Chapter 11:
Graphing Lines
Regular Math
Section 11.1: Graphing
Linear Equations
 A linear equation is an equation whose
solutions fall on a line on the coordinate
grid.
 A linear equation’s graph will always be a
straight line.
5 Steps to Graph any
Equation
1.
Choose a value for x.
2.
Substitute the x-value into the equation, and find the
corresponding y-values.
3.
Form an ordered pair with the x-value and y-value.
4.
Graph the ordered pair.
5.
Repeat the process until you have at least 3 points.

Remember – One point must be a negative point.
Graphing Equations
 Graph each equation and tell whether it is
linear.
 y = 2x – 3
 y = x squared
 y = 2/3 x
 y = -3
Try these on your own…
 Graph each equation and tell whether is it
linear.
 y = 3x -1
 Linear
 y = x cubed
 Not Linear
 y = -3/4 x
 Linear
 y=2
 Linear
Sports Application
 In bowling, the equation
h = 160 – 0.8s
represents the handicap
(h) calculated for a
bowler with average
score (s). How much will
the handicap be for each
bowler listed in the
table? Draw a graph that
represents the
relationship between the
average score and the
handicap.
Bowler
Sandi
Average
Score
145
Dominic
125
Leo
160
Sheila
140
Tawana
175
Try this one on your own…
 A lift on a ski slope rises
according to the equation a =
130t + 6250, where a is the
altitude in feet and t is the
minutes that a skier has been
on the life. Five friends are on
the lift. What is the altitude of
each person if they have been
on the ski lift for the times
listed in the table? Draw a
graph that represents the
relationships between the
time on the lift and the
altitude.
Skier
Anna
Time of
Lift
4 minutes
Tracy
3 minutes
Kwani
2 minutes
Tony
1.5
minutes
1 minute
George
Skier
Time on Lift
Altitude
Anna
4 minutes
6770 ft
Tracy
3 minutes
6640 ft
Kwani
2 minutes
6510 ft
Tony
1.5 minutes
6445 ft
George
1 minute
6380 ft
Section 11.2: Slope of a
Line
Finding Slope, Given To
Points
 Find the slope of the line that passes through
(2,5) and (8,1).
 Try this one on your own…
 Find the slope of the line that passes through (-2, 3) and (4, 6).
Finding Slope from a
Graph
 Use the graph of the
line to determine its
slope.
Try this one on your own…
 Use the graph of the
line to determine its
slope.
Parallel and Perpendicular
Slopes
 Parallel Lines have the same slope.
 Perpendicular Lines have complete
opposite slopes.
Identifying Parallel and
Perpendicular Lines by Slope
 Identifying Parallel
and Perpendicular
Lines by Slope
 Line 1: (1,9) and (-1,5)
 Line 2: (-3, -5) and (4,9)
 Line 1: (-10, 0) and
(20,6)
 Line 2: (-1, 4) and (2, 11)
Graphing a Line Using a
Point and the Slope
 Graph the line
passing through (1,1)
with slope -1/3.
 Graph the line
passing through (3,1)
with slope 2.
Section 11.3: Using Slopes
and Intercepts
 The x-intercept of a line is the value of x
where the line crosses the x-axis. (y = 0)
 The y-intercept of a line is the value of y
where the line crosses the y-axis. (x = 0)
Finding x-intercepts and y-intercepts
to Graph Linear Equations
 Find the x-intercept and y-intercept of the
line 2x + 3y = 6. Use the intercepts to
graph the equations.
 Step One: Solve for y.
 Step Two: Find the x – intercept and the y –
intercept.
 Step Three: Graph.
Try this one on your own…
 Find the x-intercept and y-intercept of the
line 4x – 3y = 12. Use the intercepts to
graph the equation.
Slope – Intercept Form
 Slope – Intercept Form : y = mx + b
 m = slope
 b = y-intercept
 Notice that y is all by itself on one side and
everything else is on the other.
Using Slope-Intercept Form
to Find Slope and y-intercept
 Write each equation in slope-intercept
form, and then find the slope and yintercept.
y=x
 7x = 3y
 2x + 5y = 8
Try these on your own…
 Write each equation in slope-intercept
form, and then find the slope and the yintercept.
 2x + y = 3
 5y = 3x
Entertainment Application
 An arcade deducts 3.5 points from your
50-point game card for each Skittle-ball
game you play. The linear equation
y = -3.5x + 50 represents the number of
points (y) on your card after (x) games.
Graph the equation using the slope and
y-intercept.
Try this one on your own…
 A video club charges $8 to join, and
$1.25 for each DVD that is rented. The
linear equation y = 1.25x + 8 represents
the amount of money (y) spent after
renting (x) DVDs. Graph the equation
using the slope and y – intercept.
Writing Slope-Intercept
Form
 Write the equation of the line that passes
through (-3,1) and (2, -1) in slopeintercept form.
 Try this one on your own…
 Write the equation of the line that passes
through (3, -4) and (-1,4) in slope-intercept
form.
Section 11.4: Point-Slope
Form
 The point-slope form of an equation of a
line with slope (m) passing through
(x1,y1) is y – y1 = m (x – x1).
Use Point-Slope Form to Identify
Information About a Line
 Use the point-slope form of each equation
to identify a point the line passes through
and the slope of the line.
 y – 9 = -2/3 (x - 21)
 m = -2/3
 Point = (21, 9)
 y – 3 = 4 (x + 7)
 m=4
 Point = (-7, 3)
Try these on your own…
 Use the point-slope form of each
equation to identify a point the line
passes through and the slope of the line.
