Today’s Date _______________
Notes for
Graphing Linear Equations
We will understand that lines represent solutions of linear equations.
We will graph linear equations.
Activity: 1) Graphing a Linear Equation
Linear Equations
A linear equation is an equation whose graph is a line. The points on the line
are solutions of the equation. A line can be represented using an equation, a
data table, or a graph.
y=x+1
Graph y = x – 3
Step 1: select three values for
the x-coordinate.
Step 2: create a data table
using the values
x
y
(x, y)
Step 3: plot the coordinates
Step 4: connect points, extending line.
Include arrows at each end and label the line.
Math 8 - 21
Today’s Date _______________
Graphing Horizontal and Vertical Lines
Graph
y = 0x + 5
The graph of y = b is
a horizontal line
passing through (0, b).
The graph of x = a is
a vertical line
passing through (a, 0).
Graph. (see graph above)
a) y = 4
b) x = -2
Practice what you know.
Math 8 - 22
Today’s Date _______________
Notes for
Slope of a Line
We will find slopes of lines by using two points.
We will find slopes of lines from tables.
Slope is the rate of change between any two points
on a line. It is the measure of the steepness of the
line.
To find the slope, m, of a line, find the ratio of the change in y (vertical change or
rise) to the change in x (horizontal change or run) between any two points on the
line.
ACTIVITY: 1) Finding the Slope of a Line
2) Using Similar Triangles
3) Drawing Lines with Given Slopes
Math 8 - 23
Today’s Date _______________
Describe the slope of the line. Then calculate the slope.
a)
b)
c)
d)
e) Calculate the value of y so that the line passing through the points (-2, 1)
and (4, y) has a slope of 2/3.
Practice what you know.
Math 8 - 24
Today’s Date _______________
Notes for
Slopes of Parallel
and
Perpendicular Lines
We will identify parallel and perpendicular lines.
On the communicator draw two parallel lines. Then slip a coordinate grid into the
communicator. What are the slopes of the two lines?
Parallel Lines and Slopes
Lines in the same plane that do not intersect are parallel
lines. Non-vertical parallel lines have the same slope.
All vertical lines are parallel.
Examples:
Which lines are parallel? How do you know?
On the communicator draw two perpendicular lines. Then slip a coordinate grid
into the communicator. What are the slopes of the two lines?
Perpendicular Lines and Slopes
Lines in the same plane that intersect at right angles are
perpendicular lines. Two non-vertical lines are perpendicular
when the product of their slopes is -1. That is the slopes have
opposite signs and are reciprocals of each other. Vertical
lines are perpendicular to horizontal lines.
Which lines are perpendicular? How do you know?
Practice what you know.
Math 8 - 25
Today’s Date _______________
Notes for
Graphing Proportional Relationships
We will write and graph proportional relationships.
Activity: 1) Identifying Proportional Relationships
2) Analyzing Proportional Relationships
3) Deriving an Equation
Direct Variation
When two quantities x and y are proportional, the
relationship can be represented by the direct
variation equation y = mx, where m is the constant
of proportionality.
The graph of y = mx is a line with a slope of m that
passes through the origin.
The cost y (in dollars) to rent x video games is represented by y = 4x. Graph the
equation and interpret the slope.
Math 8 - 26
Today’s Date _______________
The daily wage y (in dollars) of a factory worker is proportional to the number of
parts x assembled in a day. A worker who assembles 250 parts in a day earns
$75.
a) Write an equation that represents the situation.
b) How much does a worker earn who assembles 300 parts in a day?
At a track event, the distance y (in meters) traveled by Student A in x seconds is
represented by the equation y = 7x. The graph shows the distance traveled by
Student B.
a) Which student is faster?
b) Graph the equation that represents Student A
in the same coordinate plane as Student B.
Compare the steepness of the graphs. What
does this mean in the context of the problem?
Practice what you know.
