Graphs of Functions (Edexcel IGCSE Maths): Revision notes

Section 1

What does a quadratic graph look like?

A quadratic function has the form $y = ax^2 + bx + c$ . Its graph is a parabola — a symmetric U-shape.

If $a > 0$ , the parabola opens upwards (a minimum turning point).
If $a < 0$ , the parabola opens downwards (a maximum turning point).
The graph crosses the $y$ -axis at $(0, c)$ .
It crosses the $x$ -axis where $y = 0$ (the roots), found by factorising, the quadratic formula, or completing the square.
The graph is symmetric about a vertical line through the turning point.

Key termsparabolaturning pointroots

Exam tip

Always sketch quadratics by finding: the $y$ -intercept, the roots (if any), and the turning point. Three points is usually enough for a good sketch.

Common mistake

Do not assume every quadratic crosses the $x$ -axis. If the discriminant $b^2 - 4ac < 0$ , the graph never touches the $x$ -axis.

Section 2

How do I find the turning point by completing the square?

Write $y = ax^2 + bx + c$ in the form $y = a(x + p)^2 + q$ . The turning point is then $(-p, q)$ .

Method for $a = 1$ : $y = x^2 + bx + c = \left(x + \dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2 + c$ .

Example: $y = x^2 + 6x + 5$ $= (x+3)^2 - 9 + 5 = (x+3)^2 - 4$ Turning point: $(-3, -4)$ , a minimum since $a = 1 > 0$ .

If $a \neq 1$ , factor $a$ out of the $x^2$ and $x$ terms first before completing the square inside the brackets.

Key termscompleting the squareminimummaximum

Example

$y = 2x^2 - 8x + 3 = 2(x^2 - 4x) + 3 = 2\big[(x-2)^2 - 4\big] + 3 = 2(x-2)^2 - 5$ . Turning point: $(2, -5)$ , a minimum.

Common mistake

When $a \neq 1$ , students often forget to multiply the $-\left(\frac{b}{2}\right)^2$ term back by $a$ . Always check by expanding your answer back out.

Section 3

What does a cubic graph look like?

A cubic function has the form $y = ax^3 + bx^2 + cx + d$ . Key features:

If $a > 0$ , the graph rises from bottom-left to top-right (an 'S' shape).
If $a < 0$ , the graph falls from top-left to bottom-right (a reversed 'S').
A cubic can have up to two turning points (one local maximum and one local minimum), or none (if it is monotonic, like $y = x^3$ ).
It crosses the $x$ -axis at least once, and up to three times.
The $y$ -intercept is $(0, d)$ .

Key termscubicmonotonic

Think of it like this

Think of a cubic as a quadratic with an extra 'wiggle' — it can bend once more than a parabola, giving it room for two turning points instead of one.

Exam tip

To sketch $y = (x-1)(x-2)(x+3)$ , just read off the roots from the factors: $x = 1, 2, -3$ . The curve crosses the $x$ -axis at each.

Section 4

What does a reciprocal graph look like?

A reciprocal function has the form $y = \dfrac{k}{x}$ . Its graph is a hyperbola with two separate branches.

The graph never touches the $x$ -axis or $y$ -axis — these are asymptotes (lines the curve approaches but never reaches).
If $k > 0$ , the branches sit in the top-right and bottom-left quadrants.
If $k < 0$ , the branches sit in the top-left and bottom-right quadrants.
As $x \to 0$ , $y \to \pm\infty$ . As $x \to \pm\infty$ , $y \to 0$ .
There are no turning points and no $x$ - or $y$ -intercepts.

Key termsreciprocal graphhyperbolaasymptote

Common mistake

Never write $x = 0$ as a point on a reciprocal graph — division by zero is undefined, so the curve has a gap there, not a crossing point.

Section 5

What does an exponential graph look like?

An exponential function has the form $y = k^x$ (or $y = a \cdot k^x$ ), where $k > 0$ .

If $k > 1$ , the graph shows growth: it increases slowly then rapidly as $x$ increases.
If $0 < k < 1$ , the graph shows decay: it decreases rapidly then levels off as $x$ increases.
Every exponential graph $y = k^x$ passes through $(0, 1)$ , since $k^0 = 1$ .
The graph never crosses the $x$ -axis — the $x$ -axis ( $y = 0$ ) is a horizontal asymptote.
$y$ is always positive (for $k > 0$ ).

Key termsexponential graphgrowthdecay

Example

$y = 2^x$ passes through $(0,1), (1,2), (2,4), (-1, 0.5)$ — it grows increasingly steeply and never touches the $x$ -axis.

Exam tip

Population growth and radioactive decay are classic real-world exponential graph contexts in exam questions.

Must Know

Exam tip

Quadratic $y = ax^2+bx+c$ : parabola shape, one turning point, $a>0$ opens up (minimum), $a<0$ opens down (maximum).

Exam tip

Completing the square: $y = a(x+p)^2 + q$ gives turning point $(-p, q)$ directly.

Exam tip

Cubic $y=ax^3+...$ : up to two turning points, up to three $x$ -axis crossings, S-shaped.

Exam tip

Reciprocal $y = k/x$ : hyperbola with two branches, asymptotes at $x=0$ and $y=0$ , never crosses either axis.

Exam tip

Exponential $y=k^x$ : always passes through $(0,1)$ , $x$ -axis is an asymptote, growth if $k>1$ , decay if $0<k<1$ .

Exam tip

The discriminant $b^2-4ac$ tells you how many times a quadratic crosses the $x$ -axis: two ( $>0$ ), one ( $=0$ ), or none ( $<0$ ).

That's the notes covered.

Carry on to the next subtopic.