Graphing Using the Rectangular Coordinate System

1
Section 3.1

• Graphing Using the Rectangular Coordinate System

2
Objectives

• Construct a rectangular coordinate system
• Plot ordered pairs and determine the coordinates
  of a point
• Graph paired data
• Read line graphs

3
Objective 1 Construct a Rectangular Coordinate
System

• When designing the Gateway Arch in St. Louis,
  architects created a mathematical model called a
  rectangular coordinate graph.
• This graph, shown here, is drawn on a grid called
  a rectangular coordinate system.
• This coordinate system is also called a Cartesian
  coordinate system, after the 17th-century French
  mathematician René Descartes.

4
Objective 1 Construct a Rectangular Coordinate
System

• A rectangular coordinate system is formed by two
  perpendicular number lines.
• The horizontal number line is usually called the
  x-axis, and the vertical number line is usually
  called the y-axis.
• On the x-axis, the positive direction is to the
  right. On the y-axis, the positive direction is
  upward.
• Each axis should be scaled to fit the data. For
  example, the axes of the graph of the arch are
  scaled in units of 100 feet.
• The point where the axes intersect is called the
  origin. This is the zero point on each axis.
• The axes form a coordinate plane, and they divide
  it into four regions called quadrants, which are
  numbered counterclockwise using Roman numerals.

5
Objective 1 Construct a Rectangular Coordinate
System

• Each point in a coordinate plane can be
  identified by an ordered pair of real numbers x
  and y written in the form (x, y).
• The first number, x, in the pair is called the
  x-coordinate, and the second number, y, is called
  the y-coordinate.

Read as the point three, negative four or as
the ordered pair three, negative four.
6
Objective 2 Plot Ordered Pairs and Determine the
Coordinates of a Point

• The process of locating a point in the coordinate
  plane is called graphing or plotting the point.
• In the figure seen here, we use blue arrows to
  show how to graph the point with coordinates (3,
  -4).

• Since the x-coordinate, 3, is positive, we start
  at the origin and move 3 units to the right along
  the x-axis.
• Since the y-coordinate, -4, is negative, we then
  move down 4 units and draw a dot.
• This locates the point (3, -4).

7
Objective 2 Plot Ordered Pairs and Determine the
Coordinates of a Point

• In the figure, red arrows are used to show how to
  plot the point (-4, 3).
• We start at the origin, move 4 units to the left
  along the x-axis, then move up 3 units and draw a
  dot. This locates the point (-4, 3).

8
EXAMPLE 1

• Plot each point. Then state the quadrant in which
  it lies or the axis on which it lies. a. (4, 4),
  b. (-1, -7/2), c. (0, 2.5), d. (-3, 0), e. (0, 0)

• Strategy After identifying the x- and
  y-coordinates of the ordered pair, we will move
  the corresponding number of units left, right,
  up, or down to locate the point.

Why The coordinates of a point determine its
location on the coordinate plane.
9
EXAMPLE 1

• Solution

a. Since the x-coordinate, 4, is positive, we
start at the origin and move 4 units to the right
along the x-axis. Since the y-coordinate, 4, is
positive, we then move up 4 units and draw a dot.
This locates the point (4, 4). The point lies in
quadrant I.
b. To plot (-1, -7/2), we begin at the origin and
move 1 unit to the left, because the x-coordinate
is -1. Then, since the y-coordinate is negative,
we move 7/2 units, or 3½ units, down. The point
lies in quadrant III.
10
EXAMPLE 1

• Solution

c. To plot (0, 2.5), we begin at the origin and
do not move right or left, because the
x-coordinate is 0. Since the y-coordinate is
positive, we move 2.5 units up. The point lies on
the y-axis.
d. To plot (-3, 0), we begin at the origin and
move 3 units to the left, because the
x-coordinate is -3. Since the y-coordinate is 0,
we do not move up or down. The point lies on the
x-axis.
e. To plot (0, 0), we begin at the origin, and we
remain there because both coordinates are 0. The
point with coordinates (0, 0) is the origin.
11
Objective 3 Graph Paired Data

• Every day, we deal with quantities that are
  related
• The time it takes to cook a roast depends on the
  weight of the roast.
• The money we earn depends on the number of hours
  we work.
• The sales tax that we pay depends on the price of
  the item purchased.
• We can use graphs to visualize such
  relationships. For example, suppose a tub is
  filling with water, as shown below.
• Obviously, the amount of water in the tub depends
  on how long the water has been running.
• To graph this relationship, we can use the
  measurements that were taken as the tub began
  to fill.

12
Objective 3 Graph Paired Data

• The data in each row of the table can be written
  as an ordered pair and plotted on a rectangular
  coordinate system.
• Since the first coordinate of each ordered pair
  is a time, we label the x-axis Time (min). The
  second coordinate is an amount of water, so we
  label the y-axis Amount of water (gal).
• The y-axis is scaled in larger units (multiples
  of 4 gallons) because the size of the data ranges
  from 0 to 32 gallons.

13
Objective 3 Graph Paired Data

• After plotting the ordered pairs, we use a
  straightedge to draw a line through the points.
• As expected, the completed graph shows that the
  amount of water in the tub increases steadily as
  the water is allowed to run.

14
Objective 3 Graph Paired Data

• We can use the graph to determine the amount of
  water in the tub at various times.
• For example, the green dashed line on the graph
  shows that in 2 minutes, the tub will contain 16
  gallons of water.

• This process, called interpolation, uses known
  information to predict values that are not known
  but are within the range of the data.
• The blue dashed line on the graph shows that in 5
  minutes, the tub will contain 40 gallons of
  water.
• This process, called extrapolation, uses known
  information to predict values that are not known
  and are outside the range of the data.

15
Objective 4 Read Line Graphs

• Since graphs are a popular way to present
  information, the ability to read and interpret
  them is very important.

16
EXAMPLE 3
TV Shows

• The following graph shows the number of people in
  an audience before, during, and after the taping
  of a television show. Use the graph to answer the
  following questions.

a. How many people were in the audience when the
taping began? b. At what times were there
exactly 100 people in the audience? c. How long
did it take the audience to leave after the
taping ended?
17
EXAMPLE 3
TV Shows

• Strategy We will use an ordered pair of the form
  (time, size of audience) to describe each
  situation mentioned in parts (a), (b), and (c).

