A parabola is the U-shaped graph of a quadratic relationship such as y = x squared. It is symmetric about a vertical line called the axis of symmetry.

Year 9 Maths Introduction to Quadratic Equations and Graphs

The Parabola

Q: How can you tell if a parabola is concave up or down?

Look at the coefficient of x squared. If it is positive the parabola is concave up and has a minimum. If it is negative it is concave down and has a maximum.

Q: How do you find the axis of symmetry?

It is the vertical line halfway between the two x-intercepts. If the intercepts are 2 and 4, the axis is x = 3. From y = ax squared + bx + c you can also use x = minus b over 2a.

Q: How many x-intercepts can a parabola have?

Two, one or none. It has one when the vertex sits on the x-axis, and none when the whole parabola is above or below the x-axis.

20 practice questions 1 video lesson Theory + worked examples

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 1 video(s)

The Parabola - Video - Visual introduction to parabolas Watch

Theory

The graph of a quadratic such as \(y = x^2 - 6x + 8\) is a parabola. This Year 9 maths page shows how to draw one from a table of values and how to find its key features: the vertex, the axis of symmetry, whether it is concave up or down, and its intercepts. It also shows how to test whether a point is on a parabola and how a parabola models a real situation.

A parabola is the U-shaped graph of a quadratic relationship, \(y = ax^2 + bx + c\) with \(a \ne 0\). The simplest is \(y = x^2\).

The vertex (or turning point) is where the parabola turns. It is a minimum when the parabola is concave up (opens upwards, \(a > 0\)) and a maximum when it is concave down (opens downwards, \(a < 0\)).

The axis of symmetry is the vertical line through the vertex, \(x = h\). Each side of the parabola is a mirror image of the other.

The \(y\)-intercept is where the graph crosses the \(y\)-axis (put \(x = 0\)). Every parabola of this kind has exactly one. The \(x\)-intercepts are where it meets the \(x\)-axis (put \(y = 0\)). A parabola can have \(2\), \(1\) or \(0\) \(x\)-intercepts.

The axis of symmetry passes through the vertex, halfway between the \(x\)-intercepts: \(x = \dfrac{2 + 4}{2} = 3\).

In \(y = ax^2 + bx + c\), the sign of \(a\) gives the shape. Positive \(a\): a minimum turning point. Negative \(a\): a maximum.

Intercepts and the axis of symmetry:

\[y\text{-intercept: put } x = 0 \qquad x\text{-intercepts: put } y = 0 \qquad \text{axis: } x = \dfrac{x_1 + x_2}{2}\]

x = \frac{x_{1} + x_{2}}{2}

For \(y = ax^2 + bx + c\), the axis of symmetry can also be found straight from the equation:

\[x = -\dfrac{b}{2a}\]

x = - \frac{b}{2 a}

The \(y\)-intercept of \(y = ax^2 + bx + c\) is always \(c\).

In factorised form the intercepts can be read off. For \(y = (x - 2)(x - 7)\), the \(x\)-intercepts are \(x = 2\) and \(x = 7\) (each bracket is zero), and the \(y\)-intercept is \((-2) \times (-7) = 14\). In general, \(y = (x - p)(x - q)\) has \(x\)-intercepts \(p\) and \(q\) and \(y\)-intercept \(pq\). Sketching from this form is covered in Sketching Parabolas Using Intercept Form.

Simple forms. \(y = ax^2 + c\) has its vertex at \((0, c)\) and axis \(x = 0\). \(y = (x - h)^2\) has its vertex at \((h, 0)\) on the \(x\)-axis, so it has one \(x\)-intercept.

Always positive means no \(x\)-intercepts. A square is never negative, so \(x^2 + c\) with \(c > 0\) is always at least \(c\). Its graph stays above the \(x\)-axis. The same is true of \(x^2 + k^2\) for any number \(k \ne 0\).

Is a point on the parabola? Substitute its \(x\)-value and compare with its \(y\)-value. Does \((-2, 7)\) lie on \(y = x^2 + 3\)? \((-2)^2 + 3 = 7\), so yes. Does \((5, 21)\) lie on \(y = x^2 - 2x + 5\)? \(25 - 10 + 5 = 20 \ne 21\), so no.

Finding a coefficient. If \(y = kx^2\) passes through \((3, 18)\), then \(18 = k \times 3^2 = 9k\), so \(k = 2\).

How to sketch a parabola

Shape: look at the sign of the \(x^2\) term. Positive: concave up. Negative: concave down.
\(y\)-intercept: put \(x = 0\).
\(x\)-intercepts: put \(y = 0\) and solve, factorising if needed. If there are none, say so.
Axis of symmetry: halfway between the \(x\)-intercepts, or use \(x = -\dfrac{b}{2a}\).
Vertex: substitute the axis value into the equation to find the \(y\)-value.
Draw a smooth U-shaped curve through the points (not a V with a sharp point), and label the features.

