The big idea: Gradient = how steep a line is.
Go right 1 step — gradient tells you how far to move up or down: → m = 3: right 1, up 3 → m = −2: right 1, down 2 → m = 0: right 1, stay flat
Uphill = positive gradient.
Downhill = negative gradient.
Flat road = zero gradient.
- gradient (m)
- How much y changes when x increases by 1. Positive = upward slope, negative = downward slope.
- rise
- The vertical change between two points on the line (change in y).
- run
- The horizontal change between two points on the line (change in x).
| Value of m | What the line does | Example |
|---|---|---|
| m > 0 | Slopes upward left to right | m = 2 — rises steeply |
| m < 0 | Slopes downward left to right | m = −1 — gentle descent |
| m = 0 | Perfectly horizontal | y = 4 — flat line |
| undefined | Vertical line | x = 3 — no gradient defined |
[Diagram: math-gradient-visualizer] - Available in full study mode
IB notation: IB always writes gradient as m in the slope-intercept form y = mx + c.
Never write 'slope' in your final answer — IB expects the word 'gradient'.
The big idea: Given any two points on a line, you can calculate the exact gradient.
Use the formula and keep y on top, x on the bottom — every time.
- the first point — you choose which one to call 'first'
- the second point
- the change in y (rise) — goes on top
- the change in x (run) — goes on bottom
Order rule: It does not matter which point you label first — as long as you use the same order in the top and the bottom.
If you put point 2 on top, put point 2 on the bottom too.
Worked example 1 — positive gradient
Find the gradient of the line through (1, 3) and (4, 9).
Step by step
- Label the points.
- Write the formula.
- Substitute the coordinates.
- Simplify.
Final answer
The gradient is 2.
Worked example 2 — negative gradient
Find the gradient of the line through (2, 7) and (6, 3).
Step by step
- Label the points.
- Write the formula.
- Substitute the coordinates.
- Simplify.
Final answer
The gradient is −1. The line slopes downward.
The most common mistake: Do not put Δx on top and Δy on bottom — that gives you the reciprocal of the gradient.
The y change always goes on top.
Show the formula first: Write the gradient formula before you substitute numbers.
IB awards marks for the correct formula setup — even if your arithmetic has an error.
[Diagram: math-gradient-visualizer] - Available in full study mode
Practice with real exam questions
Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
The big idea: The y-intercept is where the line crosses the y-axis.
At the y-axis, x = 0 always.
So the y-intercept is the value of y when you substitute x = 0.
In the equation y = mx + c, the letter c is the y-intercept.
- the gradient — how steep the line is
- the y-intercept — where the line crosses the y-axis
This is called the slope-intercept form.
It is the most useful form for IB exam questions because m and c can be read directly without any calculation.
Reading m and c from an equation
State the gradient and y-intercept of y = −2x + 7.
Step by step
- Match the equation to the form y = mx + c.
Final answer
Gradient m = −2. The y-intercept is 7 — the line crosses the y-axis at (0, 7).
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
Watch the sign of c: If the equation is y = 4x − 3, then c = −3, not +3.
The minus sign belongs to c.
Writing c = 3 here would be wrong — IB awards marks for the correct sign.
| Equation | Gradient m | y-intercept c |
|---|---|---|
| y = 3x + 5 | 3 | 5 |
| y = −x + 2 | −1 | 2 |
| y = 0.5x − 4 | 0.5 | −4 |
| y = 6 | 0 | 6 |
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
What to do if the equation is not in y = mx + c form: Rearrange first.
For example, 2y = 6x + 4 → divide everything by 2 → y = 3x + 2.
Now m = 3 and c = 2.
The big idea: Once you can read m and c from any line, you can compare lines, decide which is steeper, and sketch them quickly.
This section brings gradient and y-intercept together.
Comparing two lines
Line A: y = 3x + 1.
Line B: y = x + 5.
Which is steeper?
Which has the higher y-intercept?
STEPS
- Read the gradients from each equation.
- Compare the gradients. Larger gradient = steeper slope. Since 3 > 1, Line A is steeper.
- Read the y-intercepts from each equation.
- Compare the y-intercepts. Since 5 > 1, Line B crosses the y-axis higher.
Final answer
Line A is steeper (m = 3 vs m = 1). Line B has the higher y-intercept (c = 5 vs c = 1).
Steepness is determined by the absolute value of the gradient.
A line with m = −4 is steeper than a line with m = 2, even though −4 is numerically smaller.
Quick sketch from m and c
Sketch the line y = 2x − 3.
STEPS
- Plot the y-intercept. Find where the line crosses the y-axis by reading c from the equation.
- Use the gradient to find a second point. The gradient m = 2 tells us the direction: for every 1 unit right, move 2 units up.
- Draw a straight line through both points. Extend the line in both directions across the graph.
Final answer
A line passing through (0, −3) and (1, −1), sloping steeply upward to the right with gradient 2.
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
Two things IB asks every time: IB almost always awards two separate marks — one for gradient, one for y-intercept.
Write each value on its own labelled line:
(a) Gradient = −3 (b) y-intercept = 7
Do not fold both into one sentence without clearly labelling each.
Method summary
- Find gradient from two points: use m = (y₂−y₁)/(x₂−x₁) — Δy on top, Δx on bottom
- Read gradient from y = mx + c: m is the coefficient of x — e.g. y = −3x + 5 → gradient = −3
- Find y-intercept: substitute x = 0, or read c directly — e.g. y = 4x + 7 → y-intercept = 7
- Sketch a line: plot c on the y-axis first, then use m to step to a second point, then draw through both