Loading…
Loading…
Recognise, sketch and analyse every non-linear graph type in IGCSE Edexcel Maths. Worked examples from grade 5 to grade 9 with key features and exam tips.
You must recognise and sketch six graph families: quadratic (U-shape or n-shape parabola), cubic (S-curve with up to two turning points), reciprocal (two curves in opposite quadrants with asymptotes on the axes), exponential (rapid growth passing through (0, 1) with an x-axis asymptote), and trigonometric (sine, cosine and tangent with their characteristic wave and asymptote patterns). For each graph, identify the roots, y-intercept, turning points and any asymptotes.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1 — Topic 3.4
A quadratic function has the general form y = ax² + bx + c. Its graph is a parabola — a smooth, symmetric U-shape (when a > 0) or an n-shape (when a < 0).
When sketching, always label the roots, y-intercept and vertex. Draw a smooth curve through these points, ensuring the parabola is symmetric about the line of symmetry.
Completing the square rewrites y = ax² + bx + c into the form y = a(x + p)² + q. This form immediately tells you the turning point is at (-p, q).
For example, y = x² - 6x + 5 becomes y = (x - 3)² - 9 + 5 = (x - 3)² - 4. The turning point is at (3, -4) and the parabola opens upwards because a = 1 > 0.
A cubic graph has a characteristic S-curve shape. When the leading coefficient is positive, the curve runs from bottom-left to top-right. A cubic can have up to two turning points and up to three roots. The basic shape y = x³ passes through the origin with a point of inflection at (0, 0).
The reciprocal graph consists of two separate curves in opposite quadrants. When k > 0, the curves sit in quadrants 1 and 3. When k < 0, they sit in quadrants 2 and 4. The key features are:
An exponential function has the form y = kx where k > 0 and k is not equal to 1. The graph shows rapid growth (when k > 1) or rapid decay (when 0 < k < 1).
Exponential functions model real-world phenomena such as population growth, compound interest and viral spread. A common exam context is modelling how a virus spreads through a community, where the number of cases doubles at regular intervals.
The three trigonometric graphs you must know for IGCSE are y = sin x, y = cos x and y = tan x. All three repeat (are periodic) and have distinct shapes.
| Feature | y = sin x | y = cos x | y = tan x |
|---|---|---|---|
| Starts at (0, ...) | 0 | 1 | 0 |
| Maximum value | 1 (at 90 degrees) | 1 (at 0 degrees) | No maximum |
| Period | 360 degrees | 360 degrees | 180 degrees |
| Asymptotes | None | None | At 90 degrees, 270 degrees, etc. |
Sine starts at 0, rises to a peak of 1 at 90 degrees, returns to 0 at 180 degrees, dips to -1 at 270 degrees and returns to 0 at 360 degrees. Cosine has the same wave shape but starts at 1 and is shifted 90 degrees to the left. Tangent has vertical asymptotes at 90 degrees and 270 degrees where the function is undefined, with the graph passing through the origin and repeating every 180 degrees.
Sketch y = x² - 4x - 5, labelling the roots, y-intercept and vertex.
Step 1 (y-intercept): Set x = 0: y = 0 - 0 - 5 = -5. The curve passes through (0, -5).
Step 2 (Roots): Solve x² - 4x - 5 = 0. Factorise: (x - 5)(x + 1) = 0, giving x = 5 and x = -1.
Step 3 (Vertex): Line of symmetry x = (-1 + 5)/2 = 2. Substitute: y = (2)² - 4(2) - 5 = 4 - 8 - 5 = -9. Vertex at (2, -9).
Step 4 (Shape): a = 1 > 0, so the parabola opens upwards (U-shape).
Step 5 (Sketch): Draw a smooth U-shape passing through (-1, 0), (0, -5), (2, -9) and (5, 0).
Answer: U-shape parabola with roots at x = -1 and x = 5, y-intercept at (0, -5), vertex at (2, -9).
Sketch the graph of y = 2/x for x not equal to 0. State the equations of both asymptotes.
