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Rate of Change and Slope

Slope measures how steep a line is and which direction it tilts. Every line has exactly one slope, and once you know it, you know how fast yy changes every time xx changes. That is why slope is also called the rate of change — the two terms describe the same idea.

The whole topic comes down to one fraction: slope=riserun\text{slope} = \dfrac{\text{rise}}{\text{run}}. The rise is how far the line goes up or down, and the run is how far it goes right. Learn to read that fraction off a graph and you can handle every slope question in this unit.

Rise over run

Pick any two points on the line. The rise is the change in yy between them — up counts as positive, down counts as negative. The run is the change in xx, always read left to right. Slope is rise divided by run.

If you have the coordinates of the two points, you do not need to count squares at all. The slope formula is m=y2−y1x2−x1\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} — the change in yy on top, the change in xx on the bottom, with the points kept in the same order in both parts.

The four types of slope

A line that rises from left to right has positive slope. A line that falls from left to right has negative slope. In the graph below, the black line has slope 11 and the blue line has slope −1-1.

Two special cases round out the list: a horizontal line has slope 00, because the rise is zero no matter how far you run. A vertical line has undefined slope, because the run is zero and you cannot divide by zero.

-4-3-2-11234-4-3-2-11234xy

Slope as a rate of change

When the axes stand for real quantities, slope becomes a rate. If a graph shows distance in miles against time in hours and the line has slope 5555, the object is moving 5555 miles per hour. Read the slope's units as the yy-units per one xx-unit, and the number will always tell you a real-world rate.

Worked examples

Example 1: counting rise over run on a graph

A line passes through (0,1)(0, 1) and (2,4)(2, 4). Find its slope.

Rise: from y=1y = 1 up to y=4y = 4rise=3\text{rise} = 3
Run: from x=0x = 0 right to x=2x = 2run=2\text{run} = 2
Divide rise by runm=32m = \dfrac{3}{2}

Answer: m=32m = \dfrac{3}{2}

Example 2: using the slope formula

Find the slope of the line through (−2,5)(-2, 5) and (4,−4)(4, -4).

Write the formulam=y2−y1x2−x1\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}
Substitute the two pointsm=−4−54−(−2)m = \dfrac{-4 - 5}{4 - (-2)}
Simplify top and bottomm=−96m = \dfrac{-9}{6}
Reduce the fractionm=−32m = -\dfrac{3}{2}

Answer: m=−32m = -\dfrac{3}{2}

Example 3: slope as a rate

A hiker's distance graph passes through (1,3)(1, 3) and (4,12)(4, 12), where xx is hours and yy is miles. What is the hiker's speed?

Change in yy (miles)12−3=912 - 3 = 9
Change in xx (hours)4−1=34 - 1 = 3
Divide to get the ratem=93=3m = \dfrac{9}{3} = 3

Answer: 33 miles per hour

Try one yourself

Common questions

Is slope the same thing as rate of change?

Yes. Slope is the graph's version of the idea, and rate of change is the real-world version. Both are the change in yy divided by the change in xx.

Does it matter which point I call (x1,y1)(x_1, y_1)?

No — as long as you keep the order the same on top and bottom. If you start with a point's yy-value on top, start with that same point's xx-value on the bottom.

What's the difference between zero slope and undefined slope?

Zero slope is a horizontal line: the rise is 00, and 00 divided by anything is 00. Undefined slope is a vertical line: the run is 00, and dividing by 00 is not allowed.

Can a slope be a fraction?

Absolutely — most are. A slope of 23\dfrac{2}{3} just means the line rises 22 for every 33 it runs. Leave slopes as reduced fractions rather than rounding to decimals.

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