Slope

Worked example

1. 5.3.w1 Through (1,2) and (3,8): $$m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$$ The slope is positive, so the line runs uphill from left to right. The 6 on top is the rise (y climbed from 2 to 8) and the 2 on the bottom is the run (x moved from 1 to 3).

2. 5.3.w2 Through (0,5) and (2,1): $$m = \frac{1 - 5}{2 - 0} = \frac{-4}{2} = -2$$ The slope is negative, so this line runs downhill. Notice the top is 1 − 5, not 5 − 1: we used the second point first on top and bottom, keeping the order matched.

3. 5.3.w3 Horizontal line y = 4: every point has y = 4, so there's no rise at all, and rise = 0 means m = 0/run = 0. Slope 0, the flat landing.

4. 5.3.w4 Vertical line x = 3: every point has x = 3, so there's no run at all, and m = rise/0 is division by zero, which has no value, so the slope is undefined, the wall.

Those last two are the pair people swap, so pin them to the picture: a flat landing has no rise, giving slope 0, while a wall has no forward run, leaving the slope undefined.

It also helps to say them in full: "zero slope" for horizontal, "undefined slope" for vertical. Try to avoid the phrase "no slope," which some people hear as "slope zero" and others as "undefined." The two full names never blur.

5. 5.3.w5 Slope in context, with units. A line on a cost (in dollars) versus time (in hours) graph passes through (0, 20) and (4, 80). Find the slope and say what it means. $$m = \frac{80 - 20}{4 - 0} = \frac{60}{4} = 15$$ The slope is 15 dollars per hour: the cost climbs $15 for each additional hour. You read its units straight off the axes: dollars on top, hours on the bottom, so dollars per hour. The intercept carries meaning too: the point (0, 20) says the cost is already $20 at 0 hours, a $20 starting charge. Constant-rate check: at (2, 50), (50 − 20)/(2 − 0) = 15 as well, so the rate really is steady. Naming the units is what turns a bare number into a rate you can read aloud, "$15 per hour," exactly the rate-of-change idea made concrete.

6. 5.3.w6 Slope two ways, graph versus formula. For the line through (1, 2) and (4, 8), compute the slope both ways and compare. From the graph (rise/run): draw the right triangle between the points, going up 6 (from y = 2 to y = 8) and across 3 (from x = 1 to x = 4), so m = rise/run = 6/3 = 2. From the formula: $$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$ Same answer, as it has to be: rise/run and (y₂ − y₁)/(x₂ − x₁) are the same subtraction, one read off a picture and one off the coordinates. When is each handier? Rise/run is quickest when you already have a graph and can count boxes; the formula is safer when the points are given as numbers, especially negatives or fractions, where counting on a grid invites mistakes. Both are correct, so use the one that fits what you're handed, and the choice is yours.

Here's a graph with that rise/run right triangle drawn in between two points, so you can see where the rise and the run come from 5.3.f1.

Here's a place people slip: reading the sign of a slope as its steepness. Slope packs two facts. The sign says which way the line tilts (minus is downhill), and the size says how steep. So a slope of −3 is steeper than a slope of 2, because 3 is the bigger size, even though −3 is the smaller number. To compare steepness, strip the signs and compare the sizes; bring the signs back only to say uphill or downhill.

And when you compute a slope, give your subtraction a quick sanity check: if you find yourself with y₂ − y₁ on top but x₁ − x₂ on the bottom, the orders don't match and the sign will come out wrong. Label your two points and subtract the same way on top and bottom. If a check ever disagrees with your answer, that mismatch is the check doing its job. Go back and re-run the subtraction slowly; it's almost always one order or one sign that slipped, not the whole method.