Gradient measures steepness: The gradient m is how steep a line is — rise ÷ run.
Picture going right one step: → m = 3 means go up 3 → m = −2 means go down 2 → m = 0 means stay flat To find it exactly, read two points off the line and divide the change in y by the change in x.
- one point on the line
- another point on the line
[Diagram: math-gradient-visualizer] - Available in full study mode
IB-style question — gradient from two points
Find the gradient of the line through A(1, 2) and B(4, 11).
Step by step
- Gradient formula — subtract in the same order, top and bottom.
- Work it out.
Final answer
m = 3 (the line rises 3 for every 1 across).
What the sign tells you
- m > 0 → uphill (left to right).
- m < 0 → downhill.
- m = 0 → horizontal line y = c.
Watch out
- Subtract in the same order top and bottom.
- A vertical line x = a has no gradient (run = 0).
- rise/run, never run/rise.
Same line, three outfits: A straight line can be written three ways — pick whichever fits the question.
Gradient–intercept
- m = gradient
- c = y-intercept
- Best for graphing
Point–gradient
- Use a point + gradient
- Best for building a line
General form
- Tidy integer form
- Gradient = −a/b
| Equation | Gradient m | y-intercept c |
|---|---|---|
| y = 3x + 5 | 3 | 5 |
| y = −x + 2 | −1 | 2 |
| y = ½x − 4 | ½ | −4 (keep the minus!) |
| y = 6 | 0 | 6 (flat line) |
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
IB-style question — switch between forms
Write y − 3 = 2(x − 1) in the form y = mx + c, then in the form ax + by + d = 0.
Step by step
- Expand the bracket.
- Make y the subject → gradient–intercept form.
- Move everything to one side → general form.
Final answer
y = 2x + 1, or equivalently 2x − y + 1 = 0.
IB-style question — gradient from ax + by + d = 0
Find the gradient of the line 4x + 3y − 12 = 0.
Step by step
- Make y the subject — move the x-term and the constant to the other side.
- Divide every term by 3.
- The gradient is the number multiplying x.
Final answer
Gradient m = −4/3. Shortcut: straight from ax + by + d = 0, the gradient is m = −a/b = −4/3 — this exact step shows up in real exams.
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Point + gradient, or two points: Given a gradient and a point: put the gradient into y = mx + c, then substitute the point to find c.
Given two points: find the gradient first, then do the same.
IB-style question — a point and a gradient
Find the equation of the line with gradient 3 that passes through (2, 5).
Give your answer as y = mx + c.
Step by step
- Start with y = mx + c, using m = 3.
- Substitute the point (2, 5): put x = 2 and y = 5.
- Solve for c.
- Write the full equation.
Final answer
y = 3x − 1.
IB-style question — two points
Find the equation of the line through P(1, 2) and Q(3, 8).
Step by step
- Gradient formula first.
- Start y = 3x + c and substitute one point, say (1, 2).
- Solve for c.
- Write the full equation.
Final answer
y = 3x − 1.
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
Faster alternative — point–gradient form: Another standard method (also in the formula booklet) is point–gradient form: y − y₁ = m(x − x₁).
Drop in the point and gradient, then expand — e.g. y − 5 = 3(x − 2) → y = 3x − 1.
Same answer; use whichever you find clearer.
(With two points, either point works.)
Set the other coordinate to zero: y-intercept: set x = 0.
x-intercept: set y = 0 and solve.
(And in y = mx + c, the number c is the y-intercept — read it straight off.)
IB-style question — both intercepts
Find where the line y = 2x − 6 crosses each axis.
Step by step
- y-intercept: put x = 0.
- x-intercept: put y = 0 and solve.
Final answer
Crosses the y-axis at (0, −6) and the x-axis at (3, 0).
Don't swap the coordinates: The y-intercept is the point (0, c) and the x-intercept is (x, 0) — the zero goes in different slots.
A common slip is writing (−6, 0) for the y-intercept.