Why The coordinates of specific points on the
graph can be used to answer each of these
questions.
18
EXAMPLE 3
TV Shows

• Solution

a. The time when the taping began is represented
by 0 on the x-axis. The point on the graph
directly above 0 is (0, 200). The y-coordinate
indicates that 200 people were in the audience
when the taping began.
b. We can draw a horizontal line passing through
100 on the y-axis. Since the line intersects the
graph twice, at (-20, 100) and at (80, 100),
there are two times when 100 people were in the
audience. The x-coordinates of the points tell us
those times 20 minutes before the taping began,
and 80 minutes after.
c. The x-coordinate of the point (70, 200) tells
us when the audience began to leave. The
x-coordinate of (90, 0) tells when the exiting
was completed. Subtracting the x-coordinates, we
see that it took 90 - 70 20 minutes for the
audience to leave.
19
Section 3.2

• Graphing Linear Equations

20
Objectives

• Determine whether an ordered pair is a solution
  of an equation
• Complete ordered-pair solutions of equations
• Construct a table of solutions
• Graph linear equations by plotting points
• Use graphs of linear equations to solve applied
  problems

21
Objective 1 Determine Whether an Ordered Pair Is
a Solution of an Equation

• We have previously solved equations in one
  variable.
• For example, x 3 9 is an equation in x.
• If we subtract 3 from both sides, we see that 6
  is the solution. To verify this, we replace x
  with 6 and note that the result is a true
  statement 9 9.
• In this chapter, we extend our equation-solving
  skills to find solutions of equations in two
  variables.
• To begin, lets consider y x - 1, an equation
  in x and y.
• A solution of y x - 1 is a pair of values, one
  for x and one for y, that make the equation true.
  To illustrate, suppose x is 5 and y is 4.
  Then we have

22
Objective 1 Determine Whether an Ordered Pair Is
a Solution of an Equation

• Since the result is a true statement, x 5 and y
  4 is a solution of y x - 1.
• We write the solution as the ordered pair (5, 4),
  with the value of x listed first.
• We say that (5, 4) satisfies the equation.

• In general, a solution of an equation in two
  variables is an ordered pair of numbers that
  makes the equation a true statement.

23
EXAMPLE 1

• Is (-1, -3) a solution of y x - 1?

• Strategy We will substitute -1 for x and -3 for
  y and see whether the resulting equation is true.

Why An ordered pair is a solution of y x - 1
if replacing the variables with the values of the
ordered pair results in a true statement.
24
EXAMPLE 1

• Solution

Conclusion Since -3 -2 is false, (-1, -3) is
not a solution of y x - 1.
25
Objective 2 Complete Ordered-Pair Solutions of
Equations

• If only one of the values of an ordered-pair
  solution is known, we can substitute it into the
  equation to determine the other value.

26
EXAMPLE 2

• Complete the solution (-5, ) of the equation
  y 2x 3.

• Strategy We will substitute the known
  x-coordinate of the solution into the given
  equation.

Why We can use the resulting equation in one
variable to find the unknown y-coordinate of the
solution.
27
EXAMPLE 2

• Solution

The completed ordered pair is (-5, 13).
28
Objective 3 Construct a Table of Solutions

• To find a solution of an equation in two
  variables, we can select a number, substitute it
  for one of the variables, and find the
  corresponding value of the other variable.
• For example, to find a solution of y x - 1, we
  can select a value for x, say, -4, substitute -4
  for x in the equation, and find y.

29
Objective 4 Graph Linear Equations by Plotting
Points

• It is impossible to list the infinitely many
  solutions of the equation y x - 1.
• However, to show all of its solutions, we can
  draw a mathematical picture of them. We call
  this picture the graph of the equation.
• To graph y x - 1, we plot the ordered pairs
  shown in the table on a rectangular coordinate
  system.
• Then we draw a straight line through the points,
  because the graph of any solution of y x - 1
  will lie on this line.
• We also draw arrowheads on either end of the line
  to indicatethat the solutions continue
  indefinitely in both directions,beyond what we
  can see on the coordinate grid.
• We call the line the graph of the equation. It
  represents all of the solutions of y x - 1.

30
Objective 4 Graph Linear Equations by Plotting
Points
31
Objective 4 Graph Linear Equations by Plotting
Points

• The equation y x - 1 is said to be linear and
  its graph is a line.
• By definition, a linear equation in two variables
  is any equation that can be written in the
  following form, where the variable terms appears
  on one side of an equal symbol and a constant
  appears on the other.
• A linear equation in two variables is an
  equation that can be written in the form Ax By
  C, where A, B, and C are real numbers and A
  and B are not both 0. This form is called
  standard form.

32
Objective 4 Graph Linear Equations by Plotting
Points

• Linear equations can be graphed in several ways.
• Generally, the form in which an equation is
  written determines the method that we use to
  graph it. To graph linear equations solved for
  y, such as y 2x 4, we can use the following
  point-plotting method.
• Graphing Linear Equations Solved for y by
  Plotting Points
• 1. Find three ordered pairs that are solutions
  of the equation by selecting three values for x
  and calculating the corresponding values of y.
• 2. Plot the solutions on a rectangular
  coordinate system.
• 3. Draw a straight line passing through the
  points. If the points do not lie on a line, check
  your calculations.

33
EXAMPLE 4

• Graph y 2x 4.

• Strategy We will find three solutions of the
  equation, plot them on a rectangular coordinate
  system, and then draw a straight line passing
  through the points.

Why To graph a linear equation in two variables
means to make a drawing that represents all of
its solutions.
34
EXAMPLE 4

• Solution

To find three solutions of this linear equation,
we select three values for x that will make the
computations easy. Then we find each
corresponding value of y.
35
EXAMPLE 4

• Solution

We enter the results in a table of solutions and
plot the points. Then we draw a straight line
through the points and label it y 2x 4.
36
EXAMPLE 4

• Solution

37
Objective 5 Use Graphs of Linear Equations to
Solve Applied Problems

• When linear equations are used to model real-life
  situations, they are often written in variables
  other than x and y.
• In such cases, we must make the appropriate
  changes when labeling the table of solutions and
  the graph of the equation.

