Palm Beach State College - Dr. Heath
3
MAT1033C Intermediate Algebra
Graphs, Linear Equations,
and Functions
3.1 The Rectangular
Coordinate System
R.1 Fractions
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.
Section 3.2, 1
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Rectangular
(or Cartesian, for Descartes)
Coordinate System
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Section 3.2, 2
EX. 1
Complete each ordered pair for 3x + 4y = 7.
y
8
(a)(5, ? )
6
4
2
-8
-6
-4
-2
0 0
0 -2 2
x
4
6
8
(b)( ? , –5)
-4
-6
-8
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Section 3.2, 3
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Section 3.2, 4
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Intercepts
A linear equation in two variables can be written
in the form
Ax + By = C,
y
y-intercept (where the line intersects
the y-axis)
where A, B, and C are real numbers
(A and B not both 0).
This form is called standard form.
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x-intercept (where the
line intersects
the x-axis)
x
Section 3.2, 5
EXAMPLE 2
Finding Intercepts
When graphing the equation of a line, find the
intercepts as follows.
Section 3.2, 6
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Finding Intercepts
Find the x- and y-intercepts of 2x – y = 6,
and graph the equation.
y
let y = 0 to find the x-intercept;
let x = 0 to find the y-intercept.
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x
Section 3.2, 7
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Section 3.2, 8
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Graphing a Horizontal Line
EXAMPLE 3
Graphing a Vertical Line
Graph x + 2 = 5.
Graph y = –3.
y
y
x
x
x
y
2
0
0
–2
–2
Section 3.2, 9
Horizontal and Vertical Lines
2.
Graphing a Line That Passes
through the Origin
Graph 3x + y = 0.
y
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.
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Section 3.2, 10
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EXAMPLE 4
To avoid confusing equations of horizontal and
vertical lines, keep the following in mind.
1.
y
2
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CAUTION
x
Section 3.2, 11
x
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Section 3.2, 12
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
3.2 The Slope of R.1
a Line
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.
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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 3.2, 13
Find the Slope of a Line Given Two Points on the Line
2
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Example 1
Section 3.2, 14
Finding the Slope of a Line
Find the slope of the line containing the
points (–3, 1) and (3, 3).
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Section 3.2, 15
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Section 3.2, 16
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Find the Slope of a Line Given the Equation of the Line
Example 2
Find the slope of each line.
a. y = 2
Find the slope of the line 4x – y = –8.
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Section 3.2, 17
Finding the Slope of Horizontal and Vertical Lines
Find the slope of each line.
b. x = 2
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Finding the Slope of Horizontal and Vertical Lines
Example 3
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Section 3.2, 18
Example 4 Finding the Slope from an Equation
Find the slope of the graph 5x – 6y = 18.
Section 3.2, 19
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Section 3.2, 20
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Using the Slope and a Point to Graph a Line
Example 5
Orientation of a Line in the Plane
Graph the line with slope –2/3 and
through the point (–5, 5).
P
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Section 3.2, 21
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Example 6
Section 3.2, 22
Determining Whether Two Lines are Parallel
Determine whether the lines passing through
(–2, 1) and (4, 5) and through (3, 0) and (0, –2)
are parallel.
A line with slope 0 is perpendicular to a line with undefined
slope.
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Section 3.2, 23
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Section 3.2, 24
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Example 7 Determining Whether Two Lines are Perpendicular
Are the lines with equations 2y = 3x – 6 and
2x + 3y = –6 perpendicular?
Example 8
Decide whether each pair of lines is parallel,
perpendicular, or neither.
8x – 2y = 4 and 5x + y = –3
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Section 3.2, 25
Given the slope m of a line and the y-intercept
b of the line, we can determine its equation.
Objectives
2.
3.
4.
5.
6.
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.
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Section 3.2, 26
Write an equation of a line given its slope and y-intercept.
3.3 Linear Equations
in Two Variables
R.1 Fractions
1.
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Section 3.2, 27
y = mx+b
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Section 3.2, 28
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Example 1
Writing an Equation of a Line
Find an equation of a line with slope –¾ and
y-intercept (0, –3).
m = ____ and b = ___.
Substitute into the slope-intercept form.
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Example 2
Section 3.2, 29
Section 3.2, 30
Graph Lines Using Slope and y-Intercept
Graph the line having slope 3/2 and y-intercept (0, 3).
m
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rise change in y 3


run change in x 2
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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.
Section 3.2, 31
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Section 3.2, 32
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
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) = ( , ) and m =
y  y1  m( x  x1 )
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Section 3.2, 33
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Section 3.2, 34
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).
Equations of Horizontal and Vertical Lines
We begin by finding the slope of the line.
y  y1
1  3
m 2

x2  x1 2  ( 1)
Use either point and substitute into the point-slope form of
the equation of a line.
y  y1  m( x  x1 )
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Section 3.2, 35
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Section 3.2, 36
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Writing Equations of Horizontal and Vertical Lines
Writing Equations of Parallel or Perpendicular Lines
Example 5
Example 6
Write an equation of the line passing through the point (–3, 3)
that satisfies the given condition.
