Sequences, functions and graphs is where algebra becomes visual. You learn to connect equations to pictures, and that connection unlocks some of the most satisfying marks in the exam.
This section of the Pearson Edexcel IGCSE Mathematics Specification A syllabus covers four topics: Sequences, Function notation (Higher only), Graphs, and Calculus (Higher only). Between them, they span territory from straightforward pattern-spotting at Foundation tier all the way to differentiation at Higher tier. Graphs in particular carry substantial weight across both papers of the 4MA1 exam and appear in nearly every session.
These edexcel igcse mathematics specification a revision notes work through each topic systematically. Whether you need sequences, functions and graphs edexcel igcse concepts explained from scratch or just need to sharpen a few weak spots, the worked examples and self-check questions below are built for that.
Sequences
A sequence is an ordered list of numbers that follows a rule. The igcse specification requires you to generate terms from term-to-term and position-to-term rules, and to find the nth term of linear (arithmetic) sequences.
Finding the nth term of a linear sequence
A linear sequence has a constant difference between consecutive terms. The nth term formula is: nth term = dn + (a - d), where d is the common difference and a is the first term. Alternatively, use nth term = a + (n - 1)d.
Common difference d = 8 - 5 = 3
First term a = 5
nth term = a + (n - 1)d = 5 + (n - 1)(3) = 5 + 3n - 3 = 3n + 2
Check: n = 1 gives 3(1) + 2 = 5. n = 2 gives 3(2) + 2 = 8. Correct.
Sum of an arithmetic series (Higher)
At Higher tier, you need to find the sum of the first n terms: Sn = n/2 x (2a + (n - 1)d), or equivalently Sn = n/2 x (first term + last term).
a = 4, d = 3, n = 20
S20 = 20/2 x (2(4) + (20 - 1)(3))
S20 = 10 x (8 + 57)
S20 = 10 x 65 = 650
Function notation (Higher)
A function maps an input to exactly one output. The edexcel specification uses two notations: f(x) = 2x + 3 and f : x maps to 2x + 3. Both mean the same thing. You should also understand domain (the set of allowed inputs) and range (the set of possible outputs).
Composite functions
fg(x) means "apply g first, then apply f to the result."
Step 1: g(4) = 42 - 3 = 16 - 3 = 13
Step 2: f(13) = 2(13) + 1 = 27
fg(4) = 27
Inverse functions
The inverse function f-1(x) reverses what f does. To find it, write y = f(x), then rearrange to make x the subject, then swap x and y.
y = (3x - 5)/2
2y = 3x - 5
2y + 5 = 3x
x = (2y + 5)/3
f-1(x) = (2x + 5)/3
Graphs
Graphs is the largest sub-topic in this section and one of the most commonly examined in the entire edexcel igcse specification. The range of skills runs from plotting points in four quadrants to graph transformations.
Straight line graphs: y = mx + c
Every straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. To find the gradient between two points (x1, y1) and (x2, y2), use m = (y2 - y1) / (x2 - x1).
m = (12 - 3) / (4 - 1) = 9 / 3 = 3
y = 3x + c
Substitute (1, 3): 3 = 3(1) + c, so c = 0
Equation: y = 3x
Parallel and perpendicular lines (Higher)
Parallel lines have the same gradient. Perpendicular lines have gradients whose product is -1. If one line has gradient m, the perpendicular gradient is -1/m.
Gradient of given line = 2
Perpendicular gradient = -1/2
y = -1/2 x + c
Substitute (6, 1): 1 = -1/2(6) + c = -3 + c, so c = 4
Equation: y = -x/2 + 4
Quadratic graphs
A quadratic graph y = ax2 + bx + c forms a parabola. If a is positive, it opens upward (U-shape). If a is negative, it opens downward (inverted U). The vertex is the turning point. To plot a quadratic, calculate y-values for a range of x-values, plot the points, and draw a smooth curve through them. Never use straight lines to join plotted points on a curve.