 Y – 7 = 3 (x – 4)
m=3
 Point = (4,7)
 Y – 1 = 1/3 ( x + 6)
 m = 1/3
 Point = (-6, 1)
Writing the Point-Slope
Form of an Equation
 Write the point-slope form of the equation
with the given slope that passes through
the indicated point.
 the line with slope -2 passing through (4,1)
 the line with slope 7 passing through (-1,3)
Try these on your own…
 Write the point-slope form of the equation
with the given slope that passes through
the indicated point.
 the line with slope 4 passing through (5, -2)
 y + 2 = 4 (x – 5)
 the line with slope -5 passing through (-3, 7)
 y – 7 = -5 (x + 3)
Medical Application
 Suppose that laser eye surgery is
modeled on a coordinate grid. The laser
is positioned at the y-intercept so that the
light shifts down 1 mm for each 40 mm it
shifts to the right. The light reaches the
center of the cornea of the eye at (125,0).
Write the equation of the light beam in
point-slope form, and find the height of
the laser.
Try this one on your own…
 A roller coaster starts by ascending 20 feet
for every 30 feet in moves forward. The
coaster starts at a point 18 feet above the
ground. Write the equation of the line that
the roller coaster travels along in point-slope
form, and use it to determine the height of
the coaster after traveling 150 feet forward.
Assume that the roller coaster travels in a
straight ling for the first 150 feet.
Section 11.5: Direct
Variation
 For direct variation, two variable
quantities are related proportionally by a
constant positive ratio. The ratio is called
constant of proportionality.
 Equation: y = kx
 k = constant
Determining Whether a
Data Set Varies Directly
 Determine whether the data set shows
direct variation.
 Shoe Sizes…
US Size
7
8
9
10
11
European
Size
39
41
43
44
45
 Determine whether the data set shows direct
variation.
 Distance Sound Travels at 20 degrees Celcius (m)
Time (s)
0
1
2
3
4
Distance
(m)
0
350
700
1050
1400
Try these on your own…
 Determine whether the data set shows direct variation.
 Adam’s Growth Chart
Age (mo)
3
6
9
12
Length (in.)
22
24
25
27
 Distance Traveled by Train
Time (Min)
10
20
30
40
Distance
(mi)
25
50
75
100
Finding Equations of
Direct Variation
 Find each equation of direct variation,
given that y varies directly with x.
 y is 52 when x is 4
 x is 10 when y is 15
 y is 15 when x is 2
Try these on your own…
 Find each equation of direct variation,
given that y varies directly with x.
 y is 54 when x is 6
 x is 12 when y is 15
 y is 8 when x is 5
Story Problem…
 Mrs. Perez has $4000 in
a CD and $4000 in a
money market account.
The amount of interest
she has earned since
the beginning of the year
is organized in the
following table.
Determine whether there
is a direct variation
between either data set
and time. If so, find the
equation of direct
variation.
Time (mo)
Interest in
CD ($)
Interest in
Money
Market ($)
0
0
0
1
17
19
2
34
37
3
51
55
4
68
73
Section 11.6: Graphing
Inequalities in Two Variables
 When the equality symbol is replaced in
a linear equation by an inequality symbol,
the statement is a linear inequality.
 A boundary line is the set of points
where the two sides of a two-variable
linear inequality are equal.
Graphing Inequalities
 Graph each inequality.
y  x 1
y  x 1
3 y  4 x  12
Try these on your own…
 Graph each inequality.
y  x 1
y  2x 1
2 y  5x  6
Science Application…
 Solar powered rovers landing on Mars in
2004 will have a range of up to 330 feet
per Martian day. Graph the relationship
between the distance a rover can travel
and the number of Martian days. Can a
rover travel 3000 feet in 8 days?
Try this one on your own…
 A successful screenwriter can write no
more than seven and a half pages of
dialogue each day. Graph the
relationship between the number of
pages the writer can write and the
number of days. At this rate, would the
writer be able to write a 200 page
screenplay in 30 days?
Section 11.7: Lines of Best
Fit
 To estimate the equation of a line of best
fit:
 Find the mean of the x-coordinates and ycoordinates. Create a new point.
 Draw a line through the new point that
appears to fit the data the best.
 Estimate the coordinates of another point on
the line.
 Find the equation of the line.
Finding a Line of Best Fit
X
2
4
5
1
3
8
6
7
Y
4
8
7
3
4
8
5
9
Try this one on your own…
X
Y
4
4
7
5
3
2
8
6
8
7
6
4
Sports Application
 Find a line of best fit
for the women’s
3000-meter speed
skating. Use the
equation of the line
to predict the winning
time in 2006.
 Let 1960 represent
year 0.
Year
Winning Time
(minutes)
1964
5.25
1968
4.94
1972
4.87
1976
4.75
1980
4.54
1984
4.41
1988
4.20
1992
4.33
1994
4.29
1998
4.12
2002
3.96
Try this one on your own…
Year
1990
Distance
(ft)
98
1992
101
1994
103
1997
106
2002
107
 Find a line of best fit
for the Main Street
Elementary annual
softball toss. Use the
equation of the line
to predict the winning
distance in 2006.
 Let x = 0 represent
the year 1990.