38
EXAMPLE 7
Cleaning Windows

• The linear equation A -0.03n 32 estimates
  the amount A of glass-cleaning solution (in
  ounces) that is left in the bottle after the
  sprayer trigger has been pulled a total of n
  times. Graph the equation and use the graph to
  estimate the amount of solution that is left
  after 500 sprays.

• Strategy We will find three solutions of the
  equation, plot them on a rectangular coordinate
  system, and then draw a straight line passing
  through the points.

Why We can use the graph to estimate the amount
of solution left after any number of sprays.
39
EXAMPLE 7
Cleaning Windows

• Solution

Since A depends on n in the equation A -0.03n
32, solutions will have the form (n, A). To
find three solutions, we begin by selecting three
values of n. Because the number of sprays cannot
be negative, and the computations to find A
involve decimal multiplication, we select 0, 100,
and 1,000. For example, if n 100, we have
Thus, (100, 29) is a solution. It indicates that
after 100 sprays, 29 ounces of cleaner will be
left in the bottle.
40
EXAMPLE 7
Cleaning Windows

• Solution

In the same way, solutions are found for n 0
and n 1000 and listed in the table. Then the
ordered pairs are plotted and a straight line is
drawn through the points.
To graphically estimate the amount of solution
that is left after 500 sprays, we draw the dashed
blue lines, as shown. Reading on the vertical
A-axis, we see that after 500 sprays, about 17
ounces of glass cleaning solution would be left.
41
Section 3.3

• Intercepts

42
Objectives

• Identify intercepts of a graph
• Graph linear equations by finding intercepts
• Identify and graph horizontal and vertical lines
• Obtain information from intercepts

43
Objective 1 Identify Intercepts of a Graph

• The graph of y 2x - 4 is shown below.
• We see that the graph crosses the x-axis at the
  point (0, -4) this point is called the
  y-intercept of the graph.
• The graph crosses the x-axis at the point (2,
  0) this point is called the x-intercept of the
  graph.

44
EXAMPLE 1

• For the graphs in figures (a) and (b), give the
  coordinates of the x- and y-intercepts.

• Strategy We will determine where each graph
  (shown in red) crosses the x-axis and the y-axis.

Why The point at which a graph crosses the
x-axis is the x-intercept and the point at which
a graph crosses the y-axis is the y-intercept.
45
EXAMPLE 1

• Solution

a. In figure (a), the graph crosses the x-axis at
(-4, 0). This is the x-intercept. The graph
crosses the y-axis at (0, 1). This is the
y-intercept.
b. In figure (b), the horizontal line does not
cross the x-axis there is no x-intercept. The
graph crosses the y-axis at (0, -2). This is the
y-intercept.
46
Objective 1 Identify Intercepts of a Graph

• From the previous examples, we see that a
  y-intercept has an x-coordinate of 0, and an
  x-intercept has a y-coordinate of 0.
• These observations suggest the following
  procedures for finding the intercepts of a graph
  from its equation.
• Finding Intercepts
• To find the y-intercept, substitute 0 for x in
  the given equation and solve for y.
• To find the x-intercept, substitute 0 for y in
  the given equation and solve for x.

47
Objective 2 Graph Linear Equations by Finding
Intercepts

• Plotting the x- and y-intercepts of a graph and
  drawing a line through them is called the
  intercept method of graphing a line.
• This method is useful when graphing linear
  equations written in the standard (general) form
  Ax By C.
• The calculations for finding intercepts can be
  simplified if we realize what occurs when we
  substitute 0 for y or 0 for x in an equation
  written in the form Ax By C.

48
EXAMPLE 4

• Graph 3x -5y 8 by finding the intercepts.

• Strategy We will let x 0 to find the
  y-intercept of the graph. We will then let y 0
  to find the x-intercept.

Why Since two points determine a line, the
y-intercept and x-intercept are enough
information to graph this linear equation.
49
EXAMPLE 4

• Solution

We find the intercepts and select x 1 to find a
check point.
50
EXAMPLE 4
Solution
The ordered pairs are plotted as shown, and a
straight line is then drawn through them.
51
Objective 3 Identify and Graph Horizontal and
Vertical Lines

• Equations such as y 4 and x -3 are linear
  equations, because they can be written in the
  general form Ax By C.
• For example, y 4 is equivalent to 0x 1y 4,
• and x -3 is equivalent to 1x 0y -3.
• We now discuss how to graph these types of linear
  equations using point-plotting.

52
EXAMPLE 6
Graph y 4

• Strategy To find three ordered-pair solutions of
  this equation to plot, we will select three
  values for x and use 4 for y each time.

Why The given equation requires that y 4.
53
EXAMPLE 6
Graph y 4

• Solution

We can write the equation in general form as 0x
y 4. Since the coefficient of x is 0, the
numbers chosen for x have no effect on y. The
value of y is always 4. For example, if x 2, we
have
54
EXAMPLE 6
Graph y 4

• Solution

One solution is (2, 4). To find two more
solutions, we choose x 0 and x -3. For any
x-value, the y-value is always 4, so we enter
(0, 4) and (-3, 4) in the table. If we plot the
ordered pairs and draw a straight line through
the points, the result is a horizontal line. The
y-intercept is (0, 4) and there is no
x-intercept.
55
EXAMPLE 7
Graph x -3

• Strategy To find three ordered-pair solutions of
  this equation to plot, we will select three
  values for x and use -3 for y each time.

Why The given equation requires that x -3.
56
EXAMPLE 7
Graph x -3

• Solution

We can write the equation in general form as x
0y -3. Since the coefficient of y is 0, the
value of has no effect on x. The value of x is
always -3 . For example, if y -2, we have
57
EXAMPLE 7
Graph x -3

• Solution

One solution is (-3, -2). To find two more
solutions, we must again select -3 for x. Any
number can be used for y. If y 0 then second
solution is (-3, 0). If y 3, a third solution
is (-3, 3). The three solutions are entered in
the table below. When we plot the ordered pairs
and draw a straight line through the points, the
result is a vertical line. The x-intercept is
(-3, 0) and there is no y-intercept.
58
Objective 3 Identify and Graph Horizontal and
Vertical Lines

• we have the following facts.
• Equations of Horizontal and Vertical Lines
• The graph of y b represents the horizontal
  line that intersects the y-axis at (0, b).

