Gradient (or slope) of a Line, and Inclination
Application: Road sign, indicating a steep gradient.
A `15%` road gradient is equivalent to `m = 0.15`.
The gradient (also known as slope) of a line is defined as
`"gradient"= text(vertical rise)/text(horizontal run`
In the following diagram, the gradient of the line AB is given by: `a/b`
In general, for the line joining the points (x1, y1) and (x2, y2), we have:
We can now write the formula for the slope of a line.
Gradient of a Line Formula
We see from the diagram above, that the gradient (usually written m) is given by:
Interactive graph - slope of a line
You can explore the concept of slope of a line in the following interactive graph (it's not a fixed image).
Drag either point A (x1, y1) or point B (x2, y2) to investigate how the gradient formula works. The numbers will update as you interact with the graph.
Notice what happens to the sign (plus or minus) of the slope when point B is above or below A.
Slope `= (y_2 - y_1)/(x_2 - x_1)`
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You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.
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Find the slope of the line joining the points (−4, −1) and (2, −5).
These are the points involved:
So the slope is:
Note the slope is negative. The line is going "down hill" as we move left to right.
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Positive and Negative Slopes
In general, a positive slope indicates the value of the dependent variable (usually y) increases as we go left to right:
The dependent variable in the above graph is the y-value, while the independent variable is x.
A negative slope means that the value of the dependent variable (usually y) is decreasing as we go left to right:
We have a line with slope m and the angle that the line makes with the x-axis is α.
From trigonometry, we recall that the tan of angle α is given by:
Now, since slope is also defined as opposite/adjacent, we have:
This gives us the result:
tan α = m
Then we can find angle α using
α = arctan m
(That is, α = tan-1 m)
This angle α is called the inclination of the line.
Find the inclination of the line with slope `2`.
Here, tan α = 2, so
NOTE: The size of angle α is (by definition) only between `0°` and `180°`.
Find the slope of the line with inclination α = 137°.