ZIMSEC O Level · Mathematics · Form 2, Form 3, Form 4
Graphs show how one quantity changes with another.
Straight lines
The equation y = mx + c gives a straight line with gradient m and y-intercept c.
- Gradient = (change in y) ÷ (change in x) = (y₂ − y₁) ÷ (x₂ − x₁).
- Parallel lines have equal gradients.
Worked example: Find the equation of the line through (1, 3) and (3, 7).
m = (7 − 3) ÷ (3 − 1) = 2. 3 = 2(1) + c → c = 1. y = 2x + 1
Quadratic graphs
y = ax² + bx + c gives a curve (parabola). Make a table of values, plot smoothly, and read off roots where the curve crosses the x-axis.
Travel graphs
- Distance–time graph: gradient = speed.
- Speed–time graph: gradient = acceleration; area under graph = distance travelled.
Functions
f(x) = 3x − 2 means “multiply by 3, then subtract 2”. f(4) = 10. The inverse f⁻¹(x) = (x + 2)/3 undoes f.
Variation
- Direct: y ∝ x means y = kx.
- Inverse: y ∝ 1/x means y = k/x.
- Joint: y varies as x and inversely as z means y = kx/z.
Worked example: y varies inversely as x². When x = 2, y = 9. Find y when x = 3.
y = k/x² → 9 = k/4 → k = 36. y = 36/9 = 4
Check yourself
- Find the gradient of the line 2y = 6x − 5.
- f(x) = 5 − 2x. Find f(−3) and f⁻¹(x).
- P varies directly as the square of Q. P = 48 when Q = 4. Find P when Q = 5.
Show answers
- y = 3x − 2.5, so gradient = 3
- f(−3) = 11; f⁻¹(x) = (5 − x)/2
- k = 3, P = 75
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