Graphing Lines with a Table
• Select (or use pre-selected) values for x
• Substitute those x values in the equation and
solve for y
• Graph the x and y values as ordered pairs
• Connect points with a line
Example
• Graph y = 2x - 1
Example
• Graph y = 2x
Example
• Graph 2x + 3y = 4
Time to work
• Worksheet!
Ch 7
Linear Equations
7.1
Slope
Slope
• Slope –
 The ratio of the rise, or vertical change, to the run,
or horizontal change
rise
slope = m =
run
Example
• Determine the slope of each line.
Example
• Determine the slope of each line.
Rate of Change
• In real-life applications, slope is the rate of
change (how much a value is changing)
Example
• The graph below shows the distance traveled
by Rebecca and Ian during a day-long bicycle
ride. Find the slope of each line. To what does
the slope refer?
Example
• A line contains the points whose coordinates
are listed in the table. Determine the slope of
the line.
Slope Formula
Example
• Determine the slope of each line.
 The line through the points at (3, 8) and (3, 4)
Example
• Determine the slope of each line.
 The line through the points at (-4, 1) and
(-3, -2)
Example
• Determine the slope of each line.
 The line through the points at (2, 5) and (3, 9)
Example
• Determine the slope of each line.
 The line through the points at (-8, 1) and (4, 1)
Types of Slope
Assignments
• #1 – due today
– P287: 1, 2, 4 – 12 even
• #2 – due next time
– P288: 13 – 27, 34, 35, 36
7.2
Writing Equations in Point-Slope
Form
Point-Slope Form
• Replace the m, x1, and y1 with the values given
Example
• Write the point-slope form of the equation of
the line passing through the given point and
having the given slope.
(-2, 7), m = -1/3
Example
• Write the point-slope form of the equation of
the line passing through the given point and
having the given slope.
(4, 0), m = 4
Example
• Write the point-slope form of the equation of
the line passing through the given point and
having the given slope.
(-3, 2), m = 2
Example
• Write the point-slope form of the equation of
the line passing through the given point and
having the given slope.
(5, 4), m = -2/3
Writing from a graph
• You can also write an equation in point-slope
form from a graph
• First find the slope of the line by counting
• Then pick a point (any point on the line will
work)
• Plug those values into the formula
Example
• Write the point-slope form of an equation of
the line below.
Example
• Write the point-slope form of an equation of
the line below.
Example
• Write the point-slope form of an equation for
the line passing through (1, 4) and (3, -5)
 Hints: find the slope first / it doesn’t matter which
point you use.
Assignments
• #1 – due today
 P293: 3 – 13
• #2 – due next time
 P293: 15 – 37
7.3
Writing Equations in Slope-Intercept
Form
Intercepts
• y-intercept –
 The point on the y-axis where the line crosses that
axis
• x-intercept –
 The point on the x-axis where the line crosses that
axis
Slope-Intercept Form
• Another form, besides point-slope
• This form helps you graph!
y = mx + b
• m – slope
• b – y-intercept (point where it crosses y-axis)
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = 3, b = -1
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = -2/3, b = 0
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = 0, b = -4
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = 2, b = 1
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = -5/3, b = 0
Example
• Write an equation in slope-int form of each
line with the given slope and y-int.
m = 0, b = -8
Example
• Write an equation of the line in slopeintercept form for the situation:
Slope 1 and passes through (2, 5)
Example
• Write an equation of the line in slopeintercept form for the situation:
Slope -3 and passes through (1, -4)
Example
• Write an equation of the line in slopeintercept form for the situation:
Passing through (-4, 4) and (2, 1)
Example
• Write an equation of the line in slopeintercept form for the situation:
Passing through (6, 2) and (3, -2)
Example
• Write an equation of the line in slopeintercept form for the situation:
Slope is ¾ and passes through (8, -2)
Example
• Write an equation of the line in slopeintercept form for the situation:
Passes through (2, 4) and (0, 5)
Assignments
• #1 – due today
 P299: 4 – 12
• #2 – due next time
 P299: 14 – 40 even, 41 – 45, 49 – 50, 53 – 57
7.4
Scatter Plots
Scatter Plots
• Scatter Plot –
 Graph where two sets of data are plotted as
ordered pairs on the same coordinate plane
 Used to see if there is a trend, pattern, or
relationship among the variables
Scatter Plots
Example
• Determine whether the
scatter plot shows a positive
relationship, negative
relationship, or no
relationship. If there is a
relationship, describe it.
• The scatter plot shows the
number of years of experience
and the salary for each
employee in a small company.
Example
• Determine whether the
scatter plot shows a positive
relationship, negative
relationship, or no
relationship. If there is a
relationship, describe it.
• The scatter plot shows the
word processing speeds of 12
students and the number of
weeks they have studied
word processing.
Example
• Determine whether the scatter plot shows a
positive relationship, negative relationship, or
no relationship. If there is a relationship,
describe it.
Example
• The table shows the
average number of
minutes a pediatric dentist
spends during each
appointment instructing
the patient in proper
dental care, and the
number of cavities for each
patient.
Example
• Make a scatter plot of the data.
Let the horizontal axis represent
instruction time and let the vertical
axis represent the number of
cavities.
• Does the scatter plot show a
relationship between instruction
time and cavities? Explain.
• Describe the independent and
dependent variables. Then state
the domain and the range.
Assignments
• P305: 4 – 8, 10 – 17, 19 – 23
7.5
Graphing Linear Equations
Graphing with Intercepts
• What are intercepts?
