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Cambridge IGCSE Mathematics 0580 · Algebra and graphs

Graphs of functions

Building a table of values, drawing an accurate graph from it, and using the graph to solve equations: roots, a horizontal line, or the intersection of a line and a curve. Core covers straight lines, quadratics and reciprocals; Extended adds powers of $x$ , sums of them, and exponential graphs for growth and decay.

C2.10E2.10Core and ExtendedFree lesson

What the syllabus says

C2.10 (Core)

1 Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms:
$a x + b$
$\pm x^{2} + a x + b$
$\frac{a}{x}$ $(x \neq 0)$
where $a$ and $b$ are integer constants.
2 Solve associated equations graphically, including finding and interpreting roots by graphical methods.

Notes and examples

e.g. find the intersection of a line and a curve.

E2.10 (Extended)

1 Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms:
$a x^{n}$ (includes sums of no more than three of these)
$a b^{x} + c$
where $n = - 2, - 1, - \frac{1}{2}, 0, \frac{1}{2}, 1, 2, 3$ ; $a$ and $c$ are rational numbers; and $b$ is a positive integer.
2 Solve associated equations graphically, including finding and interpreting roots by graphical methods.
3 Draw and interpret graphs representing exponential growth and decay problems.

Notes and examples

Examples include:
$y = x^{3} + x - 4$
$y = 2 x + \frac{3}{x^{2}}$
$y = \frac{1}{4} \times 2^{x}$ .
e.g. finding the intersection of a line and a curve.

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

Tables of values

A table of values lists $x$ values and the $y$ value the equation gives for each. Substitute each $x$ in turn. Most errors come from negative numbers, so be deliberate:

$x^{2}$ is never negative. When $x = - 3$ , $x^{2} = (- 3)^{2} = 9$ .
$- x^{2}$ is always negative or zero. When $x = - 3$ , $- x^{2} = - (9) = - 9$ : square first, then apply the minus.
$- 2 x$ when $x = - 3$ is $- 2 \times - 3 = + 6$ .

On a calculator, put negative values in brackets: type $(- 3)^{2}$ , not $- 3^{2}$ , which the calculator reads as $- 9$ .

Use the pattern to check yourself. A quadratic table is symmetrical about its middle, so if the values go 5, 0, $- 3$ , $- 4$ , $- 3$ , 0, 5, a value that breaks the symmetry is a mistake. For a straight line $y = a x + b$ the values go up (or down) by the same amount, $a$ , every time $x$ goes up by 1.

Worked example 1

Complete a table of values for $y = x^{2} - 2 x - 3$ for $x = - 2, - 1, 0, 1, 2, 3, 4$ .

Table of values for $y = x^{2} - 2 x - 3$
$x$	-2	-1	0	1	2	3	4
$y$	5	0	-3	-4	-3	0	5

$x = - 2$ : $(- 2)^{2} - 2 (- 2) - 3 = 4 + 4 - 3 = 5$ .
$x = - 1$ : $1 + 2 - 3 = 0$ .
$x = 0$ : $0 - 0 - 3 = - 3$ .
$x = 1$ : $1 - 2 - 3 = - 4$ .
$x = 2$ : $4 - 4 - 3 = - 3$ ; $x = 3$ : $9 - 6 - 3 = 0$ ; $x = 4$ : $16 - 8 - 3 = 5$ .
Check the symmetry: 5, 0, $- 3$ , $- 4$ , $- 3$ , 0, 5 is symmetrical about $x = 1$ .

Answer: $5, 0, - 3, - 4, - 3, 0, 5$

Drawing the graph

Plot every point from the table with a small cross, using the scale on the grid. Then:

A straight line $y = a x + b$ : join the points with a ruler. Three points are enough, and the third is your check.
A curve: draw one smooth curve through every point, freehand, in one sweep. Do not join the points with straight segments, and do not flatten the bottom between two equal values: a quadratic keeps curving between $x = 0$ and $x = 2$ in the table above, down to its lowest value at $x = 1$ .

Recognising the shapes:

$y = a x + b$ is a straight line with gradient $a$ , crossing the $y$ -axis at $b$ .
$y = x^{2} + a x + b$ is a U-shaped curve (a parabola) and $y = - x^{2} + a x + b$ is an upside-down U. Each has a vertical line of symmetry and crosses the $y$ -axis at $b$ .
$y = \frac{a}{x}$ is a reciprocal graph, described in the next section.

A quadratic graph and a line

The curve is $y = x^{2} - 2 x - 3$ from the table above. The dashed line is $y = x + 1$ , used in Worked example 2.

Reciprocal graphs: $y = \frac{a}{x}$

For $y = \frac{a}{x}$ you cannot use $x = 0$ , because you cannot divide by zero. So the graph has two separate branches and never crosses the $y$ -axis. It never crosses the $x$ -axis either, because $\frac{a}{x}$ is never 0.

