write an equation of a line, given its slope and a point on the line.
If we know the slope m of a line and the coordinates of a
point on the line, we can determine its equation.
If we know the slope of a line and the coordinates of a
single point on the line, we can write the equation of the
line by substituting these values into the equation above.
Section
Slide 43
of 3.1,
10443
Finding the Equation of a Line, Given the Slope and a Point
Example 3
Find an equation of the line having slope 1 and passing
through the point (2/5, 1).
Use the point-slope form of the equation of a line with
(x1, y1) = (2/5, 1) and m = 1.
y y1 m( x x1 )
2
5
y 1 1 x
y 1 x
y
x
y
x
Section
Slide 44
of 3.1,
10444
Finding an Equation of a Line, Given Two Points
Example 4
Find an equation of the line containing the points (–1, 3) and
(2, –1).
We begin by finding the slope of the line.
4
4
y 2 y1
1 3
m
3
x2 x1 2 ( 1) 3
Use either point and substitute into the point-slope form of
the equation of a line.
3 y 3
4 x 1
y y1 m( x x1 )
3y 9
4x 4
4
y 3
x ( 1)
4
x
3
y
5
3
2
5
2
1
5
3
5
Section
Slide 45
of 3.1,
10445
Section
Slide 46
of 3.1,
10446
Writing Equations of Horizontal and Vertical Lines
Equations of Horizontal and Vertical Lines
Example 5
Write an equation of the line passing through the point (–3, 3)
that satisfies the given condition.
a. The line has slope 0.
Since the slope is 0, this is a horizontal line.
The equation is y = 3.
b. The line has undefined slope.
This is a vertical line.
The equation is x = –3.
Section
Slide 47
of 3.1,
10447
Section
Slide 48
of 3.1,
10448
8
Writing Equations of Parallel or Perpendicular Lines
Writing Equations of Parallel or Perpendicular Lines
Example 6
Write an equation in slope-intercept form of the line passing
through the point (4, –7) that is parallel to the graph of
x + 2y = 6.
Continued.
Write an equation in slope-intercept form of the line passing
through the point (4, –7) that satisfies the given condition.
a. The line is parallel to the graph of x + 2y = 6.
y
Find the slope of the given line by solving for y.
x 2y 6
2y
x 6
1
y
x 3
2
A line parallel will have the same slope.
y
y1
2( y
7)
2y 14
2y
m( x
x1 )
y
1
( x 4)
2
y ( 7)
1
( x 4)
2
x 4
( 7)
x
4
x 10
1
x 5
2
Section
Slide 49
of 3.1,
10449
Writing Equations of Parallel or Perpendicular Lines
Example 6b
Write an equation in slope-intercept form of the line passing
through the point (0, 0) that is perpendicular to the graph of
2x + 3y = 7.
Find the slope of the given line by solving for y.
x 3y 7
3y
2x 7
2
7
y
x
3
3
Section
Slide 50
of 3.1,
10450
Writing Equations of Parallel or Perpendicular Lines
Continued.
Write an equation in slope-intercept form of the line passing
through the point (0, 0) that is perpendicular to the graph of
2x + 3y = 7.
Use the point (0, 0) and the point-slope form.
y
y
y1
0
A line perpendicular will have a slope of 3/2.
y
m( x
x1 )
3
( x 0)
2
3
x
2
Section
Slide 51
of 3.1,
10451
Section
Slide 52
of 3.1,
10452
Writing A Linear Equation to Describe Data
Example 7
A veterinarian charges $45 to visit a farm where cattle are
raised. The price to vaccinate each animal is $18. Write an
equation that defines the total bill that the veterinarian will
submit to vaccinate all the cattle at the farm.
Let x denote the number of cattle to be vaccinated.
3.4 Linear Inequalities
in Two Variables
R.1 Fractions
Objectives
1. Graph linear inequalities in two variables.
2. Graph the intersection of two linear inequalities.
3. Graph the union of two linear inequalities.
The cost y of only the vaccinations can be found by the linear
equation y = 18x.
There is a vet charge of $45 to visit the farm.
The total bill can be described by y = 18x + 45.
Section
Slide 53
of 3.1,
10453
Section
Slide 54
of 3.1,
10454
9
Graph Linear Inequalities in Two Variables
Graph Linear Inequalities in Two Variables.
In Section 2.1, we graphed linear inequalities in one variable on the
number line. In this section we learn to graph linear inequalities in
two variables on a rectangular coordinate system.