59
Objective 3 Identify and Graph Horizontal and
Vertical Lines

• Equations of Horizontal and Vertical Lines
• The graph of x a represents the vertical line
  that intersects the x-axis at (a, 0).

60
Objective 3 Identify and Graph Horizontal and
Vertical Lines

• Equations of Horizontal and Vertical Lines
• The graph of y 0 is the x-axis. The graph of
  x 0 is the y-axis.

61
Objective 4 Obtain Information from Intercepts

• The ability to read and interpret graphs is a
  valuable skill.
• When analyzing a graph, we should locate and
  examine the intercepts. As the following example
  illustrates, the coordinates of the intercepts
  can yield useful information.

62
EXAMPLE 8
Hybrid Mileage

• Figure (a) shows mileage data for a 2010 Toyota
  Prius Hybrid. What information do the intercepts
  give about the car?

• Strategy We will determine where the graph (the
  line in red) intersects the g-axis and where it
  intersects the m-axis.

Why Once we know the intercepts, we can
interpret their meaning.
63
EXAMPLE 8
Hybrid Mileage

• Solution

Figure (b). The g-intercept (0, 12) indicates
that when the car has been driven 0 miles, the
fuel tank contains 12 gallons of gasoline. That
is, the Prius has a 12-gallon fuel tank.
The m-intercept (600, 0) indicates that after
600 miles of city driving, the fuel tank contains
0 gallons of gasoline. Thus, 600 miles of city
driving can be done on 1 tank of gas in a Prius.
64
Section 3.4

• Slope and Rate of Change

65
Objectives

• Find the slope of a line from its graph
• Find the slope of a line given two points
• Find slopes of horizontal and vertical lines
• Solve applications of slope
• Calculate rates of change
• Determine whether lines are parallel or
  perpendicular using slope

66
Objective 1 Find the Slope of a Line from Its
Graph

• The slope of a line is a ratio that compares the
  vertical change with the corresponding horizontal
  change as we move along the line from one point
  to another.
• As an example, lets find the slope of the line
  graphed below. To begin, we select two points on
  the line, P and Q.
• One way to move from P to Q is to start at point
  P and count upward 5 grid squares. Then, moving
  to the right, we count 6 grid squares to reach
  point Q.
• The vertical change in this movement is called
  the rise. The horizontal change is called the
  run.
• Notice that a right triangle, called a slope
  triangle, is created by this process.

67
Objective 1 Find the Slope of a Line from Its
Graph

• The slope of a line is defined to be the ratio of
  the vertical change to the horizontal change. By
  tradition, the letter m is used to represent
  slope. For the line graphed on the previous page,
  we have

• The slope of the line is 5/6. This indicates that
  there is a rise (vertical change) of 5 units for
  each run (horizontal change) of 6 units.

68
EXAMPLE 1

• Find the slope of the line graphed in the figure
  (a) below.

(a)
(b)
Strategy We will pick two points on the line,
construct a slope triangle, and find the rise and
run. Then we will write the ratio of the rise to
the run.
Why The slope of a line is the ratio of the rise
to the run.
69
EXAMPLE 1

• Solution

We begin by choosing two points on the line, P
and Q, as shown in the figure. One way to move
from P to Q is to start at point P and count
downward 4 grid squares. Because this movement is
downward, the rise is -4. Then, moving right, we
count 8 grid squares to reach Q. This indicates
that the run is 8.
To find the slope of the line, we write a ratio
of the rise to the run in simplified form.
70
EXAMPLE 1

• Solution

The movement from P to Q can be reversed.
Starting at P, we can move to the right, a run of
8 and then downward, a rise of -4, to reach Q.
With this approach, the slope triangle is above
the line. When we form the ratio to find the
slope, we get the same result as before
71
Objective 2 Find the Slope of a Line Given Two
Points

• We can generalize the graphic method for finding
  slope to develop a slope formula.
• To begin, we select points P and Q on the line
  shown in the figure below. To distinguish between
  the coordinates of these points, we use subscript
  notation.
• Point P has coordinates (x1, y1), which are read
  as x sub 1 and y sub 1.
• Point Q has coordinates (x2, y2), which are read
  as x sub 2 and y sub 2.

72
Objective 2 Find the Slope of a Line Given Two
Points

• As we move from point P to point Q, the rise is
  the difference of the y-coordinates y2 - y1.
• We call this difference the change in y.
• The run is the difference of the x-coordinates
  x2 - x1.
• This difference is called the change in x.
• Since the slope is the ratio rise/run, we have
  the following formula for calculating slope.
• Slope of a Line
• The slope m of a line passing through points
  (x1, y1) and (x2, y2) is

if x2 ? x1
73
EXAMPLE 3

• Find the slope of the line that passes through
  (-2, 4) and (5, -6).

• Strategy We will use the slope formula to find
  the slope of the line.

Why We know the coordinates of two points on the
line.
74
EXAMPLE 3

• Solution

If we let (x1, y1) be (-2, 4) and (x2, y2) be (5,
-6), then
If we graph the line by plotting the two points,
we see that the line falls from left to righta
fact indicated by its negative slope.
75
Objective 2 Find the Slope of a Line Given Two
Points

• In general, lines that rise from left to right
  have a positive slope. Lines that fall from left
  to right have a negative slope.

76
Objective 2 Find the Slope of a Line Given Two
Points

• In the following figure (a), we see a line with
  slope 3 is steeper than a line with slope of 5/6,
  and a line with slope of 5/6 is steeper than a
  line with slope of 1/4. In general, the larger
  the absolute value of the slope, the steeper the
  line.
• Lines with slopes of 1 and -1 and are graphed in
  figure (b) below. When m 1, the rise and run
  are, of course, the same number. When m -1 the
  rise and run are opposites. Note that both lines
  create a 45 angle with the horizontal x-axis.

