Graphs of Functions and Sketching Curves

This guide covers subtopics 2.10 Graphs of functions and 2.11 Sketching curves, from Topic 2, Algebra and graphs, for Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series.

Where this fits in 4024

These two subtopics are grouped because they cover the same territory from two directions: 2.10 is about plotting a graph accurately from a table of values, while 2.11 is about recognising and sketching the general shape of a function type from its equation alone, without plotting every point. Both depend on the equation-solving skills from Indices and Equations.

Syllabus coverage

CAMBRIDGE O LEVEL MATHEMATICS (SYLLABUS D) 4024

• Construct tables of values, and draw, recognise and interpret graphs for functions of the form axⁿ (sums of no more than three such terms) and ab^x + c, where n ∈ {−2, −1, −½, 0, ½, 1, 2, 3}, a and c are rational numbers, and b is a positive integer (2.10)
• Solve associated equations graphically, including finding and interpreting roots by graphical methods (2.10)
• Draw and interpret graphs representing exponential growth and decay problems (2.10)
• Estimate gradients of curves by drawing tangents (2.10)
• Recognise, sketch and interpret graphs of linear, quadratic, cubic, reciprocal and exponential functions, in the general forms ax + by = c, y = ax² + bx + c, y = ax³ + b, y = ax³ + bx² + cx, y = a/x + b, and y = arˣ + b (2.11)
• Knowledge of turning points, roots and symmetry is required (2.11)
• Knowledge of vertical and horizontal asymptotes is required (2.11)
• Finding turning points of quadratics by completing the square is required (2.11)

4024 is not tiered — every candidate covers all of the above.

Plotting graphs of functions

Given a function such as y = x³ + x − 4, the standard method is to construct a table of values across the given range, calculate y for each x, plot the points, and join them with a smooth curve (not a series of straight segments).

Worked example. Complete a table of values for y = 4/x² + ¼ × 2ˣ at x = 1, 2, 3 (values rounded to 2 d.p. where needed), then use it to understand the shape of a mixed function combining a negative-power term and an exponential term.

x	1	2	3
y	4.50	2.00	2.44

The y-values fall from x = 1 to x = 2 and then rise from x = 2 to x = 3, so the curve has a minimum somewhere between x = 1 and x = 3. The table cannot say which side of x = 2 it lies: the lowest tabulated value is at x = 2, but the true minimum could be a little before or after it. Working out more values narrows it down: y(1.5) ≈ 2.48 and y(2.5) ≈ 2.05 are both above y(2) = 2.00, so the minimum lies between x = 1.5 and x = 2.5 (it is in fact near x = 2.2, where y ≈ 1.97). The shape comes from the two terms: the negative-power term dominates while x is small, pulling y down as x increases, and the exponential term takes over and pulls y back up for larger x.

The functions this syllabus covers combine terms of the form axⁿ (for the specific list of powers n given above, including negative and fractional powers) with an exponential term ab^x + c — recognising which type of term dominates for large or small x helps sketch and interpret the resulting curve sensibly.

Solving equations graphically

Once a curve is plotted, equations can be solved graphically by reading off where the curve crosses the x-axis (the roots of f(x) = 0), or by finding the intersection of two graphs (a line and a curve, for instance) to solve a pair of equations simultaneously.

Worked example. To solve x³ + x − 4 = 0 graphically, plot y = x³ + x − 4 and read off the x-value where the curve crosses the x-axis — this x-value is the root of the equation.

Exponential growth and decay

Graphs of the form y = ab^x + c represent exponential growth (b > 1) or exponential decay (0 < b < 1) — contexts such as population growth, compound interest, or radioactive decay. The curve approaches, but never quite reaches, a horizontal line at y = c as x becomes very negative (growth) or very large (decay) — this horizontal line is the curve’s horizontal asymptote.

Estimating gradients with tangents

For a curved graph, the gradient at a specific point is estimated by drawing a tangent — a straight line touching the curve at that one point only, locally — and calculating the gradient of that tangent line in the usual way (change in y over change in x, using two points read from the tangent).

Sketching curves by recognising their type

Subtopic 2.11 asks for the shape of a function to be recognised directly from its equation, without plotting a table of values:

Function type	General form	Key features
Linear	ax + by = c	Straight line
Quadratic	y = ax² + bx + c	Parabola; turning point (found by completing the square); up to 2 roots; one line of symmetry
Cubic	y = ax³ + b or y = ax³ + bx² + cx	S-shaped curve; up to 3 roots
Reciprocal	y = a/x + b	Two branches; vertical asymptote where the denominator is zero, horizontal asymptote y = b
Exponential	y = arˣ + b	Growth or decay curve; horizontal asymptote y = b

A turning point is a local maximum or minimum on a curve. For a quadratic, completing the square (from Indices and Equations) finds it directly: writing y = a(x − p)² + q gives a turning point at (p, q).

Worked example. Find the turning point of y = x² − 6x + 5 by completing the square.

y = x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4
turning point: (3, −4)

Roots are where a curve crosses the x-axis, and symmetry for a quadratic means the curve is a mirror image about the vertical line through its turning point. An asymptote is a line a curve approaches but never touches — reciprocal graphs have both a vertical asymptote (where the function is undefined) and a horizontal one; exponential graphs have a horizontal asymptote only.

Common mistakes

• Joining plotted points with straight-line segments instead of a smooth curve, for a genuinely curved function.
• Drawing an inaccurate tangent, leading to a poorly estimated gradient — a tangent must touch the curve at one point only, locally, without crossing it there.
• Forgetting an asymptote is a line the curve approaches but never reaches — sketches that show the curve touching or crossing an asymptote are incorrect.
• Missing a sign when completing the square to find a quadratic’s turning point — always expand the completed-square form back out mentally to check it matches the original expression.
• Assuming every cubic has 3 roots. Depending on the specific cubic, it may cross the x-axis once, twice (with a repeated root) or three times.

Quick revision checklist

• Constructing a table of values and plotting a smooth curve
• Solving equations graphically, by roots or by intersection of two graphs
• Recognising exponential growth vs decay, and the horizontal asymptote
• Estimating a gradient at a point using a tangent
• The five function types in 2.11, their general forms, and their key features (turning points, roots, symmetry, asymptotes)
• Finding a quadratic’s turning point by completing the square

Related resources

• Indices and Equations — the equation-solving and completing-the-square skills used throughout this page
• Graphs in Practical Situations — the previous subtopic
• Functions — the next subtopic
• Cambridge O Level Mathematics subject hub

Written against Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series. Always check the current syllabus for your examination year.