➕ Algebra 1 · Linear Functions
Slope of a Line: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Relationship status
Picture how close you feel to a friend over time. Getting closer every week is a rising line, a positive slope. Drifting apart is a falling line, a negative slope. Staying exactly the same is flat, a zero slope. And going from strangers to best friends in no time at all is a vertical jump: all rise and no run, so the slope is undefined.
How it connects: Positive means the value goes up as you move right, negative means it goes down, zero means it stays the same, and undefined means a vertical line, where the run is 0.
Spider-Man on a skyscraper
When Spider-Man climbs straight up a building, he goes up and up without moving sideways at all. There is rise but no run.
How it connects: A vertical wall has run = 0, and dividing by 0 is impossible, so its slope is undefined.
Binge-watching a series
If each episode of a series is 45 minutes, your total watch time goes up 45 minutes with every episode you watch.
How it connects: That 45 minutes per episode is the slope. “Per” is your clue that you’re looking at a rate of change.
Slow and steady wins the race
The proverb comes from the tortoise and the hare: a patient, steady pace beats a wild sprint.
How it connects: On a distance-versus-time graph, slope is speed. The tortoise’s line has a small slope but keeps climbing. The hare’s graph is steep at first, then goes flat (slope 0) while he naps, so the tortoise ends up ahead.
Slope idioms
An “uphill battle” is hard going up. A sled run that is “all downhill” heads down. If you “hit a plateau,” you stop making progress. And if sales “fell off a cliff,” they dropped very steeply.
How it connects: Uphill is positive, downhill is negative, a plateau is zero, and a cliff drop is a very steep negative slope. Only a perfectly vertical wall is undefined.
The calm horizontal line
What did the zero-slope line say to the steep line? “Relax, take it easy. You’re climbing, but I’m staying level, and I’m fine with that.”
How it connects: A horizontal line has rise = 0, so its slope is 0: no change at all.
Memory help
I before U
Which goes on top, rise or run? You rise in the morning before you run, so rise comes first, on top. Bonus trick: I comes before U in the alphabet, and rise has an I while run has a U.
HOY VUX
Horizontal lines have O (zero) slope and equations like Y = a number. Vertical lines have Undefined slope and equations like X = a number. HOY, VUX.
The hiker
You are hiking left to right. If the path climbs, your rise is positive. If it drops, your rise is negative. If it is flat, there is no rise. If it is a wall, you aren’t hiking anywhere. The farther the slope is from 0, the steeper the hike.
Slope rhyme
Rise over run, now the slope is done! Uphill positive, downhill negative, flat is zero, and a wall is undefined, not none!
Draw the staircase
Sketch a tiny staircase between the two points. The vertical side of the stair is the rise and the horizontal side is the run. Label them before you divide.
Same order, top and bottom
If you start with y2 on top, you must start with x2 on the bottom. Mixing the order flips the sign and gives the wrong slope. Also, rise goes on top. Run on top gives the reciprocal.