Graph transformations (Higher)
| Transformation | Effect on y = f(x) |
|---|---|
| y = f(x) + a | Translates the graph a units up |
| y = f(x + a) | Translates the graph a units to the left |
| y = af(x) | Stretches the graph vertically by factor a |
| y = f(ax) | Compresses the graph horizontally by factor 1/a |
A frequent mistake: students confuse the direction of the horizontal translation. y = f(x + 3) moves the graph 3 units to the left, not to the right. The sign is counterintuitive. Think of it as "what value of x gives the same output as f(0)?": x + 3 = 0 gives x = -3, so the graph moves to x = -3.
Calculus (Higher)
The edexcel igcse Higher tier includes introductory calculus: differentiation of integer powers of x. This means finding the gradient of a curve at any point, locating turning points, and applying calculus to kinematics (distance, speed, acceleration).
Differentiation rules
If y = xn, then dy/dx = nxn-1. This rule applies to each term independently.
dy/dx = 3(4)x3 - 2(3)x2 + 5(1) - 0
dy/dx = 12x3 - 6x2 + 5
Finding turning points
At a turning point, dy/dx = 0. Solve that equation to find the x-coordinates of the turning points, then substitute back into the original equation to find the y-coordinates.
dy/dx = 3x2 - 12x + 9
Set dy/dx = 0: 3x2 - 12x + 9 = 0
Divide by 3: x2 - 4x + 3 = 0
Factorise: (x - 1)(x - 3) = 0
x = 1 or x = 3
When x = 1: y = 1 - 6 + 9 + 1 = 5. Turning point at (1, 5).
When x = 3: y = 27 - 54 + 27 + 1 = 1. Turning point at (3, 1).
Since the coefficient of x3 is positive, the graph goes from bottom-left to top-right. The turning point at (1, 5) is a local maximum, and (3, 1) is a local minimum.
Kinematics application
If displacement s is given as a function of time t, then velocity v = ds/dt (differentiate displacement) and acceleration a = dv/dt (differentiate velocity). This exam expects you to differentiate and interpret the result, not to integrate.
Common mistakes
- Sequences: confusing term-to-term and position-to-term. "Add 3 each time" is term-to-term. "3n + 2" is position-to-term (the nth term formula). The exam may ask for either or both.
- Graphs: plotting points with a ruler. Quadratic and cubic curves must be smooth. Joining plotted points with straight line segments loses marks.
- Transformations: wrong direction. y = f(x - 2) shifts right, not left. y = f(x) - 2 shifts down. Mixing up horizontal and vertical shifts is very common.
- Calculus: forgetting constants differentiate to zero. The derivative of -7 is 0, not -7. The derivative of 5x is 5, not 5x.
Self-check questions
- Find the nth term of 7, 11, 15, 19, ... (Answer: 4n + 3)
- If f(x) = 4x - 1 and g(x) = x2 + 2, find gf(3). (Answer: f(3) = 11, g(11) = 123)
- Find the gradient of the line through (-2, 5) and (4, -7). (Answer: (-7 - 5)/(4 - (-2)) = -12/6 = -2)
- Differentiate y = 5x3 - x2 + 4. (Answer: dy/dx = 15x2 - 2x)
- Find the coordinates of the turning point of y = x2 - 8x + 20. (Answer: dy/dx = 2x - 8 = 0, x = 4, y = 16 - 32 + 20 = 4. Turning point: (4, 4))
- The displacement of a particle is s = 3t2 - 12t + 10. Find the velocity when t = 5. (Answer: v = 6t - 12, v(5) = 30 - 12 = 18)
For more igcse 4ma1 sequences, functions and graphs practice with instant feedback, use the Green Bridge CBT platform. The edexcel igcse mathematics specification a explained methods above, especially graph skills and calculus fundamentals, are worth practising until they become automatic. Access additional edexcel igcse mathematics specification a practice questions and edexcel igcse mathematics specification a notes on each sub-topic to build fluency ahead of the exam.
Edexcel IGCSE Mathematics Specification A revision notes on sequences, function notation, graphs and introductory calculus with full worked examples.
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