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THE SLOPE OF A LINE
Consider the line containing the points (โ4, โ1), (0, 2) and (4, 5). When we move from the point (โ4, โ1) to the point (0, 2) the y-coordinate increases by 3 units; the x-coordinate increases by 4 units.
X
Y
(-4, -1)
(0, 2)
(4, 5)
This pattern repeats when we move from the point (0, 2) to the point (4, 5). The vertical change or rise between any two points, such as (0, 2) and (4, 5), is the difference of the y- coordinates: 5 โ 2 = 3. The horizontal change or run is the difference of the x- coordinates: 4 โ 0 = 4. The ratio of the change in the y to the change in x is called the slope of the line:
change in y rise 5 2 3 slope change in x run 4 0 4
Finding Slope When Two Ordered Pairs are Given
Given any two points on a line, we'll call them (x 1 , y 1 ) and (x 2 , y 2 ), we can find the line's slope using the following formula:
2 1 2 1
change in y rise y y slope = m change in x run x x
Example 1: Find the slope of the line passing through the points (โ6, โ2) and (โ2, 4).
To find the slope, let (x 1 , y 1 ) = (โ6, โ2) and (x 2 , y 2 ) = (โ2, 4). Then substitute the values for x 1 , y 1 and x 2 , y 2 in the formula and simplify.
( ) ( )
2 1 2 1
y y 4 2 4 2 6 3 m x x 2 6 2 6 4 2
โ โ^ โ + = = = = โ โ โ โ โ +
=
The slope of a line determines its slant. When the slope is positive, the line slants upward; when the slope is negative the line slants downward.
(^4 ) 4 m 2 6 6 2 = โ = โ = โ โ
( 2 , 6 )
( 6 , 2 )
(^4 4 ) 6 2 6 4 m 6 2 2 = = = โ โ + = โ โ โ
(-2, 6 )
(-6, 2 )
X
Y
( 2 , -2)
( 2 , -6) (-6, -6) (-2, -6)
2 ( 6) 2 6 4 undefined 2 2 0 0 m โ โ + = = = โ
โ (^6 6 0 0) = 2 6 4 m 6 ( 6) 2 ( 6) = โ^ + = = โ + = โ^ โ^ โ โ โ โ
Vertical Lines: x = a
The graph of the equation x = 2 contains the points (2, 3) and (2, โ5), as the table shows.
X
Y
(2, 3)
(2, -5)
X Y
If we let (x1, y 1 ) = (2, 3) and (x 2 , y 2 ) = (2, โ5) and substitute these values into the formula we get:
2 1 2 1
y y (^5 3 ) m undefined x x 2 2 0
In general, because the x-coordinates are the same, the slope of a vertical line x = a is always undefined.
Horizontal Lines: y = b
The graph of the equation y = โ4 contains the points (โ3, โ4) and (2, โ4), as the table shows:
X
Y
(-3, - 4) (2, - 4)
X Y
If we let (x1, y 1 ) = (โ3, โ4) and (x 2 , y 2 ) = (2, โ4) and substitute these values into the formula we get:
2 1 2 1
y y (^4 44 4 ) m 0 x x 2 3 2 3 5
โ โ^ โ โ โ +
In general, because the y-coordinates are the same, the slope of any horizontal line y = b is always 0.
