In-class Handout, Section 3.3 and 3.4, June 5th, 2012, Page 1 of 2
Definitions from 3.3 Intercepts; Vertical and Horizontal Lines
𝑥-intercept
(p. 182) The 𝑥-coodinate of a point, sometimes designated 𝑎, where a graph intersects
the 𝑥-axis is an 𝑥-intercept. The 𝑦-coordinate of any 𝑥-intercept must be zero.
𝑦-intercept
(p. 182) The 𝑦-coordinate of a point, sometimes designated 𝑏, where a graph intersects
the 𝑦-axis is a 𝑦-intercept. The 𝑥-coordinate of any 𝑦-intercept must be zero.
Horizontal Line
(p. 185) The equation of a horizontal line with 𝑦-intercept 𝑏 is 𝑦 = 𝑏. Also, the slope of
any horizontal line is zero.
Vertical Line
(p. 186) The equation of a vertical line with 𝑥-intercept 𝑎 is 𝑥 = 𝑎. Also, the slope of
any vertical line is undefined.
Definitions from 3.4 Slope and Rate of Change
Slope (informal)
(p. 193) The slope is a real number that measures the “tilt” or “angle” of a line.
Slope, 𝑚
(p. 194) The slope 𝑚 of the line passing through the points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) is
𝑚=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
=
𝑦2 −𝑦1
𝑥2 −𝑥1
Please memorize!
Slope, positive
(p. 196) A line with a positive slope rises from left-to-right.
Slope, negative
(p. 196) A line with a negative slope falls from left-to-right.
Slope (word problems) (p. 198) The slope represents the rate of change between the units of the 𝑦-variable
and the units of the 𝑥-variable.
Example, Finding Intercepts, p. 182, Q29:
What is the 𝑥-intercept and the 𝑦-intercept of the line given by the equation 4𝑥 − 𝑦 = 8?



Find the 𝑥-intercept: Set 𝑦 = 0.
Find the 𝑦-intercept: Set 𝑥 = 0.
4𝑥 − 0 = 8
4𝑥 = 8
𝑥=2
So, the 𝑥-intercept is 𝑎 = 2 or (2, 0)
4∙0−𝑦 =8
−𝑦 = 8
𝑦 = −8
So, the 𝑦-intercept is 𝑏 = −8 or (0, −8)
Shortcut Method: Recognize that the equation is in Standard Form (p. 175) 𝐴𝑥 + 𝐵𝑦 = 𝐶 and use the shortcut
formulas (presented in an earlier handout).
𝑥-intercept, 𝑎 =
𝐶
𝐴
8
= 4 = 2 or (2, 0)
𝑦-intercept, 𝑏 =
𝐶
𝐵
8
= −1 = −8 or (0, −8)
Examples, Graphing Horizontal and Vertical Lines, p. 185, p. 186:
A) Graph the equation 𝑦 = −3.


If we were graphing this on a number line, the result might look like:
However, on an 𝑥𝑦-plane the equation 𝑦 = −3 has a different look (see below):
-3
𝑦-axis
Why is the graph a point on a number line but a line in the 2D 𝑥𝑦-plane? One explanation is that the line
represents all solutions to the equation 𝑦 = −3. Every point on the line has a 𝑦-coordinate of −3 and this is enough to
be a solution to the given equation 𝑦 = −3.
5
𝑦-axis
4
3
2
1
𝑥-axis
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
𝑦 = −3
-3
-4
-5
B) Graph the equation 𝑥 = −2.

If you recognize that an equation is missing a variable, in this case 𝑦, this means you can set the missing variable
to any value. The one constant is that 𝑥 = −2 no matter what the value of 𝑦 is. Using a table:
𝒙
𝒚

−2
0
−2
1
−2
2
Graphing these points should result in a vertical line (left to the student to graph).
−2
3
In-class Handout, Section 3.3 and 3.4, June 5th, 2012, Page 2 of 2
Example, Calculating the Slope, p. 194
Calculate the slope of a line passing through the points (−3, 2) and (2, −1).

One method would be the sketch these two points and use the formula 𝑚 =
𝑅𝑖𝑠𝑒
.
𝑅𝑢𝑛
The disadvantages of this
method are that a sketch may be inaccurate and if your problem has fractions or decimals then it will be more
difficult to get an accurate result.

𝑚=
The preferred method is to use the formula
o
First, label your points:

Then, plug these values into the formula:
𝑦2 −𝑦1
𝑥2 −𝑥1
(−3, 2)
𝑥1 𝑦1
𝑚=

.
Summarizing, the slope of this line is
(2, −1)
𝑥2 𝑦2
𝑦2 − 𝑦1
−1 − 2
3
=
=−
𝑥2 − 𝑥1 2 − (−3)
5
𝑚=−
3
5
Example, Calculating the Slope from a Graph, p. 196
What is the slope of the line in the graph?
5
4
3
2
1
Look for perfect intersections
through the grid.
0
-5
-4
-3
Rise
-2
-1
0
1
2
3
4
5
-1
-2
-3
Run
-4
-5
Either write the coordinates of two points and use the method of the previous example. Or use 𝑚 =
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
=
−4
.
5
Example, Sketch a Line with a Given Slope, p. 197
Sketch a line with the slope 𝑚 = 2 that passes through the point (−1, −1).

The slope cannot be used until we graph the point. So the first step is to graph the point.

From this point, we can use the slope 𝑚 = 1. Rise positive 2 (go up) and Run positive 1 (go right).

Alternatively, we can Rise negative 2 (go down) and Run negative 1 (go left) since 𝑚 = 1 = −1
2
2
−2
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
-5
(finish the sketch)
Example, Recognizing the Slope in a Word Problem
A car travels at 60 mph. Over 4 hours it will cover 240 miles.



In this problem, which number corresponds to a slope?
In the formula 𝑑 = 𝑟𝑡 which variable corresponds to the slope? (Which variable corresponds to the 𝑥variable or input; which variable corresponds to the 𝑦-variable or output)
How would you graph the distance a car travels at 60 mph for 𝑡 hours?