IGCSE Maths graphs of quadratics, cubics and reciprocals: draw, sketch and solve

What each tier expects

The key ideas

Plotting: work out each y value carefully (a calculator's table mode helps on calculator papers), plot points as small crosses and join them with a smooth curve through every point, not with straight segments. A quadratic's lowest or highest point is usually between two plotted points, so do not flatten the bottom.

Shapes to recognise: y = x^2 is a U; y = -x^2 is an n. y = x^3 rises from bottom left to top right; with extra terms it gains a hump and a dip. y = 1/x has two branches in opposite quadrants and never touches either axis; the axes are asymptotes. y = a/x + b moves the horizontal asymptote to y = b. y = 2^x passes through (0, 1), grows quickly to the right and approaches y = 0 to the left.

Solving with a graph: the roots of f(x) = 0 are where y = f(x) crosses the x-axis. To solve a different equation with the same curve, rearrange it into 'your curve = a straight line', draw that line, and read the x-coordinates of the intersections.

Transformations (Edexcel Higher): y = f(x) + a moves the graph up by a; y = f(x + a) moves it left by a; y = af(x) stretches it vertically by factor a; y = f(ax) squeezes it horizontally by factor 1/a. Cambridge 0580 does not list these transformations of y = f(x).

Worked example 1: the quadratic y = x^2 - 2x - 3 for -2 ≤ x ≤ 4

1. Table of values: x = -2, -1, 0, 1, 2, 3, 4 gives y = 5, 0, -3, -4, -3, 0, 5.
2. Check one value: x = -2 gives 4 + 4 - 3 = 5. Correct.
3. Roots (where y = 0): x = -1 and x = 3. These match the factorisation (x + 1)(x - 3).
4. Line of symmetry: halfway between the roots, x = 1. Minimum point: (1, -4).
5. Completing the square confirms it: x^2 - 2x - 3 = (x - 1)^2 - 4, so the turning point is (1, -4).

Worked example 2: choose the line to draw

1. Using the graph of y = x^2 - 2x - 3, what straight line would you draw to solve x^2 - 3x - 1 = 0?
2. Rearrange the new equation so the left side matches the curve: x^2 - 3x - 1 = 0 becomes x^2 - 2x - 3 = x - 2 (add x and subtract 2 from both sides).
3. Check: x^2 - 2x - 3 - (x - 2) = x^2 - 3x - 1. Correct.
4. Draw y = x - 2. The x-coordinates where it crosses the curve are the solutions, about -0.30 and 3.30. (The exact values are (3 ± sqrt(13))/2.)

Worked example 3: sketch y = 3/x + 2

1. Start from the shape of y = 3/x: two branches, top right and bottom left, with asymptotes x = 0 and y = 0.
2. Adding 2 lifts the whole graph by 2, so the asymptotes are x = 0 and y = 2.
3. Find the x-intercept: 3/x + 2 = 0 gives 3/x = -2, so x = -1.5. The graph crosses the x-axis at (-1.5, 0) and never meets the y-axis.
4. Label both asymptotes with their equations and mark (-1.5, 0) on the sketch.

Common mistakes that cost marks

• Calculating -x^2 for negative x wrongly: when x = -3, x^2 = 9, so -x^2 = -9, and (-3)^2 means 9, not -9.
• Joining points with a ruler for a curve, or drawing a flat bottom on a parabola.
• Drawing a reciprocal graph through the origin or joining its two branches.
• Reading intersections to the wrong accuracy. Cambridge expects values within half a small square.
• Choosing the wrong line in 'what line should you draw' questions because the rearrangement was not checked.
• Missing asymptote labels on a sketch, which Cambridge Extended expects.

Exam technique and how a tutor helps

Graph-drawing questions are long and mostly method, so they are some of the most reliable marks on Paper 4 for Cambridge and on the Edexcel papers. Use a sharp pencil, extend the curve across the whole given range, and read values carefully. On sketches, the examiner looks for the right shape in the right quadrants, the intercepts, turning points or asymptotes marked, and labelled axes.

To estimate a gradient on a curve (Cambridge E2.9, Edexcel 3.3D), draw a tangent that touches the curve at one point without crossing it, then use two points far apart on the tangent.

This topic needs a pencil and a grid, which is exactly what a shared whiteboard provides in an online lesson. A tutor can watch the student plot and draw in real time, correct technique such as straight-line joins on the spot, and then focus on the reasoning steps, like choosing which line to draw, that students find hardest.

Self-check: can you do these?

• For y = x^3 + x - 4, find y when x = 2 and when x = -1. (Answers: 6 and -6)
• State the asymptotes of y = 3/x + 2. (Answer: x = 0 and y = 2)
• Find the turning point of y = x^2 + 4x + 1. (Answer: (-2, -3))
• What line would you draw on y = x^2 + 3x - 2 to solve x^2 + x - 3 = 0? (Answer: y = 2x + 1)
• Where does y = 2^x cross the y-axis? (Answer: (0, 1))