The slope of a line is a number that describes how steep a line is. The slope formula is the ratio between the vertical change and the horizontal change of the line.

If the slope is a large number, then the line will be very steep. If it is a small number, then the line will just be a gentle incline. A perfectly flat horizontal line has a slope of 0.

If the slope is a positive number, then that means the line is sloping "uphill" from left to right. If it is a negative number, then that means the line is sloping "downhill".

There are two formulas you can use to find the slope of a line. The first one is the most useful when you are finding the slope of a line on a graph. The second one is most useful when you are finding the slope of a line from two points on the line.

\(m=\frac{rise}{run}\)

\(m=\frac{y_2-y_1}{x_2-x_1}\)

When you are finding the slope, you can choose ANY two points on the line.

So, technically, you could choose a point like (-0.5,3) on the green line.

However, the points that I've labeled are easier to work with because they have integer coordinates instead of decimals.

G = (-1,1) H = (0,5) |
A = (-4,5) B = (-3, 3) |
M = (4,1) L = (2,0) |

To find the rise of the line, choose one of your points as your starting point.

From the starting point, find the vertical distance to the second point.

If you go down, the rise will be negative. If you go up, the rise will be positive.

From G to H Rise = +5 |
From A to B Rise = -2 |
From M to L Rise = -1 |

To find the run of the line, find the horizontal distance from the starting point to the second point.

If you go left, the run will be negative. If you go right, the run will be positive.

Run = +1 |
Run = +1 |
Run = -2 |

The rise always comes first in the division problem (or the top of the fraction). If you have a hard time remembering that, you can use this memory tool:

"You first have to __RISE__ from your bed before you can go on a __RUN__."

Also, if the rise or the run is a negative number, make sure to follow the rules for dividing negative numbers when you divide them.

Slope \(=\frac{+5}{+1}=5\) |
Slope \(=\frac{-2}{+1}=-2\) |
Slope \(=\frac{-1}{-2}=\frac{1}{2}\) |

It is reasonable that the slope of the green line is +5 because it is an "uphill" line and it is relatively steep.

It is reasonable that the slope of the red line is -2 because it is a "downhill" line and it is steep but not as steep as the green line.

It is reasonable that slope of the purple line is +1/2 because it is an "uphill" graph and it is not very steep.

Positive Slope

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Negative Slope

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Zero Slope

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Undefined Slope

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Parallel Slope

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Perpendicular Slope

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