➕ Algebra 1 · Linear Functions
Slope of a Line
Understand slope as rise over run, find it from two points, and tell positive, negative, zero and undefined slopes apart, with memory tricks, diagrams and games.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by a separate AI with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): 56 findings fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- 8.EE.B.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line.
- F-IF.B.6: Calculate and interpret the average rate of change of a function over a specified interval.
By the end you can…
- Describe slope as rise over run and as a rate of change.
- Calculate the slope of a line from two points with the slope formula.
- Tell positive, negative, zero and undefined slopes apart.
Lesson (school style)
Slope tells you how steep a line is and which direction it goes. It is the ratio of how much the line rises (up or down) for every step it runs (to the right). Read every line left to right, like a book.
Slope measures steepness
Imagine walking along a line from left to right. Slope compares your vertical change to your horizontal change.
The farther the slope is from 0 (like 5 or −5), the steeper the line. A slope close to 0 (like 1/2 or −1/2) is gentle.
Rise over run
Rise is the vertical change: up is positive, down is negative. Run is the horizontal change: moving right is positive and moving left is negative.
slope = rise ÷ run
Slope is often written with the letter m.
The slope formula
If you know two points (x1, y1) and (x2, y2), then
m = (y2 − y1) ÷ (x2 − x1)
Subtract the y’s on top and the x’s on the bottom, and use the same order for both.
Four kinds of slope
- Positive: the line goes up from left to right.
- Negative: the line goes down from left to right.
- Zero: the line is perfectly flat (horizontal). The rise is 0.
- Undefined: the line is a vertical wall. The run is 0, and you cannot divide by 0.
Slope is a rate of change
In real life, slope tells you how much one thing changes for each 1 unit of another: miles per hour, dollars per ticket, percentage points of battery per hour.
When you see the word “per,” think slope.
Key words
- Slope
- A measure of how steep a line is and which way it tilts. It equals rise divided by run (undefined for vertical lines).
- Rise
- The vertical change between two points.
- Run
- The horizontal change between two points.
- Rate of change
- How much one quantity changes for each 1 unit of another quantity.
- Undefined slope
- The slope of a vertical line, where the run is 0 and division by 0 is not possible.
Worked examples
Example 1. Find the slope of the line through (1, 1) and (5, 4).
- Rise: 4 − 1 = 3.
- Run: 5 − 1 = 4.
- Slope = rise ÷ run = 3/4.
Answer: Answer: m = 3/4
Example 2. Find the slope of the line through (−2, 5) and (4, −1).
- Rise: −1 − 5 = −6.
- Run: 4 − (−2) = 6.
- Slope = −6 ÷ 6 = −1. The negative sign means the line goes downhill.
Answer: Answer: m = −1
Example 3. Find the slope of the line through (3, 7) and (9, 7), and of the line through (2, 1) and (2, 9).
- First line: rise = 7 − 7 = 0 and run = 9 − 3 = 6, so slope = 0 ÷ 6 = 0. It is horizontal.
- Second line: rise = 9 − 1 = 8 and run = 2 − 2 = 0, so slope = 8 ÷ 0, which is undefined. It is vertical.
Answer: Answer: The first slope is 0. The second slope is undefined.
Example 4. A phone battery is at 100% at 5 pm and 60% at 9 pm. What is the rate of change?
- Treat time as x and battery as y: points (5, 100) and (9, 60).
- Slope = (60 − 100) ÷ (9 − 5) = −40 ÷ 4 = −10.
- The battery loses 10 percentage points each hour.
Answer: Answer: −10 percentage points per hour
In plain English
Slope is a steepness number. It answers: for every step I take to the right, how many steps do I go up or down?
On a hiking trail that climbs 3 feet for every 6 feet of level (horizontal) distance, the slope is 3 ÷ 6 = 1/2. A steeper climb gives a bigger number. A trail going downhill gives a negative number. A flat trail gives zero. A wall straight up? You never move sideways at all, so you would have to divide by zero. That is not a number, so we call it undefined.
See it
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.
Study notes
Key ideas
- Slope = rise ÷ run = vertical change ÷ horizontal change.
- Read lines left to right: up is positive, down is negative.
- Horizontal lines have slope 0. Vertical lines have undefined slope.
- Slope is a rate of change: how much y changes for each 1 unit of x.
Steps
- Label your two points as (x1, y1) and (x2, y2).
- Subtract the y’s for the rise and the x’s for the run, in the same order.
- Divide rise by run and simplify.
- Sanity check the sign against the direction of the line.
Formulas
- Slope formula: m = (y2 − y1) ÷ (x2 − x1) (when x2 ≠ x1)
- Horizontal line: y = b, slope is 0
- Vertical line: x = a, slope is undefined
Watch out
- Keep the order of the points the same on top and bottom.
- Be careful with negatives: 4 − (−2) = 6.
- Rise is on top. If you get the reciprocal, you put the run on top.
Why it works
- Why is the slope the same between any two points on a line? A line is straight, so every rise-and-run triangle is a scaled copy of the others (they are similar triangles).
- Why is vertical slope undefined? The run is 0, and dividing by 0 has no answer.
Common mistakes
Teach it back
Explain to a friend how to find the slope between (2, 1) and (6, 9) and what the answer means.
Explaining it out loud shows what you really know. Did you…
- I found the rise (9 − 1 = 8) and the run (6 − 2 = 4).
- I divided to get slope 2.
- I explained it as: up 2 for every 1 step to the right.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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