Math 8 - 27
Today’s Date _______________
Notes for
Graphing Linear Equations
in Slope-Intercept Form
We will calculate slopes and y-intercepts of graphs of linear equations.
We will graph linear equations written in slope-intercept form.
Activity: 1) Analyzing Graphs of Lines
2) Deriving an Equation
Intercepts
The x-intercept of a line is the x-coordinate of the point
where the line crosses the x-axis. It occurs when y = 0.
The y-intercept of a line is the y-coordinate of the point
where the line crosses the y-axis. It occurs when x = 0.
Slope-Intercept Form
A linear equation written in the form y = mx + b is in
slope-intercept form. The slope of the line is m, and
the y-intercept of the line is b.
To graph a line in slope-intercept form,
1) graph the y-intercept at (0, b)
2) use the slope m as
to move from the y-intercept to another point on
the line.
Math 8 - 28
Today’s Date _______________
Graph the equation
.
Practice what you know.
Math 8 - 29
Today’s Date _______________
Notes for
Graphing Linear Equations
in Standard Form
We will graph linear equations written in standard form.
Activity: 1) Using a Table to Plot Points
2) Rewriting an Equation
Standard Form of a Linear Equation
The standard form of a linear equation is
ax + by = c
where a and b are not both zero.
Graphing Equations in Standard Form
1) Graph using slope-intercept form.
(rewrite into slope-intercept form and graph from that form)
Graph 3x – 2y = 2
Math 8 - 30
Today’s Date _______________
2) Graph using intercepts.
Graph x + 3y = -3
Practice what you know.
Math 8 - 31
Today’s Date _______________
Notes for
Writing Equations
in Slope-Intercept Form
We will write equations of lines in slope-intercept form.
Activity: 1) Writing Equations of Lines
3) Interpreting the Slope and y-intercept
Recall: Slope-Intercept Form of a Linear Equation
y = mx + b
(x, y) are representing all the points along the line
m = the slope of the line
b = the y-intercept
Steps for writing a linear equation in slope-intercept form.
1) Write the template for the slope-intercept form.
y = mx + b
2) Determine the slope of the line by either counting it on the graph or
using the slope formula.
3) Determine the y-intercept of the line by either looking at where the line
crosses the y-axis on the graph or by using the coordinate (0, y).
4) Substitute the slope into the equation at m and the y-intercept into the
equation at b. Leave the x and y as variables.
Math 8 - 32
Today’s Date _______________
Write the equation of the line represented by the graph.
a)
b)
c) Write an equation of the line that passes through the points (0, -1) and (4, -1).
Practice what you know.
Math 8 - 33
Today’s Date _______________
Notes for
Writing Equations
in Point-Slope Form
We will write equations of lines using a slope and a point.
We will write equations of lines two points.
Activity: 1) Writing Equations of Lines
2) Deriving an Equation
Point-Slope Form of a Linear Equation
A linear equation written in the form y – y1 = m(x – x1) is in
point-slope form. The line passes through the point (x1, y1),
and the slope of the line is m.
Steps for writing a linear equation in point-slope form.
1) Write the template for the point-slope form.
y – y1 = m(x – x1)
2) Determine the slope of the line by either counting it on the graph or using
the slope formula.
3) Determine a point on the line either looking at the graph or by using a
coordinate (x, y).
4) Substitute the slope into the equation at m and the point into the equation
at x1 and y1. Leave the x and y as variables.
Math 8 - 34
Today’s Date _______________
Write the equation of the line in point-slope form that passes through the given
point and has the given slope.
a) (2, 2); m =
b) (3, -6); m =
c) Write the equation of the line in slope-intercept form that passes through the
points (-3, 0) and (6, 3).
d) You are pulling down your kite at a rate of 2 feet per second. After 3
seconds, your kite is 54 feet above you.
1 – Write and graph an equation that represents the height y (in feet) of the
kite above you after x seconds.
2 – At what height was the kite flying before you began pulling it down?
Practice what you know.
Math 8 - 35