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The concept of slope in linear equations, providing three methods to find it: using the slope formula, counting rise over run, or finding the equation in slope-intercept form. It also covers horizontal and vertical lines, equations in point-slope form, and slope-intercept form. Students will learn how to find the slope and an indicated point for a line, write an equation in point-slope form, and find the equation of a line in slope-intercept form.
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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.
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
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
x y
x
y