IB AA · 4A Equations of Straight Lines
Find the gradient of the line through two points.
Gradient
m = y₂ − y₁x₂ − x₁
Rise over run. Subtract in the same order on the top and the bottom.
Sign
A line that rises as x increases has a positive gradient. A line that falls has a negative gradient.
Horizontal
Equal y-coordinates mean the rise is 0, so the gradient is 0.
Vertical
Equal x-coordinates mean the run is 0. The gradient is undefined. The equation is x = a constant.
Whole-number gradients.
m = y₂ − y₁x₂ − x₁
m = 9 − 14 − 0
m = 84
m = 2
Answer: 2
m = y₂ − y₁x₂ − x₁
m = 10 − 25 − 1
m = 84
m = 2
Answer: 2
m = y₂ − y₁x₂ − x₁
m = 12 − 04 − 0
m = 124
m = 3
Answer: 3
m = y₂ − y₁x₂ − x₁
m = 5 − 56 − 2
m = 04
m = 0
Answer: 0
m = y₂ − y₁x₂ − x₁
m = 11 − 33 − (−1)
m = 84
m = 2
Answer: 2
m = y₂ − y₁x₂ − x₁
m = 10 − 17 − 4
m = 93
m = 3
Answer: 3
Negative gradients and a zero gradient.
m = y₂ − y₁x₂ − x₁
m = −2 − 65 − 1
m = −84
m = −2
Answer: −2
m = y₂ − y₁x₂ − x₁
m = 2 − 85 − 1
m = −64
m = −32
Answer: −32
m = y₂ − y₁x₂ − x₁
m = 5 − 13 − (−3)
m = 46
m = 23
Answer: 23
m = y₂ − y₁x₂ − x₁
m = 5 − (−4)8 − 2
m = 96
m = 32
Answer: 32
m = y₂ − y₁x₂ − x₁
m = −2 − 74 − (−2)
m = −96
m = −32
Answer: −32
m = y₂ − y₁x₂ − x₁
m = −5 − 46 − 0
m = −96
m = −32
Answer: −32
Fractions, and one vertical line.
m = y₂ − y₁x₂ − x₁
m = 6 − 34 − (−2)
m = 36
m = 12
Answer: 12
m = y₂ − y₁x₂ − x₁
m = 3 − 24 − 1
m = 13
Answer: 13
m = y₂ − y₁x₂ − x₁
m = 4 − 1−1 − (−5)
m = 34
Answer: 34
m = y₂ − y₁x₂ − x₁
m = 9 − 32 − 2
The run is 0, so the gradient is undefined.
The line is vertical: x = 2
Answer: undefined, x = 2
m = y₂ − y₁x₂ − x₁
m = 1 − (−2)2 − (−4)
m = 36
m = 12
Answer: 12
m = y₂ − y₁x₂ − x₁
m = −1 − 5−1 − 3
m = −6−4
m = 32
Answer: 32
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