IGCSE · Practice questions · Sketching curves: roots, intercepts and turning points

Shape first, points second.

Six original Cambridge-style questions on sketching curves: intercepts, roots, symmetry, turning points and asymptotes.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Shape, intercepts, turning points, asymptotes.

Shape: decide from the highest power and its sign (U or ∩; rising or falling cubic; two-branch reciprocal; exponential).
\(y\)-intercept: put \(x = 0\).
Roots: put \(y = 0\) and solve, usually by factorising.
Quadratic turning point (Extended): complete the square, \(y = (x + p)^2 + q\) turns at \((-p, q)\).
Asymptotes (Extended): dashed lines, labelled; the curve gets close but never touches.
Label every key point with its coordinates.

Recall

[2 marks]

Sketch the line \(3x + 4y = 24\), labelling where it crosses each axis.

\(x = 0\): \(4y = 24\), \(y = 6\), so \((0, 6)\). ✓
\(y = 0\): \(3x = 24\), \(x = 8\), so \((8, 0)\). Join with a straight line. ✓

Recall

[3 marks]

Sketch \(y = x^2 - 2x - 8\). Label the roots, the \(y\)-intercept and the line of symmetry.

\(x^2 - 2x - 8 = (x - 4)(x + 2)\), so roots \(x = 4\) and \(x = -2\). ✓
\(y\)-intercept: \(-8\). ✓
Line of symmetry halfway between the roots: \(x = 1\). U-shaped curve. ✓

Calculation

[3 marks]

(Extended) (a) Write \(x^2 - 10x + 18\) in the form \((x + p)^2 + q\).

(b) Hence write down the turning point of \(y = x^2 - 10x + 18\), and state whether it is a maximum or a minimum.

(a) \(x^2 - 10x + 18 = (x - 5)^2 - 25 + 18 = (x - 5)^2 - 7\). ✓
(b) Turning point \((5, -7)\). ✓
The \(x^2\) term is positive (U shape), so it is a minimum. ✓

Analysis

[3 marks]

(Extended) Sketch \(y = 4x - x^3\), showing where the curve crosses the axes.

\(4x - x^3 = x(4 - x^2) = x(2 - x)(2 + x)\). ✓
Roots: \(x = -2, 0, 2\). The \(y\)-intercept is 0. ✓
The \(x^3\) term is negative, so the curve comes down from top-left, turns, passes through the origin, turns again and falls to bottom-right. ✓

Analysis

[3 marks]

(Extended) Sketch \(y = \dfrac{6}{x} - 3\). Give the equations of the asymptotes and the coordinates of any point where the curve crosses an axis.

Asymptotes: \(x = 0\) and \(y = -3\). ✓
\(x\)-intercept: \(\dfrac{6}{x} = 3\), so \(x = 2\): the point \((2, 0)\). ✓
There is no \(y\)-intercept because \(x = 0\) is not allowed. Branches lie top-right and bottom-left of the point where the asymptotes cross. ✓

Analysis

[4 marks]

(Extended) A cup of coffee cools so that its temperature, \(T\) °C, after \(t\) minutes is \(T = 70 \times 0.8^{t} + 20\).

(a) Find the temperature when \(t = 0\) and when \(t = 3\).

(b) State the equation of the asymptote of the graph and explain what it means for the coffee.

(a) \(t = 0\): \(70 \times 1 + 20 = 90\) °C. ✓
\(t = 3\): \(70 \times 0.512 + 20 = 55.84 \approx 55.8\) °C. ✓
(b) As \(t\) increases, \(0.8^t \to 0\), so \(T \to 20\): asymptote \(T = 20\). The coffee cools towards room temperature, 20 °C, but never drops below it. ✓
(c) A decreasing curve starting at \((0, 90)\), falling steeply at first, then levelling off above the dashed line \(T = 20\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.