Functions and Graphs is worth ~35 marks in Paper 1, and almost every sub-question follows one of five patterns. Drill the patterns, not random examples.
How functions are examined in Paper 1 (23 October 2026)
NSC Mathematics Paper 1 is written on Friday 23 October 2026 at 09:00: 3 hours for 150 marks. Functions and graphs carry about 35 of those marks, more than any other Paper 1 topic except differential calculus, so this is where steady practice pays most.
- Sketching parabolas, hyperbolas and exponential graphs with their intercepts, asymptotes and turning points.
- Finding a function's equation from a given graph.
- Inverses of linear, quadratic and exponential functions, including the logarithmic inverse.
- Transformations, and reading the domain and range from a graph.
Practice 1 — Parabola turning point
Given f(x) = −2x² + 8x − 3. Find the turning point. Use x = −b/2a = 2, then f(2) = 5. Turning point: (2, 5).
Practice 2 — Hyperbola asymptotes
Given g(x) = 2/(x − 3) + 1. Vertical asymptote: x = 3. Horizontal asymptote: y = 1. Domain: x ≠ 3. Range: y ≠ 1.
Practice 3 — Exponential function
Given h(x) = 3·2^x − 6. Find the x-intercept: 3·2^x = 6 → 2^x = 2 → x = 1. Horizontal asymptote: y = −6.
Practice 4 — Inverse function
Find the inverse of f(x) = 2x + 3. Swap and solve: x = 2y + 3 → y = (x − 3)/2. So f⁻¹(x) = (x − 3)/2.
Practice 5 — Transformations
If f(x) = x², describe f(x + 2) − 3. The graph shifts 2 units left and 3 units down. New turning point: (−2, −3).
Practice 6 — Average gradient
For f(x) = x² − 4x − 5, find the average gradient between x = 1 and x = 4. f(1) = −8 and f(4) = −5, so the average gradient is (−5 − (−8)) / (4 − 1) = 1.
Practice 7 — A hyperbola's equation
A hyperbola has asymptotes x = 2 and y = −1 and passes through (3; 1). Then y = a/(x − 2) − 1, and substituting the point gives 1 = a − 1, so a = 2 and y = 2/(x − 2) − 1.
Practice 8 — A logarithmic inverse
For g(x) = (½)ˣ, swap x and y: x = (½)ʸ, so g⁻¹(x) = log_½ x, defined for x > 0. Both graphs are decreasing, and they are reflections of each other in the line y = x.
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