Home / IGCSE Mathematics 0580 / Sketching curves

Cambridge IGCSE Mathematics 0580 · Algebra and graphs

Sketching curves

Recognising a graph from its equation and sketching it with its key features: intercepts, roots and symmetry for straight lines and quadratics. Extended students also find turning points by completing the square, and sketch cubic, reciprocal and exponential curves with their asymptotes.

C2.11E2.11Core and ExtendedFree lesson

What the syllabus says

C2.11 (Core)

Recognise, sketch and interpret graphs of the following functions:
(a) linear
(b) quadratic.

Notes and examples

Knowledge of symmetry and roots is required.
Knowledge of turning points is not required.

E2.11 (Extended)

Recognise, sketch and interpret graphs of the following functions:
(a) linear
(b) quadratic
(c) cubic
(d) reciprocal
(e) exponential.

Notes and examples

Functions will be equivalent to:
$a x + b y = c$
$y = a x^{2} + b x + c$
$y = a x^{3} + b$
$y = a x^{3} + b x^{2} + c x$
$y = \frac{a}{x} + b$
$y = a r^{x} + b$
where $a$ , $b$ and $c$ are rational numbers and $r$ is a rational, positive number.
Knowledge of turning points, roots and symmetry is required.
Knowledge of vertical and horizontal asymptotes is required.
Finding turning points of quadratics by completing the square is required.

Quoted from the Cambridge IGCSE Mathematics 0580 syllabus for 2025 to 2027.

A sketch is not a plotted graph

When a question says sketch, you do not need a table of values or graph paper. A sketch shows the right shape in the right place, with the important points labelled:

where the graph crosses the $y$ -axis (put $x = 0$ );
where it crosses the $x$ -axis, the roots (put $y = 0$ );
any line of symmetry.

Extended sketches also show turning points and asymptotes. Write the coordinates or values next to each point. A smooth curve of the correct shape with the key values labelled earns the marks; a sketch with no values usually earns only one.

Straight lines

A linear graph is a straight line. In the form $y = m x + c$ :

$m$ is the gradient: positive slopes up from left to right, negative slopes down, and the size says how steep.
$c$ is where it crosses the $y$ -axis, the point $(0, c)$ .

To sketch a line, find the two intercepts. Put $x = 0$ to get the $y$ -intercept and $y = 0$ to get the $x$ -intercept, then join them and extend the line.

A line is often given as $a x + b y = c$ , for example $2 x + 3 y = 12$ . The intercepts are quick: $x = 0$ gives $3 y = 12$ , so $y = 4$ ; $y = 0$ gives $2 x = 12$ , so $x = 6$ . To find the gradient, rearrange to $y = \dots$ : $3 y = 12 - 2 x$ , so $y = 4 - \frac{2}{3} x$ and the gradient is $- \frac{2}{3}$ .

Worked example 1

Sketch the line $2 x + 3 y = 12$ , showing where it crosses the axes.

$x = 0$ : $3 y = 12$ , so $y = 4$ . The line crosses the $y$ -axis at $(0, 4)$ .
$y = 0$ : $2 x = 12$ , so $x = 6$ . It crosses the $x$ -axis at $(6, 0)$ .
Join the two points with a straight line, extended beyond them, and label both.

Answer: A straight line through $(0, 4)$ and $(6, 0)$

Quadratic curves

A quadratic graph $y = a x^{2} + b x + c$ is a smooth curve called a parabola.

If the $x^{2}$ term is positive, the curve is U-shaped: it opens upwards.
If the $x^{2}$ term is negative, it is an upside-down U: it opens downwards.
It crosses the $y$ -axis at $(0, c)$ , because putting $x = 0$ leaves only $c$ .
It has a vertical line of symmetry.

The roots are where $y = 0$ . A quadratic can cross the $x$ -axis twice, touch it once, or miss it. For curves like $y = x^{2} - 9$ , the roots come from $x^{2} = 9$ , so $x = 3$ or $x = - 3$ .

Symmetry and roots go together. The line of symmetry is exactly halfway between the two roots. If the roots are $x = - 1$ and $x = 7$ , the line of symmetry is $x = \frac{- 1 + 7}{2} = 3$ . Points at the same height are mirror images in that line: if $(0, - 7)$ is on the curve, so is $(6, - 7)$ , because 0 is 3 to the left of the line and 6 is 3 to the right.

