Formula sheet
Gradient (intercepts)
This shortcut is given in the exam. You must know it applies only when you already know both intercepts (where the line cuts each axis), and remember the minus sign in front, it is minus the y-intercept divided by the x-intercept, so a line cutting both axes on the positive side has a negative gradient.
What the symbols mean
- m the gradient of the straight line; a pure number with no units
- y-intercept the y-value where the line crosses the y-axis (the point (0, y-intercept))
- x-intercept the x-value where the line crosses the x-axis (the point (x-intercept, 0))
Given in the exam, or memorise?
This shortcut is given in the exam. You must know it applies only when you already know both intercepts (where the line cuts each axis), and remember the minus sign in front, it is minus the y-intercept divided by the x-intercept, so a line cutting both axes on the positive side has a negative gradient.
Why it works
This is the two-point gradient formula applied to the two points where the line meets the axes.
- A line crosses the x-axis at (x-intercept, 0) and the y-axis at (0, y-intercept).
- Putting these into m = (y₂ − y₁)/(x₂ − x₁) gives (y-intercept − 0)/(0 − x-intercept).
- That simplifies to y-intercept over (−x-intercept), i.e. −(y-intercept)/(x-intercept).
Worked example 1
A straight line cuts the x-axis at 4 and the y-axis at 8. Find its gradient.
- x-intercept = 4 and y-intercept = 8.
- m = −(y-intercept)/(x-intercept) = −(8)/(4)
- = −2
Worked example 2
A line has an x-intercept of −3 and a y-intercept of 6. Find its gradient.
- x-intercept = −3 and y-intercept = 6.
- m = −(y-intercept)/(x-intercept) = −(6)/(−3)
- = −(−2)
- = 2
Where students go wrong
- Dropping the minus sign, the formula is minus the y-intercept over the x-intercept, not just y-intercept over x-intercept.
- Putting the x-intercept on top and the y-intercept on the bottom (the formula upside down).
- Sign slips when an intercept is negative, e.g. −(6)/(−3) = +2, not −2.
- Using this formula when only one intercept is known, it needs both intercepts.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
When should I use this instead of the two-point gradient formula?
Use it when the question gives you the two intercepts (where the line crosses each axis) directly, so you can skip writing out coordinates. If you are given two general points instead, use m = (y₂ − y₁)/(x₂ − x₁).
Both give the same gradient for the same line.
Why is there a minus sign in the formula?
Because the two intercept points are (x-intercept, 0) and (0, y-intercept). Substituting them into rise over run puts the y-intercept over minus the x-intercept, which is where the minus comes from.
If you ever forget it, deriving from the two-point formula recovers the sign quickly.
What if the line passes through the origin?
Then both intercepts are 0, so the formula gives 0/0, which is undefined, it cannot be used. A line through the origin has the form y = mx, so find its gradient from any other point it passes through using the two-point formula instead.