Teacher Key
Slope and Intercept: Grapher
Instructor
Directions
The Teacher Key contains the same strategies and possible solutions available to the
student in the computer-based activity AL168. Use the Key to assist in lesson planning and
as a quick glance into the activity.
Lesson at a Glance
In this lesson, you will explore the slope and y-intercept of linear equations.
Review the computer-based activity AL168 to access the Grapher and the interactive
version of the lesson.
Warm-up—Catch Me if You Can
Directions
Detective Joe is on a thrilling mission. A gang of five thieves is at large, and Joe has to
catch them all. Help Detective Joe match the thieves to their real identity and to the
rewards posted for their capture. Place an X in a box if it is not a match. Place an O in the
box if it is a match.
Clues:
• Eagle Eye got her name from being able to spot a rich tourist.
• Bugger’s reward money is 10 times as much as Herb’s and 50 times as much as Rick’s.
• Herb’s alias refers to his fast driving.
• Drama Mama threw a fit when he realized that he had the lowest reward money posted.
• Phil’s reward is the median of all the posted rewards.
© 2006 by CompassLearning, Inc.
Algebra AL168
Case Studies
Directions
Hannah and Pete are getting ready to learn about graphing linear equations. Join them by
reading each case and experimenting with the Grapher.
Case Study 1—Grappling with the Grapher
Hannah and Pete notice that there are several formats for entering linear equations into the Grapher:
the standard form, the point-slope form, the slope-intercept form, and the table form.
They don’t know which form to select when graphing an equation with a y-intercept of 3 and an
x-intercept of 2.
Which form would you recommend and why? Describe the steps needed to draw the line using
the Grapher.
Strategy
Questions to Ask Yourself
• What is an intercept?
• What can you assume about the ordered
pair for an x-intercept?
• What can you assume about the ordered
pair for a y-intercept?
Possible Solution
I would recommend using the table form
because it is the most direct way of taking
the given information and graphing it.
Strategy
List the information needed for each entry
format. Check each format to see if the
intercepts are sufficient.
Since the y-intercept is 3, this means that
one of the points on the line is (0, 3). Since
the x-intercept is 2, this means that another
point on the line is (2, 0).
To draw any line, two points are needed.
Therefore, Hannah and Pete should select
the table form for a line and enter the
coordinates (0, 3) and (2, 0).
© 2006 by CompassLearning, Inc.
Algebra AL168
Case Study 2—Perception of Interception
Pete plots several lines, and
each line intersects the y-axis
as well as the x-axis. He
concludes that all lines have
two intercepts, one x-intercept
and one y-intercept.
Hannah is sure that this is
incorrect, but she cannot figure
out how to show this on the
Grapher.
What do you think? Is Pete or Hannah correct? Provide examples of equations to support your reasoning.
Strategy
Questions to Ask Yourself
• When would a line never intersect with
another line?
• When would a line never intersect with
an axis?
• Which forms do not require an intercept
to be entered?
Strategy
Think of the axes as lines and the intercepts
as points of intersection. Find Grapher
formats that allow you to draw a line on an
axis.
© 2006 by CompassLearning, Inc.
Possible Solution
Hannah is correct. Every line does not have
2 intercepts.
All of Pete’s examples have a positive or
negative slope. These lines will always have
2 intercepts.
If a line is on an axis, it will have an infinite
amount of intercepts. An example is y 0.
This line has one y-intercept and an infinite
amount of x-intercepts.
If a line is parallel to an axis, it will have
only one intercept. An example is y 3.
This line has one y-intercept and no
x-intercept.
Algebra AL168
Case Study 3—Slippery Slopes
Hannah is graphing linear functions written in the slope-intercept form, y mx b. She knows that m
represents the slope and b represents the y-intercept.
She plots several functions to determine how changing m and b affects a line.
y mx b
y 2x 3
Experiment with the Grapher and describe what happens as m and b are changed. Were you surprised by
the result? Why or why not?
Strategy
Questions to Ask Yourself
• What happens as m and b increase?
• What happens if m and b are negative values?
• What happens if m and b are fractional values?
Strategy
Look for a pattern. Find how the line moves for each incremental change.
© 2006 by CompassLearning, Inc.