Step 1 Draw the graph of the straight line that is the
boundary. Make the line solid if the inequality
involves , or . Make the line dashed if the
inequality involves < or >.
Step 2 Choose a test point. Choose any point not on
the line, and substitute the coordinates of this
point in the inequality.
Step 3 Shade the appropriate region. Shade the
region that includes the test point if it satisfies
the original inequality. Otherwise, shade the
region on the other side of the boundary line.
Section
Slide 55
of 3.1,
10455
Section
Slide 56
of 3.1,
10456
Graphing a Linear Inequality
Graph Linear Inequalities
Example 1
Continued.
Graph the inequality 2x + 3y
6.
Graph the inequality 2x + 3y
The inequality 2x + 3y ≤ 6 means that
2x + 3y < 6 or 2x + 3y = 6.
The graph of 2x + 3y = 6 is a line. This boundary line
divides the plane into two regions. The graph of the
solutions of the inequality 2x + 3y < 6 will include only one
of these regions. We find the required region by checking
a test point.
We choose any point not on the boundary line. Because
(0, 0) is easy to substitute, we often use it.
6.
Check (0, 0)
2x + 3y ≤ 6
2(0) + 3(0) ≤ 6
0+0≤6
0≤6
True.
Since the last statement is true,
we shade the region that
includes the test point (0, 0).
Section
Slide 57
of 3.1,
10457
Graph a Linear Inequality with Boundary Through the Origin
Section
Slide 58
of 3.1,
10458
Graphing the Intersection of Two Inequalities
Example 2
Example 3
Graph the inequality y – 3x < 0.
y 3x 0
y 3x
Graph 3 x
Begin by graphing y = 3x,
using a dashed line.
4y
12 and y
2.
Graph each of the two inequalities separately.
Thus, we shade the region
containing (1,1).
Shade the common area.
5
Since (0, 0) is on the boundary
line, choose a different test
point. Here, we choose (1,1).
4
3
2
1
-5 -4 -3 -2 -1
1 < 3(1)
-1
1
2
3
4
5
-2
1<3
True
-3
-4
-5
Section
Slide 59
of 3.1,
10459
Section
Slide 60
of 3.1,
10460
10
Graphing the Union of Two Inequalities
Example 4
Graph 3 x
4y
12 or y
3.5 Introduction to
Relations
R.1
Fractionsand Functions
Objectives
2.
Graph each of the two inequalities separately.
The graph of the union includes all points in either inequality.
Shade the common area.
1. Distinguish between independent and dependent
variables.
2. Define and identify relations and functions.
3. Find domain and range.
4. Identify functions defined by graphs and
equations.
Section
Slide 61
of 3.1,
10461
Section
Slide 62
of 3.1,
10462
Independent and Dependent Variables
Independent and Dependent Variables
We often describe one quantity in terms of another. We can indicate the
relationship between these quantities by writing ordered pairs in which the
first number is used to arrive at the second number. Here are some
examples.
(5, $11)
We often describe one quantity in terms of another. We can
indicate the relationship between these quantities by writing
ordered pairs in which the first number is used to arrive at the
second number. Here are some examples.
5 gallons of gasoline
(8, $17.60)
8 gallons of gasoline
will cost $11. The total cost
depends on the number of
gallons purchased.
(the number of gallons, the total cost)
depends on
will cost $17.60. Again, the
total cost depends on the
number of gallons purchased.
Section
Slide 63
of 3.1,
10463
Section
Slide 64
of 3.1,
10464
Independent and Dependent Variables
Independent and Dependent Variables
We often describe one quantity in terms of another. We can indicate the
relationship between these quantities by writing ordered pairs in which the
first number is used to arrive at the second number. Here are some
examples.
(10, $150)
We often describe one quantity in terms of another. We can indicate the
relationship between these quantities by writing ordered pairs in which the
first number is used to arrive at the second number. Here are some
examples.
Working for 10 hours,
(15, $225)
Working for 15 hours,
you will earn $150. The total
gross pay depends on the
number of hours worked.
(the number of hours worked, the total gross pay)
depends on
you will earn $225. The total
gross pay depends on the
number of hours worked.
Section
Slide 65
of 3.1,
10465
Section
Slide 66
of 3.1,
10466
11
Independent and Dependent Variables
Define and identify relations and functions.
We often describe one quantity in terms of another. We can indicate the
relationship between these quantities by writing ordered pairs in which the
first number is used to arrive at the second number. Here are some
examples.
Relation
A relation is any set of ordered pairs.
A special kind of relation, called a function, is very important in
mathematics and its applications.
