2.4PureCore

Quadratic Graphs

The graph of a quadratic is a smooth U-shaped curve called a parabola. To sketch one well you need its shape, its intercepts and its turning point — all of which come straight from the equation.

30 min Video by Zeeshan Zamurred Quadratics

Edexcel AS Level Maths: 2.4 Quadratic GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Identify whether a parabola opens up or down
Find where a quadratic crosses the axes
Locate the turning point (vertex) and line of symmetry
Sketch a quadratic showing all key features

Shape: which way up?

The sign of $a$ in $y = a x^{2} + b x + c$ controls the shape. A positive $a$ gives a $\cup$ shape with a minimum; a negative $a$ gives a $\cap$ shape with a maximum.

a > 0 \Rightarrow \cup (minimum), a < 0 \Rightarrow \cap (maximum)

Always decide the shape first — it guides everything else.

Finding the intercepts

The $y$ -intercept is just $c$ (set $x = 0$ ). The $x$ -intercepts (roots) come from solving $a x^{2} + b x + c = 0$ by factorising or the formula.

y

-intercept: set

x = 0

, giving

y = - 8

x

-intercepts:

x^{2} - 2 x - 8 = (x - 4) (x + 2) = 0

Answercrosses at

(0, - 8)

(4, 0)

and

(- 2, 0)

The turning point and symmetry

Completing the square gives the vertex directly. The parabola is symmetric about a vertical line through the vertex, so the line of symmetry is $x = - p$ for $a (x + p)^{2} + q$ .

y = a (x + p)^{2} + q \Rightarrow vertex (- p, q), line of symmetry x = - p

The vertex sits exactly halfway between the two roots.

Tip — Quick check: the line of symmetry is the average of the two x-intercepts.

Putting a sketch together

A good sketch shows: the correct shape, both axis intercepts, and the turning point labelled. You do not need graph paper — just the key features in the right places.

1Shape:

a = 1 > 0

, so a

\cup

with a minimum.

2Intercepts:

(0, - 8)

(4, 0)

(- 2, 0)

3Symmetry at

x = 1

(midpoint of

- 2

and

4

); substitute to get vertex

(1, - 9)

Answer

\cup

curve, roots

- 2

and

4

, vertex

(1, - 9)

Formula recap

a > 0 \Rightarrow \cup, a < 0 \Rightarrow \cap

Shape from the sign of a.

y -intercept = c

Set x = 0.

symmetry: x = \frac{root _{1} + root _{2}}{2}

Halfway between the roots.

Common mistakes to avoid

Assuming every parabola has two x-intercepts.

A parabola may cross the x-axis twice, once (a repeated root) or not at all.

Mixing up the y-intercept and the turning point.

The y-intercept is where x = 0; the vertex is the curve’s lowest or highest point.

Key takeaways

Sign of a sets the shape: + is a ∪, − is a ∩.
y-intercept is c; x-intercepts come from solving = 0.
Vertex and line of symmetry come from completed-square form.
A full sketch labels shape, both intercepts and the turning point.

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