77
Objective 3 Find Slopes of Horizontal and
Vertical Lines

• In this section, we calculate the slope of a
  horizontal line and we show that a vertical line
  has no defined slope.
• The y-coordinates of any two points on a
  horizontal line will be the same, and the
  x-coordinates will be different.
• Thus, the numerator of y2 - y1 / x2 - x1 will
  always be zero, and the denominator will always
  be nonzero. Therefore, the slope of a horizontal
  line is 0.

78
EXAMPLE 4

• Find the slope of the line y 3.

• Strategy We will find the coordinates of two
  points on the line.

Why We can then use the slope formula to find
the slope of the line.
79
EXAMPLE 4

• Solution

The graph of y 3 is a horizontal line. To find
its slope, we select two points on the line (-2,
3) and (3, 3). If (x1, y1) is (-2, 3) and (x2,
y2) is (3, 3), we have
The slope of the line y 3 is 0.
80
Objective 3 Find Slopes of Horizontal and
Vertical Lines

• The y-coordinates of any two points on a vertical
  line will be different, and the x-coordinates
  will be the same.
• Thus, the numerator of y2 - y1 / x2 - x1 will
  always be nonzero, and the denominator will
  always be 0. Therefore, the slope of a vertical
  line is undefined.

81
Objective 4 Solve Applications of Slope

• The concept of slope has many applications.
• For example,
• Architects use slope when designing ramps and
  roofs.
• Truckers must be aware of the slope, or grade, of
  the roads they travel.
• Mountain bikers ride up rocky trails and snow
  skiers speed down steep slopes.

82
EXAMPLE 6
Architecture

• Pitch is the incline of a roof expressed as a
  ratio of the vertical rise to the horizontal run.
  Find the pitch of the roof shown in the
  illustration.

• Strategy We will determine the rise and the run
  of the roof from the illustration. Then we will
  write the ratio of the rise to the run.

Why The pitch of a roof is its slope, and the
slope of a line is the ratio of the rise to the
run.
Solution A level is used to create a slope
triangle. The rise of the slope triangle is given
as 7 inches. Since a ratio is a quotient of two
quantities with the same units, we will express
the length of the one-foot-long level as 12
inches.
83
Objective 5 Calculate Rates of Change

• We have seen that the slope of a line compares
  the change in y to the change in x. This is
  calledthe rate of change of y with respect to x.
  
• In our daily lives, we often make many
  suchcomparisons of the change in one quantity
  with respect to another.
• For example, we might speak of snow melting at
  the rate of 6 inches per day or a tourist
  exchanging money at the rate of 12 pesos per
  dollar.

84
EXAMPLE 7
Banking

• A bank offers a business account with a fixed
  monthly fee, plus a service charge for each check
  written. The relationship between the monthly
  cost y and the number x of checks written is
  graphed below. At what rate does the monthly cost
  change?

85
EXAMPLE 7
Banking

• Strategy We will use the slope formula to
  calculate the slope of the line and attach the
  proper units to the numerator and denominator.

Why We know the coordinates of two points on the
line.
Solution Two points on the line are (25, 12) and
(75, 16). If we let (x1, y1) (25, 12) and (x2,
y2) (75, 16), we have
The monthly cost of the checking account
increases 2 for every 25 checks written.
86
Objective 6 Determine Whether Lines Are Parallel
or Perpendicular Using Slope

• Two lines that lie in the same plane but do not
  intersect are called parallel lines.
• Parallel lines have the same slope and different
  y-intercepts. For example, the lines graphed in
  figure (a) are parallel because they both have
  slope -2/3.

87
Objective 6 Determine Whether Lines Are Parallel
or Perpendicular Using Slope

• Lines that intersect to form four right angles
  (angles with measure 90) are called
  perpendicular lines.
• If the product of the slopes of two lines is -1,
  the lines are perpendicular. This means that the
  slopes are negative (or opposite) reciprocals.
• In figure (b) on the previous slide, we know that
  the lines with slopes 4/5 and -5/4 are
  perpendicular because

88
Objective 6 Determine Whether Lines Are Parallel
or Perpendicular Using Slope

• Slopes of Parallel and Perpendicular Lines
• 1. Two lines with the same slope are parallel.
• 2. Two lines are perpendicular if the product of
  the slopes is -1 that is, if their slopes are
  negative reciprocals.
• 3. A horizontal line is perpendicular to any
  vertical line, and vice versa.

89
EXAMPLE 8

• Determine whether the line that passes through
  (7, -9) and (10, 2) and the line that passes
  through (0, 1) and (3, 12) are parallel,
  perpendicular, or neither.

• Strategy We will use the slope formula to find
  the slope of each line.

Why If the slopes are equal, the lines are
parallel. If the slopes are negative reciprocals,
the lines are perpendicular. Otherwise, the lines
are neither parallel nor perpendicular.
90
EXAMPLE 8

• Solution

To calculate the slope of each line, we use the
slope formula.
Since the slopes are the same, the lines are
parallel.
91
Section 3.5

• Slope-Intercept Form

92
Objectives

• Use slopeintercept form to identify the slope
  and y-intercept of a line
• Write a linear equation in slopeintercept form
• Write an equation of a line given its slope and
  y-intercept
• Use the slope and y-intercept to graph a linear
  equation
• Recognize parallel and perpendicular lines
• Use slopeintercept form to write an equation to
  model data

93
Objective 1 Use SlopeIntercept Form to Identify
the Slope and y-Intercept of a Line

• To explore the relationship between a linear
  equation and its graph, lets consider y 2x
  1. To graph this equation, three values of x were
  selected (-1, 0, and 1), the corresponding
  valuesof y were found, and the results were
  entered in the table. Then theordered pairs were
  plotted and a straight line was drawn through
  them, as shown below.

To find the slope of the line, we pick two points
on the line,(-1, -1) and (0, 1), and draw a
slope triangle and count grid squares
94
Objective 1 Use SlopeIntercept Form to Identify
the Slope and y-Intercept of a Line

• From the equation and the graph, we can make two
  observations
• The graph crosses the y-axis at 1. This is the
  same as the constant term in y 2x 1.
• The slope of the line is 2. This is the same as
  the coefficient of x in y 2x 1.
• This illustrates that the slope and y-intercept
  of the graph of y 2x 1 can be determined from
  the equation.