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.
a. The line has slope 0.
Find the slope of the given line by solving for y.
b. The line has undefined slope.
A line parallel will have the same slope.
y  y1  m( x  x1 )
1
y  ( 7)   ( x  4)
2
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Section 3.2, 37
Writing Equations of Parallel or Perpendicular Lines
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Section 3.2, 38
Writing Equations of Parallel or Perpendicular Lines
Example 6b
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.
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Section 3.2, 39
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.
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Section 3.2, 40
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
3.4 Linear Inequalities
in Two Variables
R.1 Fractions
Graph Linear Inequalities in Two Variables
Objectives
1. Graph linear inequalities in two variables.
2. Graph the intersection of two linear inequalities.
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Section 3.2, 41
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Section 3.2, 42
Graphing a Linear Inequality
Graph Linear Inequalities in Two Variables.
Example 1
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.
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Section 3.2, 43
Graph the inequality 2x + 3y  6.
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Section 3.2, 44
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Graph a Linear Inequality with Boundary Through the Orig
Graphing the Intersection of Two Inequalities
Example 2
Example 3
Graph the inequality y – 3x < 0.
Graph 3 x  4 y  12 and y  2.
Graph each of the two inequalities separately.
Shade the common area.
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Section 3.2, 45
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Section 3.2, 46
3.5 Introduction to
R.1Relations
Fractionsand Functions
Define and identify relations and functions.
Objectives
Relation
1. Define and identify relations and functions.
2. Find domain and range.
A relation is any set of ordered pairs.
A special kind of relation, called a function, is very important in
mathematics and its applications.
Function
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.
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Section 3.2, 47
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Section 3.2, 48
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Determining Whether Relations Are Functions
Example 1
Mapping Relations
Tell whether each relation defines a function.
L = { (2, 3), (–5, 8), (4, 10) }
F
M = { (–3, 0), (–1, 4), (1, 7), (3, 7) }
G
1
2
–3
5
4
3
–1
6
–2
0
N = { (6, 2), (–4, 4), (6, 5) }
Section 3.2, 49
Copyright © 2014, 2010, 2006 Pearson Education, Inc.
NOTE
y
y
–2
6
0
0
2
–6
Functions
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.
Tables and Graphs
x
Section 3.2, 50
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x
O
(Input x)
(Output y)
2
–5
(Input x)
Table of the
function, F
4
Graph of the function, F
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Section 3.2, 51
y = –3x–12
–6
15
(Output y)
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Section 3.2, 52
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Example 2
Give the domain and range of each relation.
Tell whether the relation defines a function.
Domain and Range
In a relation, the set of all values of the independent
variable (x) is the domain.
(a) { (3, –8), (5, 9), (5, 11), (8, 15) }
The set of all values of the dependent
variable (y) is the range.
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Section 3.2, 53
y
–2
3
1
3
2
3
Section 3.2, 54
Example 3
Give the domain and range of each relation.
Tell whether the relation defines a function.
x
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Give the domain and range of each relation.
(a)
y
(–3, 2)
(2, 1)
O
x
(4, –1)
(0, –3)
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Section 3.2, 55
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Section 3.2, 56
Palm Beach State College - Dr. Heath
MAT1033C Intermediate Algebra
Finding Domains and Ranges from Graphs
Finding Domains and Ranges from Graphs
Continued.
Continued.
Give the domain and range of each relation.
(b)
Give the domain and range of each relation.
(c)
y
O
y
x
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x
O
Section 3.2, 57
Section 3.2, 58
Copyright © 2014, 2010, 2006 Pearson Education, Inc.
Finding Domains and Ranges from Graphs
Vertical Line Test
Continued.
Give the domain and range of each relation.
(d)
If every vertical line intersects the graph of a relation in no more than
one point, then the relation represents a function.
y
(a)
O
y
(b)
y
x
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x
Section 3.2, 59
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x
Section 3.2, 60
Palm Beach State College - Dr. Heath
Example 4
MAT1033C Intermediate Algebra
Using the Vertical Line Test
Use the vertical line test to determine whether each relation is a function.
(a)
Use the vertical line test to determine whether each relation is a function.
(b)
y
(–3, 2)
Using the Vertical Line Test
Continued.
y
(2, 1)
O
x
O
(4, –1)
x
(0, –3)
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Section 3.2, 61
Using the Vertical Line Test
Continued.
O
Use the vertical line test to determine whether each relation is a function.
(d)
y
x
Copyright © 2014, 2010, 2006 Pearson Education, Inc.
y
O
Section 3.2, 63
Section 3.2, 62
Using the Vertical Line Test
Continued.
Use the vertical line test to determine whether each relation is a function.
(c)
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x
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Section 3.2, 64