Core candidates do not need to find the turning point (the lowest or highest point). It is enough to draw the curve rounded at the bottom or top, on the line of symmetry.

Worked example 2

Sketch $y = x^{2} - 9$ and $y = 4 - x^{2}$ on the same axes, showing where each curve meets the axes.

$y = x^{2} - 9$ : the $x^{2}$ is positive, so a U shape. $x = 0$ gives $y = - 9$ . $y = 0$ gives $x^{2} = 9$ , so $x = \pm 3$ .
$y = 4 - x^{2}$ : the $x^{2}$ is negative, so an upside-down U. $x = 0$ gives $y = 4$ . $y = 0$ gives $x^{2} = 4$ , so $x = \pm 2$ .
Both are symmetrical about the $y$ -axis, the line $x = 0$ , because their roots are equal distances either side of 0.
Label $(0, - 9)$ , $(\pm 3, 0)$ , $(0, 4)$ and $(\pm 2, 0)$ on the sketch.

Answer: See the sketch: a U through $(- 3, 0)$ , $(0, - 9)$ , $(3, 0)$ and an upside-down U through $(- 2, 0)$ , $(0, 4)$ , $(2, 0)$

Worked example 3

A quadratic curve crosses the $x$ -axis at $(- 2, 0)$ and $(6, 0)$ and the $y$ -axis at $(0, - 12)$ . (a) Write down the equation of its line of symmetry. (b) Write down the coordinates of one more point on the curve.

(a) Halfway between the roots: $\frac{- 2 + 6}{2} = 2$ . The line of symmetry is $x = 2$ .
(b) $(0, - 12)$ is 2 units to the left of $x = 2$ . Its mirror image is 2 units to the right, at $x = 4$ , at the same height.
So $(4, - 12)$ is on the curve.

Answer: (a) $x = 2$ (b) $(4, - 12)$

Check yourself

A quadratic graph crosses the $x$ -axis at $x = - 5$ and $x = 1$ . Write down the equation of its line of symmetry.

Answer it in the portal, where your answer is checked as you type →

Check yourself

Write down the coordinates of the point where $y = x^{2} + 3 x - 10$ crosses the $y$ -axis.

Answer it in the portal, where your answer is checked as you type →

Turning points by completing the square Extended

Extended candidates find the turning point of a quadratic exactly, by completing the square. Write $x^{2} + b x + c$ as $(x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c$ .

Why it works: in $y = (x - p)^{2} + q$ , the square $(x - p)^{2}$ is never negative, and it is 0 only when $x = p$ . So the smallest value of $y$ is $q$ , reached at $x = p$ . The minimum point is $(p, q)$ and the line of symmetry is $x = p$ .

When the $x^{2}$ coefficient is $a \neq 1$ , take $a$ out of the $x$ terms first: $y = a (x - p)^{2} + q$ . The turning point is still $(p, q)$ . If $a$ is negative, $a (x - p)^{2}$ is never positive, so $(p, q)$ is a maximum.

For a full Extended sketch of a quadratic, show the $y$ -intercept, the roots (by factorising, or by the formula when it does not factorise) and the turning point.

Worked example 4 Extended

Sketch $y = x^{2} - 6 x + 5$ , showing the turning point and where the curve meets the axes.

Complete the square: half of $- 6$ is $- 3$ , so $x^{2} - 6 x + 5 = (x - 3)^{2} - 9 + 5 = (x - 3)^{2} - 4$ .
Minimum point $(3, - 4)$ ; line of symmetry $x = 3$ .
Roots: $x^{2} - 6 x + 5 = (x - 1) (x - 5) = 0$ , so $x = 1$ or $x = 5$ . Check: halfway between them is 3.
$y$ -intercept: $(0, 5)$ . Draw a U shape through these points.

Answer: A U shape with minimum $(3, - 4)$ , roots 1 and 5, $y$ -intercept 5

Worked example 5 Extended

Find the turning point of $y = - 2 x^{2} - 4 x + 6$ and say whether it is a maximum or a minimum. Find where the curve crosses the axes.