95
Objective 1 Use SlopeIntercept Form to Identify
the Slope and y-Intercept of a Line

• These observations suggest the following form of
  an equation of a line.
• Slope-Intercept Form of the Equation of a Line
• If a linear equation is written in the form y
  mx b, the graph of the equation is a line with
  slope m and y-intercept (0, b).

• When an equation of a line is written in
  slopeintercept form, the coefficient of the
  x-term is the lines slope and the constant term
  gives the y-coordinate of y-intercept.

96
Objective 1 Use SlopeIntercept Form to Identify
the Slope and y-Intercept of a Line
97
Objective 2 Write a Linear Equation in
SlopeIntercept Form

• The equation of any nonvertical line can be
  written in slopeintercept form. To write a
  linear equation in two variables in
  slopeintercept form, solve the equation for y.

98
EXAMPLE 1

• Find the slope and y-intercept of the line with
  the given equation.
• a. 8x y 9, b. x 4y 16, c. -9x - 3y 11

• Strategy We will write each equation in
  slopeintercept form, by solving for y.

Why When the equations are written in
slopeintercept form, the slope and y-intercept
of their graphs become apparent.
99
EXAMPLE 1

• Solution

a. The slope and y-intercept of the graph of 8x
y 9 are not obvious because the equation is not
in slopeintercept form. To write it in y mx
b form, we isolate y.
This is the given equation.
To isolate y on the left side, subtract 8x from
both sides. Since we want the right side of the
equation to have the form mx b, we show the
subtraction from that side as 8x - 9 rather than
9-8x.
Since m 8 and , b 9, the slope is 8 and the
y-intercept is (0, 9).
100
EXAMPLE 1

• Solution

b. To write the equation in slopeintercept form,
we solve for y.
This is the given equation.
To isolate the term 4y on the left side, subtract
x from both sides. Write the subtraction before
the constant term 16.
On the left side. combine like terms x - x 0.
To isolate , undo the multiplication by 4 by
dividing both sides by 4, term-by-term.
Since m -1/4 and b 4, the slope is -1/4 and
the y-intercept is (0, 4).
101
EXAMPLE 1

• Solution

c. To write the equation in y mx b form, we
isolate y on the left side.
This is the given equation.
To isolate the term -3y on the left side, add
9x to both sides. Write the addition before the
constant term 11.
To isolate , undo the multiplication by -3 by
dividing both sides by -3, term-by-term.
Since m -3 and b -11/3, the slope is -3 and
the y-intercept is (0, -11/3).
102
Objective 3 Write an Equation of a Line Given
Its Slope and y-Intercept

• If we are given the slope and y-intercept of a
  line, we can write an equation of the line by
  substituting for m and b in the slopeintercept
  form.

103
EXAMPLE 2

• Write an equation of the line with slope -1 and
  y-intercept (0, 9).

• Strategy We will use the slopeintercept form, y
  mx b, to write an equation of the line.

Why We know the slope of the line and its
y-intercept.
104
EXAMPLE 2

• Solution

If the slope is -1 and the y-intercept is (0, 9),
then m -1 and b 9.
The equation of the line with slope -1 and
y-intercept (0, 9) is y -x 9.
105
Objective 4 Use the Slope and y-Intercept to
Graph a Linear Equation

• If we know the slope and y-intercept of a line,
  we can graph the line.

106
EXAMPLE 4

• Use the slope and y-intercept to graph y 5x - 4.

• Strategy We will examine the equation to
  identify the slope and the y-intercept of the
  line to be graphed. Then we will plot the
  y-intercept and use the slope to determine a
  second point on the line.

Why Once we locate two points on the line, we
can draw the graph of the line.
107
EXAMPLE 4

• Solution

Since y 5x - 4 is written in y mx b form,
we know that its graph is a line with a slope of
5 and a y-intercept of (0, -4). To draw the
graph, we begin by plotting the y-intercept. The
slope can be used to find another point on the
line.
If we write the slope as the fraction 5/1, the
rise is 5 and the run is 1. From (0, -4), we move
5 units upward (because the numerator, 5, is
positive) and 1 unit to the right (because the
denominator, 1, is positive). This locates a
second point on the line, (1, 1). The line
through (0, -4) and (1, 1) is the graph of y 5x
- 4.
108
EXAMPLE 4

• Solution

An alternate way to find another point on the
line is to write the slope in the form -5/-1. As
before, we begin at the y-intercept (0, -4).
Since the rise is negative, we move 5 units
downward, and since the run is negative, we then
move 1 unit to the left. We arrive at (-1, -9),
another point on the graph of y 5x - 4.
109
Objective 5 Recognize Parallel and Perpendicular
Lines

• The slopeintercept form enables us to quickly
  identify parallel and perpendicular lines.

110
EXAMPLE 6

• Are the graphs of y -5x 6 and x - 5y -10
  parallel, perpendicular, or neither?

• Strategy We will find the slope of each line and
  then compare the slopes.

Why If the slopes are equal, the lines are
parallel. If the slopes are negative reciprocals,
the lines are perpendicular. Otherwise, the lines
are neither parallel nor perpendicular.
111
EXAMPLE 6

• Solution

The graph of y -5x 6 is a line with slope -5.
To find the slope of the graph of x - 5y -10,
we will write the equation in slopeintercept
form.
This is the second given equation.
To isolate the term -5y on the left side,
subtract x from both sides.
To isolate , undo the multiplication by -5 by
dividing both sides by -5 term-by-term.
The graph of y x/5 2 is a line with slope
1/5. Since the slopes -5 and 1/5 are negative
reciprocals, the lines are perpendicular.
112
Objective 6 Use SlopeIntercept Form to Write an
Equation to Model Data

• To make the equation more descriptive, we replace
  x and y in y mx b with two other variables.

113
EXAMPLE 7
Group Discounts

• To promote group sales for an Alaskan cruise, a
  travel agency reduces the regular ticket price of
  4,500 by 5 for each person traveling in the
  group.
• a. Write a linear equation that determines the
  per-person cost c of the cruise, if p people
  travel together.
• b. Use the equation to determine the per-person
  cost if 55 teachers travel together.

