Graphs · Slope · 4.2

Slope from two points.

For any two points, slope = (y₂ − y₁) ÷ (x₂ − x₁). From (1, 2) to (4, 8): (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2. Keep the order consistent in top and bottom.

01 / THE IDEA

See it work.

Watch the method

Slope from two points

(1,2)→(4,8): (8−2)/(4−1) = 2.

1 / 2
-6-666
1

Rise over run

Rise = change in y = 6; run = change in x = 3.

63\frac{6}{3}
2

Simplify

The slope is rise ÷ run.

m=2m = 2
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number, e.g. 5.

streak 0
best 0
-6-666
Your answer:
— / WHY IT WORKS

The idea underneath.

The slope formula

Subtract the y’s for the top and the x’s for the bottom, in the same order.

Why it works

Δy over Δx.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Find the slope of the line through (-2,-6) and (-5,1).

1

Rise over run

Rise = change in y = 7; run = change in x = -3. 7/-3

2

Simplify

The slope is rise ÷ run. m = -7/3

EX 2medium

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

1

Rise over run

Rise = change in y = -2; run = change in x = 0. -2/0

2

Simplify

The slope is rise ÷ run. m = undefined

EX 3hard

Find the slope of the line through (1,-4) and (2,-1).

1

Rise over run

Rise = change in y = 3; run = change in x = 1. 3/1

2

Simplify

The slope is rise ÷ run. m = 3

04 / COMMON TRAPS

Where students slip.

Trap

Mixing the order top vs bottom.

Fix

Subtract in the same order in both.

Trap

Swapping x and y.

Fix

y’s on top, x’s on the bottom.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Find the slope of the line through (-3,-2) and (3,5).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
C2

Find the slope of the line through (-1,-1) and (-3,5).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
C3

Find the slope of the line through (-1,1) and (5,-1).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
C4

Find the slope of the line through (-3,-4) and (2,3).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
Apply

Apply (4)

A1

Find the slope of the line through (1,4) and (1,6).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
A2

Find the slope of the line through (2,1) and (-1,-5).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
A3

Find the slope of the line through (2,6) and (-4,-6).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
A4

Find the slope of the line through (2,1) and (6,2).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
Extension

Extension (4)

E1

Find the slope of the line through (4,2) and (-1,5).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
E2

Find the slope of the line through (3,3) and (5,5).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
E3

Find the slope of the line through (3,-6) and (5,6).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.
E4

Find the slope of the line through (5,5) and (2,-1).

Need a hint?
  • Subtract the y’s, then the x’s.
  • Keep the order consistent.