Topic 5 of 11 · Form 3
Sequences, functions and graphs
nth term of a linear sequence: dn + (first term − d). Quadratic sequences have a constant second difference.
Notes for this topic 1 file, in order
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Sequences
nth term of a linear sequence: dn + (first term − d). Quadratic sequences have a constant second difference.
Worked example: Find the nth term of 5, 8, 11, 14, …
d = 3 → 3n + 2
Functions
Function notation f(x); composite functions fg(x) = f(g(x)); inverse functions f⁻¹(x).
Worked example: f(x) = 3x − 4. Find f⁻¹(x).
y = 3x − 4 → x = (y + 4)/3 → f⁻¹(x) = (x + 4)/3
Graphs
- Straight lines y = mx + c: gradient m, intercept c; parallel lines have equal gradients; perpendicular gradients multiply to −1.
- Quadratic (parabolas), cubic, reciprocal (y = k/x) and exponential graphs; solving equations graphically.
- Gradient of a curve by drawing a tangent.
- Distance–time (gradient = speed) and speed–time graphs (gradient = acceleration; area = distance).
Worked example: Find the equation of the line through (1, 3) and (4, 12).
m = 9/3 = 3; 3 = 3(1) + c → c = 0 → y = 3x
Exam tip: Plot points accurately and join quadratic curves smoothly – never with ruled segments.
Check yourself
- Find the 20th term of 7, 11, 15, …
- g(x) = x² + 1 and f(x) = 2x. Find fg(3).
- What is the gradient of a line perpendicular to y = 2x + 5?
Show answers
- 4n + 3 → 83
- g(3) = 10; f(10) = 20
- −½
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