As $x$ gets larger, $\frac{a}{x}$ gets closer and closer to 0, so the curve gets closer to the $x$ -axis without touching it. As $x$ gets close to 0, $\frac{a}{x}$ becomes very large, so the curve climbs up beside the $y$ -axis.

If $a$ is positive, the branches are in the top-right and bottom-left quarters.
If $a$ is negative, they are in the top-left and bottom-right quarters.

The graph is symmetrical: $\frac{6}{- 2} = - 3$ is the negative of $\frac{6}{2} = 3$ . Draw each branch as a separate smooth curve, and never join the two branches across the gap at $x = 0$ .

A reciprocal graph

The graph of $y = \frac{6}{x}$ for $- 6 ⩽ x ⩽ 6$ , $x \neq 0$ .

Solving equations from a graph

Every point on the graph of $y = f (x)$ is a pair of values that fits the equation. That is why a graph can solve equations:

Roots. The solutions of $f (x) = 0$ are the $x$ -coordinates where the graph crosses the $x$ -axis, because there $y = 0$ .
Equal to a number. To solve $f (x) = k$ , draw the horizontal line $y = k$ and read the $x$ -coordinates where it meets the curve.
A line and a curve. Where the graphs of two equations cross, the point fits both, so the $x$ -coordinate solves the equation made by setting them equal.

To solve a new equation with a graph you already have, rearrange the new equation so that one side is the equation you have drawn. The other side is the line to draw. For example, with the graph of $y = x^{2} - 2 x - 3$ drawn, the equation $x^{2} - 3 x - 4 = 0$ can be written $x^{2} - 2 x - 3 = x + 1$ (add $x + 1$ to both sides), so draw $y = x + 1$ .

Graph readings are approximate: give them to the accuracy the grid allows, usually 1 decimal place, and give every solution in the range drawn. Count the crossings: a line can meet a quadratic twice, once or not at all, and the number of crossings is the number of solutions.

Worked example 2

Use the graph of $y = x^{2} - 2 x - 3$ to solve (a) $x^{2} - 2 x - 3 = 0$ , (b) $x^{2} - 2 x - 3 = 2$ and (c) $x^{2} - 2 x - 3 = x + 1$ .

(a) The curve crosses the $x$ -axis at $x = - 1$ and $x = 3$ .
(b) Draw the horizontal line $y = 2$ . It meets the curve at $x \approx - 1.4$ and $x \approx 3.4$ .
These are approximate: the exact values are $1 \pm \sqrt{6}$ , which are $- 1.449$ and $3.449$ , so readings from $- 1.5$ to $- 1.4$ and from $3.4$ to $3.5$ would score.
(c) Draw the line $y = x + 1$ (it passes through $(0, 1)$ and $(3, 4)$ ). It meets the curve at $(- 1, 0)$ and $(4, 5)$ .
So $x = - 1$ or $x = 4$ . Check $x = 4$ : $16 - 8 - 3 = 5$ and $4 + 1 = 5$ .

Answer: (a) $x = - 1$ or $3$ (b) $x \approx - 1.4$ or $3.4$ (c) $x = - 1$ or $4$

Check yourself

A graph of $y = x^{2} - 4 x$ crosses the $x$ -axis twice. Write down the solutions of $x^{2} - 4 x = 0$ .

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

Check yourself

Find the value of $y = \frac{6}{x}$ when $x = - 4$ .

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

Graphs of $a x^{n}$ and sums of them Extended

Extended students draw graphs of $y = a x^{n}$ for $n = - 2, - 1, - \frac{1}{2}, 0, \frac{1}{2}, 1, 2, 3$ , and of sums of up to three such terms, such as $y = x^{3} - 3 x + 1$ or $y = x^{2} - \frac{3}{x}$ . Use the rules of indices to see what each term means:

$x^{0} = 1$ , so $a x^{0}$ is the constant $a$ : a horizontal line.
$x^{\frac{1}{2}} = \sqrt{x}$ , which only exists for $x ⩾ 0$ .
$x^{- 1} = \frac{1}{x}$ and $x^{- 2} = \frac{1}{x^{2}}$ : neither exists at $x = 0$ . Since $x^{2}$ is positive, $\frac{1}{x^{2}}$ is positive on both sides of the $y$ -axis.
$x^{- \frac{1}{2}} = \frac{1}{\sqrt{x}}$ , which exists only for $x > 0$ .
$x^{3}$ keeps the sign of $x$ : the curve rises from bottom left to top right.