Generalizing, if the value of the variable y depends on the value of the
variable x, then y is called the dependent variable and x is the
independent variable.
Function
Independent variable
A function is a relation in which, for each value of the first component
of the ordered pairs, there is exactly one value of the second component.
(x, y)
Dependent variable
Section
Slide 67
of 3.1,
10467
Section
Slide 68
of 3.1,
10468
Determining Whether Relations Are Functions
Example 1
Mapping Relations
Tell whether each relation defines a function.
L = { (2, 3), (–5, 8), (4, 10) }
F
G
M = { (–3, 0), (–1, 4), (1, 7), (3, 7) }
1
2
–3
5
Relations L and M are functions, because for each different x-value there
is exactly one y-value.
4
3
In relation N, the first and third ordered pairs have the same x-value
paired with two different y-values (6 is paired with both 2 and 5), so N is a
relation but not a function. In a function, no two ordered pairs can have
the same first component and different second components.
F is a function.
N = { (6, 2), (–4, 4), (6, 5) }
y
–2
6
0
0
2
–6
–2
0
Section
Slide 70
of 3.1,
10470
Using an Equation to Define a Relation or Function
y
x
6
G is not a function.
Section
Slide 69
of 3.1,
10469
Tables and Graphs
–1
Relations and functions can also be described using rules. Usually, the rule
is given as an equation. For example, from the previous slide, the chart and
graph could be described using the following equation.
y = –3x
x
O
Dependent variable
Table of the
function, F
Independent variable
An equation is the most efficient way to define a relation or function.
Graph of the function, F
Section
Slide 71
of 3.1,
10471
Section
Slide 72
of 3.1,
10472
12
Functions
Domain and Range
NOTE
Another way to think of a function relationship is to think of the
independent variable as an input and the dependent variable as an
output. This is illustrated by the input-output (function) machine (below)
for the function defined by y = –3x.
In a relation, the set of all values of the independent variable (x) is the
domain. The set of all values of the dependent variable (y) is the range.
(Input x) (Output y)
–6
2
–5
4
2
(Input x)
y = –3x –12
–6
15
–5
15
4
–12
(Output y)
Section
Slide 73
of 3.1,
10473
Finding Domains and Ranges of Relations
Example 2
Give the domain and range of each relation. Tell whether the relation defines
a function.
Section
Slide 74
of 3.1,
10474
Finding Domains and Ranges of Relations
Continued.
Give the domain and range of each relation. Tell whether the relation defines
a function.
(b)
(a) { (3, –8), (5, 9), (5, 11), (8, 15) }
6
M
1
The domain, the set of x-values, is {3, 5, 8}; the range, the set of y-values,
is {–8, 9, 11, 15}. This relation is not a function because the same x-value 5 is
paired with two different y-values, 9 and 11.
–9
N
The domain of this relation is {6, 1, –9}. The range is {M, N}.
This mapping defines a function – each x-value corresponds to exactly one
y-value.
Section
Slide 75
of 3.1,
10475
Section
Slide 76
of 3.1,
10476
Finding Domains and Ranges from Graphs
Finding Domains and Ranges of Relations
Continued.
Example 3
Give the domain and range of each relation. Tell whether the relation defines
a function.
(c)
x
y
–2
3
1
3
2
3
Give the domain and range of each relation.
y
(a)
(–3, 2)
(2, 1)
O
This is a table of ordered pairs, so the domain is the set of x-values,
{–2, 1, 2}, and the range is the set of y-values, {3}. The table defines a
function because each different x-value corresponds to exactly one y-value
(even though it is the same y-value).
Section
Slide 77
of 3.1,
10477
The domain is the set of
x-values, {–3, 0, 2 , 4}. The
range, the set of y-values, is
{–3, –1, 1, 2}.
x
(4, –1)
(0, –3)
Section
Slide 78
of 3.1,
10478
13
Finding Domains and Ranges from Graphs
Continued.
Give the domain and range of each relation.
Give the domain and range of each relation.
y
(b)
Finding Domains and Ranges from Graphs
Continued.
Range
O
The x-values of the points on the
graph include all numbers between
–7 and 2, inclusive. The y-values
include all numbers between –2 and
2, inclusive. Using interval notation,
y
(c)
The arrowheads indicate that the
line extends indefinitely left and right,
as well as up and down. Therefore,
both the domain and range include
all real numbers, written (-∞, ∞).
x
x
O
the domain is [–7, 2];
the range is [–2, 2].
