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 Quadratics

Edexcel AS Level Maths: 2.4 Quadratic GraphsCurated by Kepler, not produced by us. Watch it 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 $y = ax^2 + bx + c$ 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 \text{ (minimum)}, \qquad a < 0 \Rightarrow \cap \text{ (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 $ax^2 + bx + c = 0$ by factorising or the formula.

Example 1Find where

y = x^2 - 2x - 8

crosses the axes.

y

-intercept: set

x = 0

, giving

y = -8

x

-intercepts:

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

Answer

crosses 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\; \text{vertex } (-p, q), \;\text{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.

Example 1Sketch

y = x^2 - 2x - 8

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, \quad a < 0 \Rightarrow \cap

Shape from the sign of a.

y\text{-intercept} = c

Set x = 0.

\text{symmetry: } x = \tfrac{\text{root}_1 + \text{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The graph of $y = -x^{2} + 3x + 1$ is:
- a ∩ shape with a maximum
- a ∪ shape with a minimum
- a straight line
- a ∪ shape with a maximum
Show answer and explanation
Answer: a ∩ shape with a maximum
$a = -1 < 0$ , so the parabola opens downwards.
Where does $y = x^{2} + 5x + 6$ cross the $y$ -axis?
- $(0, 6)$
- $(6, 0)$
- $(0, 5)$
- $(0, -6)$
Show answer and explanation
Answer: $(0, 6)$
Set $x = 0$ : $y = 6$ .
The roots of $y = (x - 1)(x - 7)$ are:
- $x = 1, 7$
- $x = -1, -7$
- $x = 1, -7$
- $x = 7$ only
Show answer and explanation
Answer: $x = 1, 7$
Each bracket equals zero at $x = 1$ and $x = 7$ .
A parabola has roots at $x = -2$ and $x = 6$ . Its line of symmetry is:
- $x = 2$
- $x = 4$
- $x = -4$
- $x = 8$
Show answer and explanation
Answer: $x = 2$
Average of the roots: $(-2 + 6)/2 = 2$ .
The vertex of $y = (x + 3)^{2} - 4$ is:
- $(-3, -4)$
- $(3, -4)$
- $(-3, 4)$
- $(3, 4)$
Show answer and explanation
Answer: $(-3, -4)$
Vertex at $(-p, q)$ with $p = 3$ , $q = -4$ .
How many times does $y = x^{2} + 1$ cross the $x$ -axis?
- Never
- Once
- Twice
- Three times
Show answer and explanation
Answer: Never
$x^2 + 1 = 0$ has no real solutions, so the curve sits entirely above the x-axis.
The minimum point of $y = x^{2} - 6x + 5$ is at $x =$ ?
- $3$
- $-3$
- $6$
- $5$
Show answer and explanation
Answer: $3$
Roots at 1 and 5; symmetry at $x = 3$ (also from completing the square).
Which feature does the y-intercept give you directly?
- The constant term c
- The vertex
- The line of symmetry
- The value of a
Show answer and explanation
Answer: The constant term c
At $x = 0$ , $y = c$ , so the y-intercept is the constant term.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.