Take $- 2$ out of the $x$ terms: $- 2 (x^{2} + 2 x) + 6$ .
Complete the square inside: $x^{2} + 2 x = (x + 1)^{2} - 1$ .
So $y = - 2 ((x + 1)^{2} - 1) + 6 = - 2 (x + 1)^{2} + 2 + 6 = - 2 (x + 1)^{2} + 8$ .
The coefficient $- 2$ is negative, so the turning point $(- 1, 8)$ is a maximum.
Roots: $- 2 x^{2} - 4 x + 6 = - 2 (x^{2} + 2 x - 3) = - 2 (x + 3) (x - 1)$ , so $x = - 3$ or $x = 1$ . $y$ -intercept: $(0, 6)$ .

Answer: Maximum at $(- 1, 8)$ ; roots $- 3$ and $1$ ; $y$ -intercept $6$

Check yourself Extended

Find the coordinates of the turning point of $y = x^{2} + 8 x + 10$ .

Answer it in the portal, where your answer is checked as you type →

Cubic curves Extended

A cubic has an $x^{3}$ term. Two forms are examined.

$y = a x^{3} + b$ . This is the curve $y = x^{3}$ stretched by $a$ and moved up by $b$ . With $a > 0$ it rises from bottom left to top right, flattening for a moment as it passes through $(0, b)$ . It has rotational symmetry about $(0, b)$ and one root, where $a x^{3} = - b$ , so $x = \sqrt[3]{- \frac{b}{a}}$ . With $a < 0$ it falls from top left to bottom right.

$y = a x^{3} + b x^{2} + c x$ . There is no constant term, so the curve passes through the origin. Take out $x$ and factorise: $y = x (a x^{2} + b x + c)$ . The roots are $x = 0$ and the roots of the quadratic, so there can be up to three.

With $a > 0$ the curve comes up from the bottom left and ends at the top right; with $a < 0$ it starts top left and ends bottom right.
Between neighbouring roots the curve turns: one turning point is a local maximum and the other a local minimum. On a sketch, place them roughly between the roots; their exact positions are not needed.

Worked example 6 Extended

Sketch $y = x^{3} - x^{2} - 6 x$ , showing where it meets the axes.

Take out $x$ : $y = x (x^{2} - x - 6) = x (x - 3) (x + 2)$ .
Roots: $x = - 2$ , $x = 0$ and $x = 3$ . The curve meets the $y$ -axis at the origin.
The $x^{3}$ coefficient is positive, so the curve starts at the bottom left and ends at the top right.
It rises through $- 2$ , turns between $- 2$ and 0, comes down through 0, turns between 0 and 3, and rises through 3.

Answer: A cubic through $(- 2, 0)$ , $(0, 0)$ and $(3, 0)$ , rising from bottom left to top right

Reciprocal curves and asymptotes Extended

An asymptote is a straight line that a curve gets closer and closer to without reaching.

For $y = \frac{a}{x} + b$ :

$x = 0$ is impossible, so the $y$ -axis, $x = 0$ , is a vertical asymptote.
As $x$ becomes very large (positive or negative), $\frac{a}{x}$ gets close to 0, so $y$ gets close to $b$ . The line $y = b$ is a horizontal asymptote.
The curve is $y = \frac{a}{x}$ moved up by $b$ . For $a > 0$ the branches are top right and bottom left of where the asymptotes cross; for $a < 0$ , top left and bottom right.
There is no $y$ -intercept. The $x$ -intercept is where $\frac{a}{x} + b = 0$ , so $x = - \frac{a}{b}$ (when $b \neq 0$ ).

Draw asymptotes as dashed lines, label them with their equations, and make the branches bend towards them without touching or crossing.

Exponential curves Extended

For $y = a r^{x} + b$ with $r$ positive:

At $x = 0$ , $r^{0} = 1$ , so the $y$ -intercept is $a + b$ .
$r^{x}$ is always positive and approaches 0 on one side, so the line $y = b$ is a horizontal asymptote.
If $r > 1$ and $a > 0$ : growth. The curve approaches $y = b$ on the left and rises ever more steeply on the right.
If $0 < r < 1$ and $a > 0$ : decay. The curve falls steeply on the left and levels off towards $y = b$ on the right.
A negative $a$ reflects the curve: it lies below the asymptote instead of above it.