For a sum, work out each term separately in the table and add, or key in the whole expression carefully. A table often has decimals; give them to 1 or 2 decimal places as the question says. Where a term like $\frac{3}{x}$ appears, leave out $x = 0$ and expect the curve to shoot up or down beside the $y$ -axis.

Some powers of x Extended

For positive $x$ : $y = x^{\frac{1}{2}}$ rises ever more slowly, while $y = x^{- 1}$ and $y = x^{- 2}$ fall towards the $x$ -axis. All three pass through $(1, 1)$ , because 1 to any power is 1.

Worked example 3 Extended

Complete a table of values for $y = x^{3} - 3 x + 1$ for $- 2 ⩽ x ⩽ 2$ , in steps of 0.5. Draw the graph and use it to solve $x^{3} - 3 x + 1 = 0$ .

Table of values for $y = x^{3} - 3 x + 1$
$x$	-2	-1.5	-1	-0.5	0	0.5	1	1.5	2
$y$	-1	2.125	3	2.375	1	-0.375	-1	-0.125	3

$x = - 1.5$ : $(- 1.5)^{3} - 3 (- 1.5) + 1 = - 3.375 + 4.5 + 1 = 2.125$ .
$x = 0.5$ : $0.125 - 1.5 + 1 = - 0.375$ . The other values are worked the same way and are shown in the table.
Plot the points and draw a smooth S-shaped curve through them (the graph is shown after Worked example 4).
The curve crosses the $x$ -axis three times: at about $x = - 1.9$ , $x = 0.3$ and $x = 1.5$ .

Answer: $x \approx - 1.9$ , $0.3$ or $1.5$

Worked example 4 Extended

Using the graph of $y = x^{3} - 3 x + 1$ for $- 2 ⩽ x ⩽ 2$ , find the equation of the straight line you need to draw to solve $x^{3} - 4 x + 2 = 0$ . Then solve the equation.

Make the left side match the graph: $x^{3} - 4 x + 2 = 0$ is $x^{3} - 3 x + 1 - x + 1 = 0$ .
So $x^{3} - 3 x + 1 = x - 1$ . Draw the line $y = x - 1$ .
The line meets the curve at about $x = 0.5$ and $x = 1.7$ .
The cubic has a third solution, $x \approx - 2.2$ , but it is outside $- 2 ⩽ x ⩽ 2$ , so this graph cannot show it. Give only the solutions in the range drawn.

Answer: Line $y = x - 1$ ; $x \approx 0.5$ or $1.7$

The cubic and the line Extended

The curve $y = x^{3} - 3 x + 1$ and the dashed line $y = x - 1$ from Worked example 4.

Check yourself Extended

The graph of $y = x^{2} - 2 x - 3$ is drawn. Write down the equation of the straight line you would draw to solve $x^{2} - 4 x + 1 = 0$ .

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

Exponential graphs: $y = a b^{x} + c$ Extended

In an exponential function the variable is in the power. With $b$ a positive whole number such as 2 or 3:

When $x = 0$ , $b^{0} = 1$ , so the curve crosses the $y$ -axis at $y = a + c$ .
As $x$ decreases, $b^{x}$ gets closer and closer to 0 but never reaches it, so $y$ gets closer to $c$ . The horizontal line $y = c$ is an asymptote: the curve approaches it without touching.
As $x$ increases, $b^{x}$ grows faster and faster, so (for positive $a$ ) the curve rises more and more steeply.

Negative values of $x$ give fractions: $2^{- 2} = \frac{1}{4}$ . Every step of 1 in $x$ multiplies $b^{x}$ by $b$ , which is a good check on a table.

Worked example 5 Extended

Complete a table of values for $y = 3 \times 2^{x} - 4$ for $x = - 2, - 1, 0, 1, 2$ . Draw the graph and use it to solve $3 \times 2^{x} - 4 = 5$ .

Table of values for $y = 3 \times 2^{x} - 4$
$x$	-2	-1	0	1	2
$y$	-3.25	-2.5	-1	2	8

$x = - 2$ : $3 \times \frac{1}{4} - 4 = 0.75 - 4 = - 3.25$ . $x = - 1$ : $3 \times \frac{1}{2} - 4 = - 2.5$ .
$x = 0$ : $3 \times 1 - 4 = - 1$ . $x = 1$ : $6 - 4 = 2$ . $x = 2$ : $12 - 4 = 8$ .
Draw a smooth curve, flattening towards $y = - 4$ on the left.
Draw the horizontal line $y = 5$ . It meets the curve at about $x = 1.6$ .

Answer: $x \approx 1.6$

An exponential graph Extended

The curve $y = 3 \times 2^{x} - 4$ . The dashed line $y = - 4$ is its asymptote.