• Strategy We will determine the slope and the
  y-intercept of the graph of the equation from the
  given facts about the cruise.

Why If we know the slope and y-intercept, we can
use the slopeintercept form, y mx b, to
write an equation to model the situation.
114
EXAMPLE 7
Group Discounts

• Solution

a. We will let p represent the number of people
traveling in the group and c represent the
per-person cost of the cruise. Since the cost
depends on the number of people in the group, the
linear equation that models this situation is
Since the per-person cost of the cruise steadily
decreases as the number of people in the group
increases, the rate of change of -5 per person
is the slope of the graph of the equation. Thus,
m is -5.
115
EXAMPLE 7
Group Discounts

• Solution

If 0 people take the cruise, there will be no
discount and the per-person cost of the cruise
will be 4,500. Written as an ordered pair of the
form (p, c), we have (0, 4500). When graphed,
this point would be the c-intercept. Thus, b is
4,500.
Substituting for m and b in the slopeintercept
form cmp b of the equation, we obtain the
linear equation that models the pricing
arrangement. A graph of the equation for groups
of up to 100 (c ?100) is shown on the right.
c -5p 4,500
m - 5 and b 4,500.
116
EXAMPLE 7
Group Discounts

• Solution

b. To find the per-person cost of the cruise for
a group of 55 people, we substitute 55 for p and
solve for c.
Do the multiplication.
Do the addition.
If a group of 55 people travel together, the
cruise will cost each person 4,225.
117
Section 3.6

• Point-Slope Form

118
Objectives

• Use pointslope form to write an equation of a
  line
• Write an equation of a line given two points on
  the line
• Write equations of horizontal and vertical lines
• Use a point and the slope to graph a line
• Write linear equations that model data

119
Objective 1 Use PointSlope Form to Write an
Equation of a Line

• Refer to the line graphed on the left, with slope
  3 and passing through the point (2, 1). To
  develop a new form for the equation of a line, we
  will find the slope of this line in another way.
• If we pick another point on the line with
  coordinates (x, y), we can find the slope of the
  line by substituting the coordinates of the
  points (x, y) and (2, 1) into the slope formula.

120
Objective 1 Use PointSlope Form to Write an
Equation of a Line

• Since the slope of the line is 3, we can
  substitute 3 for m in the previous equation.

• We then multiply both sides by x - 2 to clear the
  equation of the fraction.

121
Objective 1 Use PointSlope Form to Write an
Equation of a Line

• The resulting equation displays the slope of the
  line and the coordinates of one point on the line

• In general, suppose we know that the slope of a
  line is m and that the line passes through the
  point (x1, y1). Then if (x, y) is any other point
  on the line, we can use the definition of slope
  to write (y - y1) / (x - x1) m.

122
Objective 1 Use PointSlope Form to Write an
Equation of a Line

• If we multiply both sides by x - x1 to clear the
  equation of the fraction, we have y - y1 m(x -
  x1).
• This form of a linear equation is called
  pointslope form. It can be used to write the
  equation of a line when the slope and one point
  on the line are known.

123
EXAMPLE 1

• Find an equation of a line that has slope -8 and
  passes through (-1, 5). Write the answer in
  slopeintercept form.

• Strategy We will use the pointslope form, y -
  y1 m(x - x1), to write an equation of the line.

Why We are given the slope of the line and the
coordinates of a point that it passes through.
124
EXAMPLE 1

• Solution

Because we are given the coordinates of a point
on the line and the slope of the line, we begin
by writing the equation of the line in the
pointslope form. Since the slope is -8 and the
given point is (-1, 5), we have m -8, x1 -1,
and y1 5.
Brackets are used to enclose x - (-1).
To write this equation in slopeintercept form,
we solve for y.
This is the simplified pointslope form.
This is the requested slopeintercept form.
125
EXAMPLE 1

• Solution

Check results To verify this result, we note
that m -8. Therefore, the slope of the line is
-8, as required. To see whether the line passes
through (-1, 5), we substitute -1 for x and 5 for
y in the equation. If this point is on the line,
a true statement should result.
126
Objective 2 Write an Equation of a Line Given
Two Points on the Line

• In the next example, we show that it is possible
  to write the equation of a line when we know the
  coordinates of two points on the line.

127
EXAMPLE 2

• Find an equation of the line that passes through
  (-2, 6) and (4, 7). Write the equation in
  slopeintercept form.

128
EXAMPLE 2

• Strategy We will use the pointslope form, y -
  y1 m(x - x1), to write an equation of the line.

Why We know the coordinates of a point that the
line passes through and we can calculate the
slope of the line using the slope formula.
129
EXAMPLE 2

• Solution

To find the slope of the line, we use the slope
formula.
Either point on the line can serve as (x1, y1).
If we choose (4, 7), we have
130
EXAMPLE 2

• Solution

To write this equation in slopeintercept form,
we solve for y.
The equation of the line that passes through (-2,
6) and (4, 7) is y (1/6)x 19/3.
131
Objective 3 Write Equations of Horizontal and
Vertical Lines

• We have previously graphed horizontal and
  vertical lines. We will now discuss how to write
  their equations.

132
EXAMPLE 3

• Write an equation of each line and graph it.
• a. A horizontal line passing through (-2, -4).
• b. A vertical line passing through (1, 3).

• Strategy We will use the appropriate form,
  either y b or x a, to write an equation of
  each line.

Why These are the standard forms for the
equations of a horizontal and a vertical line.
133
EXAMPLE 3

• Solution

a. The equation of a horizontal line can be
written in the form y b. Since the y-coordinate
of (-2, -4) is -4, the equation of the line is y
-4. The graph is shown in the figure.
b. The equation of a vertical line can be written
in the form x a. Since the x-coordinate of (1,
3) is 1, the equation of the line is x 1. The
graph is shown in the figure.
134
Objective 4 Use a Point and the Slope to Graph a
Line

• If we know the coordinates of a point on a line,
  and if we know the slope of the line, we can use
  the slope to determine a second point on the line.

135
EXAMPLE 4

• Graph the line with slope 2/5 that passes through
  (-1, -3).

