To find the slope on a graph, pick any two points on the line and use “rise over run”: how much the line goes up or down divided by how much it goes across.

What slope means

  • Slope tells you how steep a line is and whether it’s going up or down as you move left to right.
  • If the line slants up from left to right, the slope is positive; if it slants down, the slope is negative.
  • A perfectly horizontal line has slope 0, and a vertical line has an undefined slope.

Think of walking up or down a hill: the steeper the hill, the larger the slope (in size).

Step‑by‑step: rise over run

Use this anytime you have a straight line drawn on a coordinate grid.

  1. Choose two clear points on the line.
    • Prefer points where the line hits exact grid intersections (so you can read whole number coordinates).
  1. Write their coordinates.
    • Call them (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​).
  1. Find the rise (change in y).
    • Compute y2−y1y_2-y_1y2​−y1​: up is positive, down is negative.
  1. Find the run (change in x).
    • Compute x2−x1x_2-x_1x2​−x1​: right is positive, left is negative.
  1. Divide rise by run.
    • Slope m=riserun=y2−y1x2−x1m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}m=runrise​=x2​−x1​y2​−y1​​.
 * Simplify the fraction if you can (for example, 63=2\dfrac{6}{3}=236​=2).

Tiny story example

Imagine a line passing through the points (1,2)(1,2)(1,2) and (3,6)(3,6)(3,6) on a graph.

  • Rise: 6−2=46-2=46−2=4 (you go up 4).
  • Run: 3−1=23-1=23−1=2 (you go right 2).
  • Slope: m=42=2m=\dfrac{4}{2}=2m=24​=2.

This means every time x increases by 1, y increases by 2.

Visual sense of positive, negative, zero, undefined

  • Positive slope : line climbs as you move right (like a hill going up).
  • Negative slope : line falls as you move right (like going downhill).
  • Zero slope : flat horizontal line; no rise, only run.
  • Undefined slope : vertical line; run is 0, so you’d divide by 0, which is not allowed.

Example quick checks:

  • A horizontal line through (1,3)(1,3)(1,3) and (5,3)(5,3)(5,3): rise =0=0=0, run =4=4=4, slope =0=0=0.
  • A vertical line through (2,1)(2,1)(2,1) and (2,4)(2,4)(2,4): rise =3=3=3, run =0=0=0, slope undefined.

Short “formula view” vs “counting blocks” view

You can think of finding slope in two equally valid ways:

  • Formula view
    • Pick two points, plug into m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​, and simplify.
  • Counting‑blocks view (what many teachers use on the board)
    • From one point to the other, count how many squares you move up or down (rise) and left or right (run).
* Then put rise over run as a fraction and reduce it.

Both always give the same answer for a straight line.

Common mistakes to avoid

  • Swapping points inconsistently: if you start with point A as (x1,y1)(x_1,y_1)(x1​,y1​), be sure you always subtract in the same order for x and y (either “second minus first” for both or “first minus second” for both).
  • Forgetting the sign: if you go down instead of up, the rise is negative, which changes the slope.
  • Using a point that’s not exactly on a grid intersection; that can lead to messy decimals or mistakes.

Very quick recap

  • Slope is how steep a line is: m=riserun=y2−y1x2−x1m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}m=runrise​=x2​−x1​y2​−y1​​.
  • Use any two clear points on the line, count up/down and left/right, then do rise ÷ run.
  • Upward lines → positive slope; downward → negative; horizontal → 0; vertical → undefined.

Information gathered from public forums or data available on the internet and portrayed here.