Exponential growth and decay Extended

Many real quantities change by the same percentage each time period, not by the same amount: a population growing by 4% a year, a car losing 20% of its value a year, a medicine leaving the body. Each period multiplies the value by the same factor, so the formula has the form $y = a \times r^{t}$ , where $a$ is the starting value.

$r > 1$ is growth: an increase of 4% means $r = 1.04$ .
$0 < r < 1$ is decay: a decrease of 20% means $r = 0.8$ .

A decay graph falls quickly at first and then levels off, because 20% of a smaller value is a smaller amount. It never reaches zero. A straight line would mean the same amount lost every year, which is a different situation. Questions ask you to draw the graph from a table, read off a value, or find when the quantity reaches a target, such as half its starting value.

A decay graph Extended

A car bought for £12 000 loses 20% of its value each year, so its value after $t$ years is $V = 12 000 \times {0.8}^{t}$ .

Worked example 6 Extended

Use the graph of $V = 12 000 \times {0.8}^{t}$ to estimate when the car is worth half its original value.

Half of £12 000 is £6000.
Go across from 6000 to the curve, then down to the $t$ axis.
The reading is about 3.1 years. Check: $12 000 \times {0.8}^{3} = 6144$ , a little over 6000, so just over 3 years is right.

Answer: About $3.1$ years

Check yourself Extended

Find the value of $y = 3 \times 2^{x} - 4$ when $x = - 3$ . Give your answer as a fraction or a decimal.

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

Key points

Substitute carefully: square negatives in brackets, and $- x^{2}$ is the negative of $x^{2}$ .
Join points of a straight line with a ruler; join points of a curve with one smooth freehand curve.
$y = \frac{a}{x}$ has two branches and meets neither axis.
Roots are where the graph crosses the $x$ -axis. To solve $f (x) = k$ , draw $y = k$ .
To solve a new equation, rearrange it so that one side is the graph you have; the other side is the line to draw.
Extended: $x^{- 1}$ , $x^{- 2}$ and $x^{- \frac{1}{2}}$ do not exist at $x = 0$ ; $x^{\frac{1}{2}}$ needs $x ⩾ 0$ .
Extended: $y = a b^{x} + c$ crosses the $y$ -axis at $a + c$ and approaches $y = c$ . Growth and decay by a percentage are exponential, not linear.

Where marks are lost

Working out $(- 3)^{2}$ as $- 9$ , or $- x^{2}$ at $x = - 3$ as $+ 9$ .
Joining the points of a curve with straight lines, or drawing a flat bottom between two equal values.
Joining the two branches of a reciprocal graph across $x = 0$ .
Giving only one solution when the line crosses the curve twice, or giving a solution outside the range of the graph.
Reading the $y$ -coordinate when the question asks for the solution, which is the $x$ -coordinate.
Drawing the wrong line because of a sign error in the rearrangement. Check by subtracting the two equations.
Extended: treating $2^{- 2}$ as $- 4$ instead of $\frac{1}{4}$ .
Extended: drawing exponential decay as a straight line.

Related IGCSE Mathematics 0580 lessons

C2.11 · E2.11

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.

Graphs of functions

Tables of values

Worked example 1

Drawing the graph

A quadratic graph and a line

Reciprocal graphs: $y = \frac{a}{x}$

A reciprocal graph

Solving equations from a graph

Worked example 2

Check yourself

Check yourself

Graphs of $a x^{n}$ and sums of them Extended

Some powers of x Extended

Worked example 3 Extended

Worked example 4 Extended

The cubic and the line Extended

Check yourself Extended

Exponential graphs: $y = a b^{x} + c$ Extended

Worked example 5 Extended

An exponential graph Extended

Exponential growth and decay Extended

A decay graph Extended

Worked example 6 Extended

Check yourself Extended

Key points

Where marks are lost

Related IGCSE Mathematics 0580 lessons

Sketching curves

Graphs in practical situations

Differentiation

Proportion

Graphs of functions

Tables of values

Worked example 1

Drawing the graph

A quadratic graph and a line

Reciprocal graphs: y=ax

A reciprocal graph

Solving equations from a graph

Worked example 2

Check yourself

Check yourself

Graphs of axn and sums of them Extended

Some powers of x Extended

Worked example 3 Extended

Worked example 4 Extended

The cubic and the line Extended

Check yourself Extended

Exponential graphs: y=abx+c Extended

Worked example 5 Extended

An exponential graph Extended

Exponential growth and decay Extended

A decay graph Extended

Worked example 6 Extended

Check yourself Extended

Key points

Where marks are lost

Related IGCSE Mathematics 0580 lessons

Sketching curves

Graphs in practical situations

Differentiation

Proportion

Reciprocal graphs: $y = \frac{a}{x}$

Graphs of $a x^{n}$ and sums of them Extended

Exponential graphs: $y = a b^{x} + c$ Extended