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MA 15200 Lesson 22 Sections 2.3 and 2.
I Slope of a Line
A measure of the ‘steepness’ of a line is called the slope of the line. Slope compares a vertical change (called the rise ) to the horizontal change (called the run ) when moving from one point to another point along a line.
Slope is a ratio of vertical change to horizontal change.
If a non-vertical line contains points ( x 1 (^) , y 1 )and( x 2 , y 2 ), the slope of
the line is the ratio described by 2 1 2 1
change in y change in x
rise y y m run x x
*Note: Always be consistent in the order of the coordinates.
There are 3 ways to find slope.
- Using the slope formula (above)
- Counting rise over run (when shown a graph)
- Finding the equation in slope-intercept form (later in lesson)
If a line is horizontal, the numerator in the slope formula will be 0 (the y coordinates of all points of a horizontal line are the same). The slope of a horizontal line is 0.
If a line is vertical, the denominator in the slope formula will be 0 (the x coordinates of all points of a vertical line are the same). A number with a zero denominator is not defined or undefined. The slope of a vertical line is undefined.
There are 4 types of slopes. Positive Negative Zero Undefined
When given two points, it does not matter which one is called point 1 and which point 2. 2 1 1 2 2 1 1 2
y y y y x x x x
line rises left to right
line falls left to right horizontal line
vertical line
Never say ‘no slope’ to define the slope of a vertical line. No slope could be interpreted as 0 or undefined.
Ex 1: Find the slope of a line containing each pair of points. Describe if the line rises from left to right, falls from left to right, is horizontal, or is vertical. ) (2, 3), ( 6, 12)
a P Q
b P Q
c P Q
d P Q
e P Q
II Equations of Lines, Point-Slope Form
Begin with the slope formula and drop the subscript 2’s, putting them back as regular variables.
2 1 1 1 1 2 1 1
cross multiply ( )
y y y y m m y y m x x x x x x
This is known as the point-slope form of the equation of a line.
Point-Slope Form If a line contains the point ( x 1 (^) , y 1 ) and has the slope m ,
then the equation in point-slope form is y − y 1 (^) = m x ( − x 1 ).
Ex 2: a ) Write an equation in point-slope form for a line with a slope of 3
and
through the point (2, 12).
When using point-slope form, substitute values for x 1 (^) , y 1 , and m. Never substitute for x and y. These are the variables of the equation.
Ex 5: Find an equation of a line with slope
and point (0, 6) 8
− − in slope-intercept
form.
Ex 6: Find an equation in slope-intercept form for a line with the following slope and point 3 , ( 6, 1) 2
m = P − −
If a line has a slope m and a y -intercept of b (point (0, b )), then the equation of the line can be written as y = mx + b. This is known as slope-intercept form of the equation of a non-vertical line. This can also be written as f ( ) x = mx + b and is a linear function.
Ex7: Find the slope of each line given its equation.
b x y
a y x
IV Graphing a Line using slope and y-intercept
- Plot the y-intercept on the y -axis (0, b )
- Obtain a second point using the slope m. Write m as a fraction and use rise over run, starting at the y -intercept. (Note: If the slope is negative, let the rise be negative and the run positive. Move down and then right. If you let the run be negative and rise positive, move up and then left.)
- Connect the two points to draw the line. Put arrows at each end to indicate the line continues indefinitely in both directions.
Ex 8: Graph each line.
1 2 2
y = x +
y = − 3 x − 4
V Equations and Graphs of Horizontal or Vertical Lines
It a line is horizontal, the slope-intercept form is written y = 0 x + b or y = b. A vertical
line cannot be written in slope-intercept form because there is no possible number for m. However, a vertical line would have points all with the same x -coordinate. So a vertical line can be written as x = a , where a is the x -intercept.
If a and b are real numbers, then
x
y
x
y
a x y
b x y
c x
Using Intercepts to Graph a Line
- Plot the x -intercept.
- Plot the y -intercept.
- Draw a line through the two points that are the intercepts. Ex 6: Find the intercepts and use them to graph the line. 4 x − 3 y − 12 = 0
x y
x
y
