This mini-lesson walks you through the whole of the Edexcel 4MA1 topic Sequences, Functions & Graphs: the nth term of sequences, function notation, straight-line graphs, and the shapes of quadratic, cubic, reciprocal and exponential graphs, right up to tangents and distance-time graphs.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Items marked Higher are on the Higher tier only. Press Start when you're ready.
A sequence is a list of numbers in order; each number is a term. A term-to-term rule tells you how to get from one term to the next one.
Watch out: a term-to-term rule (like "+3") lets you find the next term, but not the 100th term directly. For that you need the nth term rule — coming up next.
A linear (arithmetic) sequence goes up by the same amount each time. Its nth term is always dn + b, where d is the common difference.
Step 1. Common difference d = 7 − 3 = 4, so the rule contains 4n.
Step 2. Compare 4n with the sequence: 4×1 = 4, but the 1st term is 3, so subtract 1.
Step 3. nth term = 4n − 1. Check: n = 4 gives 4×4 − 1 = 15 ✓
Misconception: the nth term rule is not the term-to-term rule. Here "+4" is term-to-term; "4n − 1" is the nth term. Only the nth term lets you jump straight to, say, the 50th term (4×50 − 1 = 199).
If the first differences change but the second differences are constant, the sequence is quadratic — its nth term contains an².
Step 1. First differences: 5, 7, 9, 11. Second differences: 2, 2, 2 → constant, so quadratic.
Step 2. 2a = 2, so a = 1 → the rule starts with n².
Step 3. Subtract n²: 3−1=2, 8−4=4, 15−9=6, 24−16=8 → 2, 4, 6, 8 has nth term 2n.
Step 4. nth term = n² + 2n. Check: n = 5 gives 25 + 10 = 35 ✓
Higher only. The constant second difference is the flag for a quadratic sequence. Halve it to get the coefficient of n².
Tap a sequence on the left, then its correct nth term on the right.
A function is a rule that turns an input into one output. We write it f(x). To evaluate f at a number, substitute that number for x everywhere.
Misconception: f(x) does not mean f × x. The brackets show the input to the function, not multiplication. f(3) is the output when x = 3.
A composite function fg(x) means "do g first, then feed the result into f". An inverse function f⁻¹(x) undoes f — it reverses the rule.
Composite fg(2): do g first: g(2) = 2² = 4. Then f(4) = 3×4 + 1 = 13.
Inverse f⁻¹(x): write y = 3x + 1, swap to x = 3y + 1, rearrange: y = (x − 1)/3.
So f⁻¹(x) = (x − 1)/3. Check: f(2) = 7 and f⁻¹(7) = 6/3 = 2 ✓
Higher only. Order matters: fg(x) is usually not the same as gf(x). Always apply the function nearest x first.
Points are written as coordinates (x, y). Between two points A(x₁, y₁) and B(x₂, y₂):
Midpoint: ( (1+5)/2 , (2+10)/2 ) = (3, 6).
Gradient: (10 − 2) ÷ (5 − 1) = 8 ÷ 4 = 2.
Every straight line has the form y = mx + c: m is the gradient (steepness) and c is the y-intercept (where it crosses the y-axis).
To find the equation from a graph: read c off the y-axis, then use a gradient triangle for m.
Misconception: the perpendicular gradient is the negative reciprocal, not just the negative. If m = 2, perpendicular is −½ (not −2). Flip and change the sign.
You must recognise the shape of each type of graph from its equation:
A linear graph (y = mx + c) is a straight line. These are the standard curves in the 4MA1 spec.
A quadratic y = x² − 2x − 3 makes a parabola. Its roots (where y = 0) are where the curve crosses the x-axis — reading these off solves the equation.
To solve a different equation like x² − 2x − 3 = 2, draw the line y = 2 and read the x-values where it meets the curve.
Tap an equation, then tap the box for its graph shape.
On a curve, the gradient keeps changing. To estimate it at a point, draw a tangent (a straight line just touching the curve) and find its gradient. The area under a graph can be estimated by counting squares or splitting it into triangles and trapeziums.
Higher only. On a speed-time graph, the gradient is the acceleration and the area under the graph is the distance travelled.
On a distance-time graph, the gradient is the speed. A steeper line means faster; a horizontal line means the object has stopped.
Misconception: on a distance-time graph, a flat line means stationary (not "moving at steady speed"). A straight sloped line is steady speed; a horizontal line is stopped.
Linear nth term: dn + b (d = common difference)
Quadratic nth term: 2a = second difference (Higher)
Function: f(x) is the output for input x — not f × x
Composite / inverse: fg = g first then f; f⁻¹ undoes f (Higher)
Straight line: y = mx + c · midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Parallel: equal m · Perpendicular: m₁m₂ = −1 (Higher)
Graphs: use crossings for solutions; tangent = gradient, area under = distance (Higher)
You've covered the whole Edexcel 4MA1 topic — sequences, functions and graphs. Press Finish to see your score.
You've worked through Sequences, Functions & Graphs for Edexcel International GCSE Maths A. 🎉
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Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.