• Strategy First, we will plot the given point
  (-1, -3). Then we will use the slope to find a
  second point that the line passes through.

Why Once we determine two points that the line
passes through, we can draw the graph of the line.
136
EXAMPLE 4

• Solution

We begin by plotting the point (-1, -3). From
there, we move 2 units up (rise) and then 5 units
to the right (run), since the slope is 2/5. This
locates a second point on the line, (4, -1).
We then draw a straight line through the two
points.
From (1, 3) draw the rise and run parts of the
slope triangle for m 2/5 to find another point
on the line.
Plot the given point (1, 3).
Use a straightedge to draw a line through the
points.
137
Objective 5 Write Linear Equations That Model
Data

• In the next example, we will see how the
  pointslope form can be used to write linear
  equations that model certain realworld
  situations.

138
EXAMPLE 6
Studying Learning

• In a series of 40 trials, a rat was released in a
  maze to search for food. Researchers recorded the
  trial number and the time that it took the rat to
  complete the maze on a scatter diagram shown
  below. After the 40th trial, they drew a line
  through the data to obtain a model of the rats
  performance. Write an equation of the line in
  slopeintercept form.

139
EXAMPLE 6
Studying Learning

• Strategy From the graph, we will determine the
  coordinates of two points on the line.

Why We can write an equation of a line when we
know the coordinates of two points on the line.
(See Example 2.).
140
EXAMPLE 6
Studying Learning

• Solution

We begin by writing a pointslope equation. The
line passes through several points we will use
(4, 24) and (36, 16) to find the slope.
Any point on the line can serve as (x1, y1). We
will use (4, 24).
To write this equation in slopeintercept form,
solve for y.
1 24 25.
Conclusion A linear equation that models the
rats performance on the maze is y -¼x 25,
where x is the number of the trial and y is the
time it took, in seconds.
141
Section 3.7

• Graphing Linear Inequalities

142
Objectives

• Determine whether an ordered pair is a solution
  of an inequality
• Graph a linear inequality in two variables
• Graph inequalities with a boundary through the
  origin
• Solve applied problems involving linear
  inequalities in two variables

143
Objective 1 Determine Whether an Ordered Pair Is
a Solution of an Inequality

• Recall that an inequality is a statement that
  contains one of the symbols lt, , gt, or .
• Inequalities in one variable, such as x 6 lt 8
  and 5x 3 4x, were solved in Section 2.7.
• Because they have an infinite number of
  solutions, we represented their solution sets
  graphically, by shading intervals on a number
  line.
• We now extend that concept to linear inequalities
  in two variables, as we introduce a procedure
  that is used to graph their solution sets.

144
Objective 1 Determine Whether an Ordered Pair Is
a Solution of an Inequality

• If the symbol in a linear equation in two
  variables is replaced with an inequality symbol,
  we have a linear inequality in two variables.
• Some examples of linear inequalities in two
  variables are x - y 5, 4x 3y lt -6, and y gt
  2x and x lt 3
• As with linear equations, an ordered pair (x, y)
  is a solution of an inequality in x and y if a
  true statement results when the values of the
  variables are substituted into the inequality.

145
EXAMPLE 1

• Determine whether each ordered pair is a solution
  of x - y 5. Then graph each solution
• a. (4, 2), b. (0, -6), c. (1, -4)

• Strategy We will substitute each ordered pair of
  coordinates into the inequality.

Why If the resulting statement is true, the
ordered pair is a solution.
146
EXAMPLE 1

• Solution

a. For (4, 2)
Because 2 5 is true, (4, 2) is a solution of x
- y 5. We say that (4, 2) satisfies the
inequality.
b. For (0, -6)
Because 6 5 is false, (0, -6) is not a solution.
147
EXAMPLE 1

• Solution

c. For (1, -4)
Because 5 5 is true, (1, -4) is a solution, and
we graph it as shown.
148
Objective 2 Graph a Linear Inequality in Two
Variables

• In Example 1, we graphed two of the solutions of
  x - y 5.
• Since there are infinitely more ordered pairs (x,
  y) that make the inequality true, it would not be
  reasonable to plot all of them. Fortunately,
  there is an easier way to show all of the
  solutions.
• The graph of a linear inequality is a picture
  that represents the set of all points whose
  coordinates satisfy the inequality.
• In general, such graphs are regions bounded by a
  line. We call those regions half-planes, and we
  use a two-step procedure to find them.

149
Objective 2 Graph a Linear Inequality in Two
Variables

• Graphing Linear Inequalities in Two Variables
• 1. Replace the inequality symbol with an equal
  symbol and graph the boundary line of the
  region. If the original inequality allows the
  possibility of equality (the symbol is either
  or ), draw the boundary line as a solid line. If
  equality is not allowed (lt or gt), draw the
  boundary line as a dashed line.
• 2. Pick a test point that is on one side of the
  boundary line. (Use the origin if possible.)
  Replace x and y in the inequality with the
  coordinates of that point. If a true statement
  results, shade the side that contains that point.
  If a false statement results, shade the other
  side of the boundary.

150
EXAMPLE 3

• Graph 4x 3y lt -6

• Strategy We will graph the related equation 4x
  3y -6 to establish the boundary line between
  two regions of the coordinate plane. Then we will
  determine which region contains points that
  satisfy the given inequality.

Why The graph of a linear inequality in two
variables is a region of the coordinate plane on
one side of a boundary line.
151
EXAMPLE 3

• Solution

To find the boundary line, we replace the
inequality symbol with an equal symbol and
graph 4x 3y -6 using the intercept method.
Since the inequality symbol lt does not include an
equal symbol, the points on the graph of 4x 3y
-6 will not be part of the graph of 4x 3y lt
-6. To show this, we draw the boundary line as a
dashed line.
152
EXAMPLE 3

• Solution

To determine which half-plane to shade, we
substitute the coordinates of a point that lies
on one side of the boundary line into 4x 3y lt
-6. We choose the origin (0, 0) as the test point
because the calculations are easy when they
involve 0. We substitute 0 for x and 0 or y in
the inequality.
153
EXAMPLE 3

• Solution

Since 0 lt -6 is a false statement, the point (