Step 1 (Identify type): y = 2/x is a reciprocal function with k = 2 > 0.
Step 2 (Asymptotes): The x-axis (y = 0) and the y-axis (x = 0) are both asymptotes.
Step 3 (Quadrants): Since k > 0, the two curves sit in quadrant 1 (where both x and y are positive) and quadrant 3 (where both are negative).
Step 4 (Key points): When x = 1, y = 2. When x = 2, y = 1. When x = -1, y = -2. When x = -2, y = -1.
Step 5 (Sketch): Draw two smooth curves in quadrants 1 and 3, approaching but never touching either axis.
Answer: Two curves in quadrants 1 and 3. Asymptotes: x = 0 and y = 0.
Sketch the graph of y = 3x. State the y-intercept and the equation of the asymptote.
Step 1 (y-intercept): Set x = 0: y = 3⁰ = 1. The curve passes through (0, 1).
Step 2 (Growth pattern): x = 1 gives y = 3, x = 2 gives y = 9, x = 3 gives y = 27. The values increase rapidly.
Step 3 (Negative x): x = -1 gives y = 1/3, x = -2 gives y = 1/9. Values approach 0 but never reach it.
Step 4 (Asymptote): The x-axis (y = 0) is a horizontal asymptote. The curve approaches it from above as x becomes very negative.
Step 5 (Sketch): Draw a smooth curve that hugs the x-axis on the left, passes through (0, 1) and rises steeply to the right.
Answer: Exponential growth curve through (0, 1). Asymptote: y = 0.
Exponential functions model scenarios where a quantity doubles at regular intervals. If a virus infects 3 new people for every person already infected, the total cases follow y = 3x. After 5 rounds of infection that is 3⁵ = 243 cases. Understanding exponential curves helps students interpret real-world data and is a common context in IGCSE exam questions.
Confusing sketch and plot: A "sketch" requires the correct shape with labelled key features. A "plot" requires a table of values and accurately plotted points joined by a smooth curve. Using the wrong approach loses marks.
Drawing asymptotes through the curve: Asymptotes are lines the curve approaches but never crosses or touches. Draw them as dashed lines and ensure your curve gets close to, but does not meet, the asymptote.
Forgetting tangent asymptotes: The tan graph has vertical asymptotes at 90 degrees and 270 degrees (and every 180 degrees thereafter). Students often draw a continuous curve through these points, which is incorrect.
Misidentifying the sign of a: When a < 0 the parabola is an n-shape (opens downward). Students frequently draw a U-shape regardless of the sign. Always check the coefficient of x² first.
Label everything: Always label roots, y-intercept, turning point coordinates and asymptote equations on your sketch. Unlabelled features do not earn marks.
Check the sign of a: Before drawing any quadratic, look at the coefficient of x². Positive means U-shape, negative means n-shape. This single check prevents the most common graphing error.
Use completing the square for the vertex: On Higher-tier papers, finding the turning point by completing the square often earns more method marks than reading from a plotted graph.
Draw smooth curves: Use a single continuous motion rather than short, jagged strokes. Examiners penalise sketches that consist of straight-line segments joined together.
Worksheets & Answers
Free graph sketching practice
Formula Sheet
Visual guide to key formulae
All IGCSE Topics
Browse all six topic areas
IGCSE Tutors in Dubai
Specialist in-home IGCSE support
Maths Tutors in Dubai
Expert maths tutoring at home
Contact GetYourTutors — IGCSE Maths Graphs: Quadratic, Cubic, Exponential & Trigonometric
Phone: (+971) 4-313-2715 | Mobile: 050-947-9432
WhatsApp: 050-947-9432
Email: info@getyourtutors.com
Emirates Towers, Office Tower, Level 41, Sheikh Zayed Road, PO Box 31003, Dubai, UAE
Last updated: March 2026
Everything you need to know about our private tutoring services in Dubai.
Our specialist IGCSE maths tutors build visual fluency with every graph type. In-home sessions across Dubai.