Algebra AL168
Possible Solution
A line with a positive y-intercept crosses the y-axis above
the x-axis. A line with a negative y-intercept crosses the
y-axis below the x-axis.
The line moves up and down relative to the value of the
y-intercept.
A line with a positive slope rises from left to right. However,
a line with a negative slope falls from left to right.
As the absolute value of a slope or m increases, the line
rotates closer and closer to being vertical.
As the absolute value of a slope or m decreases, the line rotates closer to being horizontal.
I was surprised that moving the line up and down also made the line appear to move left and right. I
also noticed that the lines with the same slope never intersect each other, but that lines with
different slopes always intersect each other.
© 2006 by CompassLearning, Inc.
Algebra AL168
Case Study 4—Setting the Standard
Pete and Hannah have learned that the standard form of a linear equation is Ax By C, where A, B,
and C are integers and A 0. And, they wonder how this form is related to others.
Help them discover the relationship between the standard form and the slope-intercept form.
Find the relationship between the values of A, B, and C of the standard form with m and b of the slopeintercept form.
Strategy
Questions to Ask Yourself
• What do the values of m and b represent?
• How would you solve the equation, Ax By C, for y ?
• What is the resulting coefficient of x?
Strategy
Check the answer by substituting values for the two forms. Enter the equations into the Grapher
and compare the drawn lines.
© 2006 by CompassLearning, Inc.
Algebra AL168
Possible Solution
When the equation Ax By C is solved for y, it is transformed into the slope-intercept form.
Start with the standard form.
Ax By C
Subtract Ax from both sides.
By -Ax C
Divide both sides by B.
-A
C
y x B
B
-A
The value is m or the slope of the line.
B
C
The value is b or the y-intercept.
B
These values can be tested by entering corresponding equations into the Grapher.
© 2006 by CompassLearning, Inc.
Algebra AL168
Case Study 5—Lining Up
Hannah and Pete are challenged to write an equation of a line that passes through three points.
They are given the ordered pairs (2, 3), (4, 7), and (-1, -1).
Help Hannah and Pete write an equation that contains these points.
Or, if these points cannot lie on the same line, change a single coordinate in one of the ordered pairs so
that they do.
Describe your strategy.
Strategy
Questions to Ask Yourself
• What is the relationship between any pair of coordinates in a function?
• How do you verify if a point is on a line?
Strategy
Visualize the coordinates and the lines. Use the Grapher to plot points. Use the table form to see if
each pair of coordinates forms the same line. Test your answers by graphing your equation and
plotting the points.
© 2006 by CompassLearning, Inc.
Algebra AL168
Possible Solution
The ordered pairs do not lie on the same line. One way to see this is by drawing a line through (2, 3)
and (4, 7), and plotting (-1, -1) separately.
One strategy is to write an equation using two points, such as (2, 3) and (4, 7).
First, find the slope by finding the difference in the y-coordinates divided by the difference in the
x-coordinates.
y
73
Slope 2
x
42
Then, use the point-slope equation with the calculated slope and one of the given coordinates. Here,
(2, 3) is used. Then, the equation is simplified into the slope-intercept form, y 2x 1.
(y y1) m(x x1)
(y 3) 2(x 2)
y 3 2x 4
y 2x 1
Point-slope form
Slope-intercept form
Lastly, the third coordinate (-1, -1) has to be altered to lie on the line. By solving the linear equation
when x -1, the coordinate becomes (-1, -3).
The Grapher confirms that the points (2, 3), (4, 7), and (-1, -3) are on the line y 2x 1.
© 2006 by CompassLearning, Inc.
Algebra AL168
Summary—Ordering It Right
Pete and Hannah understand linear equations, slope, and intercept much better after investigating
these cases with the Grapher.
Their favorite lessons included the following:
• An intercept is the intersection point between the line and an axis.
• Slopes determine the steepness of lines.
• To graph a line, you only need two points or a point and a slope.
• If two lines overlap each other exactly, they are equivalent equations.
Think About This
Although lines can be drawn in any direction, functions have specific rules. In a function, each value
of x has one and only one value of y.
Are all equations that can be expressed in the slope-intercept form, y mx b, functions?
© 2006 by CompassLearning, Inc.
Algebra AL168