Domain
Section
Slide 79
of 3.1,
10479
Section
Slide 80
of 3.1,
10480
Finding Domains and Ranges from Graphs
Continued.
Agreement on Domain
Give the domain and range of each relation.
y
(d)
x
O
The arrowheads indicate that the
graph extends indefinitely left and
right, as well as upward. The domain
is (-∞, ∞).Because there is a least yvalue, –1, the range includes all
numbers greater than or equal to –1,
written [–1, ∞).
The domain of a relation is assumed to be all real numbers that produce
real numbers when substituted for the independent variable.
Section
Slide 81
of 3.1,
10481
Example 4
Vertical Line Test
If every vertical line intersects the graph of a relation in no more than
one point, then the relation represents a function.
(a)
y
(b)
Section
Slide 82
of 3.1,
10482
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
y
(a)
y
This relation is a function.
(–3, 2)
x
Not a function – the same
x-value corresponds to two
different y-values.
x
Function – each x-value
corresponds to only one
y-value.
Section
Slide 83
of 3.1,
10483
(2, 1)
O
x
(4, –1)
(0, –3)
Section
Slide 84
of 3.1,
10484
14
Continued.
Using the Vertical Line Test
Continued.
Use the vertical line test to determine whether each relation is a function.
y
(b)
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
y
(c)
This graph fails the vertical line test
since the same x-value corresponds
to two different y-values; therefore,
it is not the graph of a function.
O
This relation is a function.
x
O
Section
Slide 85
of 3.1,
10485
x
Section
Slide 86
of 3.1,
10486
Identifying Functions from Their Equations
Continued.
Using the Vertical Line Test
Example 5
Use the vertical line test to determine whether each relation is a function.
y
(d)
This relation is a function.
O
Decide whether each relation defines a function and give the
domain.
(a) y = x – 5
In the defining equation, y = x – 5, y is always found by
subtracting 5 from x. Thus, each value of x corresponds to just
one value of y and the relation defines a function; x can be any
real number, so the domain is (–∞, ∞).
x
Section
Slide 87
of 3.1,
10487
Section
Slide 88
of 3.1,
10488
Identifying Functions from Their Equations
Continued.
Decide whether each relation defines a function and give the domain.
(b)
y=
Identifying Functions from Their Equations
Continued.
3x – 1
Decide whether each relation defines a function and give the domain.
(c)
For any choice of x in the domain, there is exactly one corresponding
value for y (the radical is a nonnegative number), so this equation defines a
function. Since the equation involves a square root, the quantity under the
radical sign cannot be negative. Thus,
y2 = x
The ordered pair (9, 3) and (9, –3) both satisfy this equation. Since one
value of x, 9, corresponds to two values of y, 3 and –3, this equation does
not define a function. Because x is equal to the square of y, the values of x
must always be nonnegative. The domain of the relation is [0, ∞).
3x – 1 ≥ 0
3x ≥ 1
x ≥1
3
,
and the domain of the function is [1 , ∞).
3
Section
Slide 89
of 3.1,
10489
Section
Slide 90
of 3.1,
10490
15
Identifying Functions from Their Equations
Continued.
Decide whether each relation defines a function and give the domain.
(d)
Identifying Functions from Their Equations
Continued.
y≥x–3
By definition, y is a function of x if every value of x leads to exactly one
value of y. Here a particular value of x, say 4, corresponds to many values
of y. The ordered pairs (4, 7), (4, 6), (4, 5), and so on, all satisfy the
inequality. Thus, an inequality never defines a function. Any number can
be used for x so the domain is the set of real numbers (–∞, ∞).
Decide whether each relation defines a function and give the domain.
(e) y =
3
x+4
Given any value of x in the domain, we find y by adding 4, then dividing
the result into 3. This process produces exactly one value of y for each value
in the domain, so this equation defines a function. The domain includes all
real numbers except those that make the denominator 0. We find these
numbers by setting the denominator equal to 0 and solving for x.
x+4=0
x = –4
The domain includes all real numbers except –4, written (–∞, –4) U (–4, ∞).
Section
Slide 91
of 3.1,
10491
Section
Slide 92
of 3.1,
10492
Variations of the Definition of Function
1. A function is a relation in which, for each value of the first component
of the ordered pairs, there is exactly one value of the second
component.
2. A function is a set of ordered pairs in which no first component is
repeated.
3. A function is an equation (rule) or correspondence (mapping) that
assigns exactly one range value to each distinct domain value.
Section
Slide 93
of 3.1,
10493
16