– Point where the line crosses the x- and y-axes
•Find the intercepts
and plot them, draw
a line between
•Point of y-intercept
is always (o, y)
•Point of x-intercept
is always (x, 0)
Example
• Determine the x-intercept and y-intercept of
the graph of the line 2y – x = 8. Then graph.
Example
• Determine the x-intercept and y-intercept of
the graph of the line 3x – 2y = 12. Then graph.
Example
• Determine the x-intercept and y-intercept of
the graph of the line x + y = 2. Then graph.
Example
• Determine the x-intercept and y-intercept of
the graph of the line 3x + y = 3. Then graph.
Example
• Determine the x-intercept and y-intercept of
the graph of the line 4x – 5y = 20. Then graph.
Example
• Suppose to ship a package it costs $2.05
for the first pound and $1.55 for each
additional pound. This can be represented
by y = 2.05 + 1.55x. Determine the slope
and y-intercept of the graph of the
equation.
Example
• Determine the slope and y-intercept of the
graph 6x – 9y = 18.
Example
• Determine the slope and y-intercept of the
graph of 4x + 3y = 6.
Example
• Graph the equation using slope intercept
form.
2
y  x5
3
Example
• Graph the equation using slope intercept
form.
1
y  x2
5
Example
• Graph the equation using slope intercept
form.
1
y  x3
2
Example
• Graph the equation using slope intercept
form.
3x  y  4
Example
• Graph the equation using slope intercept
form.
y  3
Example
• Graph the equation using slope intercept
form.
x4
Example
• Graph the equation using slope intercept
form.
y  1
Example
• Graph the equation using slope intercept
form.
x3
Assignments
• P314: 7 – 11, 24 – 34 even, 36 – 38, 43 – 49
7.6
Families of Linear Graphs
Review
• Slope formula:
• Point-Slope Form:
• Slope-Intercept Form:
Linear Graphs
Example
• Graph the pair of equations. Describe any
similarities or differences. Explain why they
are a family of graphs.
1
y  x2
2
1
y   x 1
2
Example
• Graph the pair of equations. Describe any
similarities or differences. Explain why they
are a family of graphs.
y  5x  1
y  x 1
Example
• Graph the pair of equations. Describe any
similarities or differences. Explain why they
are a family of graphs.
y  2x 1
y  2x  5
Example
• Graph the pair of equations. Describe any
similarities or differences. Explain why they
are a family of graphs.
y  x 1
y  3x  1
Example
• Gretchen and Max each have a savings
account and plan to save $20 per month. The
current balance in Gretchen’s account is $150
and the balance in Max’s account is $100.
Then y = 20x + 150 and y = 20x + 100
represent how much money each has in their
account, respectively, after x months.
Compare and contrast the graphs of the
equations.
Parent Graphs
• The simplest of graphs in a family
• Questions:
 How does changing the slope affect the line?
 How does changing the y-int affect the line?
Example
• Change y = -3x – 1 so that the graph of the
new equation fits each description.
 Same y-intercept, less steep positive slope.
 Same slope, y-intercept is shifted down 2 units.
Example
• Change y = 2x + 1 so that the graph of the new
equation fits each description.
– Same slope, shifted down 1 unit
– Same y-intercept, less steep positive slope
Assignments
• #1 – due today
 P319: 1, 4 – 10 even
• #2 – due next time
 P319: 12 – 30 even, 31, 34 – 39
7.7
Parallel and Perpendicular Lines
Parallel
• Two lines are parallel if
they never intersect
• What would have to be
true about the lines so
that they would never
intersect?
• They have the same
slope!!
Parallel Lines
Example
• Determine whether the graphs of the
equations are parallel.
y  3x  4
9 x  3 y  12
Example
• Determine whether the graphs of the
equations are parallel.
y  2x
7  2x  y
Example
• Determine whether the graphs of the
equations are parallel.
y  3x  3
2 y  6x  5
Parallelogram
• A four-sided figure with two sets of parallel
sides
Example
• Determine whether figure EFGH is a
parallelogram.
Example
• Determine whether figure ABCD is a
parallelogram.
Example
• Write an equation in slope-intercept form of
2
the line that is parallel to the graph y  x  3
3
of and passes through the point at (-3, 1).
Example
• Write an equation in slope-intercept form of
the line that is parallel to the graph y  6 x  4
of and passes through the point at (2, 3).
Example
• Write an equation in slope-intercept form of
the line that is parallel to the graph 3x  2 y  9
of and passes through the point at (2, 0).
Perpendicular Lines
Example
• Determine whether the graphs of the
equations are perpendicular.
y  2 x  4
1
y  x3
2
Example
• Determine whether the graphs of the
equations are perpendicular.
1
y  x2
5
y  5x  1
Example
• Determine whether the graphs of the
equations are perpendicular.
y  4 x  3
4y  x 5
Example
• Write an equation in slope-intercept form of
the line that is perpendicular to the graph of
y  2. x  5 and passes through the point at (2, -3).
Example
• Write an equation in slope-intercept form of
the line that is perpendicular to the graph of
and passes through the point at (0, 0).
Example
• Write an equation in slope-intercept form of
the line that is perpendicular to the graph of
2 x  3. y  2 and passes through the point at (3, 0).
Assignments
• #1 – due today
 P325: 2 – 14
• #2 – due today
 P326: 16 – 38 even, 42 – 50