An exponential curve has no vertical asymptote and no line of symmetry. It crosses the $x$ -axis once if the curve and the asymptote are on opposite sides of it, and not at all otherwise.

Worked example 7 Extended

Sketch $y = \frac{6}{x} - 2$ , showing its asymptotes and where it crosses an axis.

Vertical asymptote $x = 0$ ; horizontal asymptote $y = - 2$ . Draw both dashed.
$a = 6$ is positive, so the branches sit top right and bottom left of the point $(0, - 2)$ where the asymptotes cross.
$x$ -intercept: $\frac{6}{x} - 2 = 0$ , so $\frac{6}{x} = 2$ and $x = 3$ . The point is $(3, 0)$ .
There is no $y$ -intercept, because $x = 0$ is impossible.

Answer: Asymptotes $x = 0$ and $y = - 2$ ; crosses the $x$ -axis at $(3, 0)$

Worked example 8 Extended

Sketch $y = 3 \times {0.5}^{x} + 1$ , showing its asymptote and $y$ -intercept.

$y$ -intercept: $3 \times 1 + 1 = 4$ , the point $(0, 4)$ .
Asymptote: ${0.5}^{x}$ approaches 0 as $x$ increases, so $y$ approaches 1. Draw $y = 1$ dashed.
$r = 0.5$ is between 0 and 1 and $a = 3$ is positive: decay. The curve falls from the top left, passes through $(0, 4)$ and levels off just above $y = 1$ .
It never crosses the $x$ -axis, because every $y$ value is greater than 1.

Answer: A decreasing curve through $(0, 4)$ , approaching $y = 1$

Check yourself Extended

Write down the equation of the horizontal asymptote of $y = 5 \times 3^{x} - 2$ .

Answer it in the portal, where your answer is checked as you type →

Key points

A sketch shows the shape and labels the key points: intercepts and roots, plus (Extended) turning points and asymptotes.
Lines: find both intercepts by putting $x = 0$ and $y = 0$ .
Quadratics: positive $x^{2}$ is a U, negative is an upside-down U; the line of symmetry is halfway between the roots.
Extended: $y = a (x - p)^{2} + q$ has turning point $(p, q)$ , a minimum if $a > 0$ and a maximum if $a < 0$ .
Extended: cubics $a x^{3} + b x^{2} + c x$ pass through the origin; factorise to find the other roots.
Extended: $y = \frac{a}{x} + b$ has asymptotes $x = 0$ and $y = b$ ; $y = a r^{x} + b$ has asymptote $y = b$ and $y$ -intercept $a + b$ .

Where marks are lost

Leaving the key values off a sketch. Label every intercept, root and turning point.
Drawing an upside-down U for a positive $x^{2}$ term, or the other way round.
Drawing a quadratic with a pointed bottom like a V. It is rounded.
Extended: getting the sign wrong in the turning point. $(x - 3)^{2} - 4$ has its minimum at $x = + 3$ .
Extended: forgetting the root $x = 0$ of a cubic with no constant term.
Extended: letting a branch of a reciprocal or an exponential curve touch or cross its asymptote.
Extended: giving the asymptote of $y = a r^{x} + b$ as $y = 0$ instead of $y = b$ .

Related IGCSE Mathematics 0580 lessons

E2.12

All IGCSE Mathematics 0580 lessons →

An original lesson written to the Cambridge IGCSE Mathematics 0580 syllabus (2025 to 2027). The worked examples and questions are our own, not Cambridge questions. Back to every IGCSE Mathematics 0580 lesson.

Sketching curves

A sketch is not a plotted graph

Straight lines

Worked example 1

Quadratic curves

Worked example 2

Worked example 3

Check yourself

Check yourself

Turning points by completing the square Extended

Worked example 4 Extended

Worked example 5 Extended

Check yourself Extended

Cubic curves Extended

Worked example 6 Extended

Reciprocal curves and asymptotes Extended

Exponential curves Extended

Worked example 7 Extended

Worked example 8 Extended

Check yourself Extended

Key points

Where marks are lost

Related IGCSE Mathematics 0580 lessons

Differentiation

Graphs of functions

Functions

Graphs in practical situations