How to read features from a graph

Find the lowest or highest point: that is the vertex. The axis is \(x =\) its \(x\)-coordinate.
Read where the curve crosses each axis: the \(x\)-intercepts and the \(y\)-intercept.
Decide the shape: opening upwards is concave up, downwards is concave down.

Example 1 — A table of values

For \(y = 4 - x^2\):
(i) complete a table of values for \(x = -3\) to \(3\)
(ii) describe the main features of its graph.

Solution

(i) Square \(x\) first, then subtract from \(4\). For \(x = -3\): \(4 - 9 = -5\).

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(y\)	\(-5\)	\(0\)	\(3\)	\(4\)	\(3\)	\(0\)	\(-5\)

(ii) The \(y\)-values are symmetric about \(x = 0\), and the largest value is \(4\).

\(\text{vertex}\)	\(\)	\(\boldsymbol{(0, 4)}\text{, a maximum}\)
\(\text{axis}\)	\(\)	\(\boldsymbol{x = 0}\)
\(x\text{-intercepts}\)	\(\)	\(\boldsymbol{x = -2,\ x = 2}\)
\(\text{shape}\)	\(\)	\(\text{concave down } (a = -1)\)

y = 4 - x^{2}

Example 2 — Axis and vertex from the equation

For \(y = x^2 - 8x + 7\):
(i) find the axis of symmetry and the vertex
(ii) find the \(y\)-intercept and the \(x\)-intercepts.

Solution

(i) Here \(a = 1\) and \(b = -8\). Use \(x = -\dfrac{b}{2a}\), then substitute to find \(y\).

\(x\)	\(=\)	\(-\dfrac{-8}{2 \times 1} = \boldsymbol{4}\)
\(y\)	\(=\)	\(4^2 - 8 \times 4 + 7\)
\(\)	\(=\)	\(16 - 32 + 7 = -9\)

The axis of symmetry is \(\boldsymbol{x = 4}\) and the vertex is \(\mathbf{(4, -9)}\). Since \(a = 1 > 0\), it is a minimum turning point.

(ii) Put \(x = 0\) for the \(y\)-intercept, and \(y = 0\) for the \(x\)-intercepts.

\(y\)	\(=\)	\(0 - 0 + 7 = \boldsymbol{7}\)
\(x^2 - 8x + 7\)	\(=\)	\(0\)
\((x - 1)(x - 7)\)	\(=\)	\(0\)
\(x\)	\(=\)	\(\boldsymbol{1} \text{ or } \boldsymbol{7}\)

vertex (4, - 9)

Check: the axis \(x = 4\) is halfway between \(1\) and \(7\).

Example 3 — Touching or missing the axis

How many \(x\)-intercepts does each parabola have?
(i) \(y = (x - 3)^2\)
(ii) \(y = -x^2 - 2\)

Solution

(i) \((x - 3)^2 = 0\) only when \(x = 3\). The vertex \((3, 0)\) sits on the \(x\)-axis.

\(\text{vertex}\)	\(\)	\((3, 0)\)
\(y\text{-intercept}\)	\(\)	\((0 - 3)^2 = 9\)
\(\text{answer}\)	\(\)	\(\textbf{one } x\textbf{-intercept}\)

(ii) \(x^2\) is never negative, so \(-x^2 \le 0\) and \(y = -x^2 - 2 \le -2\). The graph never reaches \(y = 0\).

\(\text{vertex}\)	\(\)	\((0, -2)\text{, concave down}\)
\(\text{answer}\)	\(\)	\(\textbf{no } x\textbf{-intercepts}\)

y \leq - 2

Example 4 — A parabola in context

A netball is thrown so that its height is \(h = -x^2 + 8x\) metres, where \(x\) m is the horizontal distance travelled.
(i) How far does it travel before it reaches the ground (\(h = 0\))?
(ii) What is its greatest height?

Solution

(i) Factorise and use the null factor law.

\(-x^2 + 8x\)	\(=\)	\(0\)
\(x(8 - x)\)	\(=\)	\(0\)
\(x\)	\(=\)	\(0 \text{ or } 8\)

\(x = 0\) is where it starts, so it travels \(\mathbf{8}\) m.

(ii) The greatest height is at the vertex, on the axis of symmetry halfway between \(0\) and \(8\).

\(x\)	\(=\)	\(4\)
\(h\)	\(=\)	\(-(4)^2 + 8 \times 4 = -16 + 32 = \mathbf{16}\textbf{ m}\)

h = 16

Common pitfalls

Square before you change the sign. For \(y = 4 - x^2\) with \(x = -3\): \(4 - (-3)^2 = 4 - 9 = -5\), not \(4 + 9\).

The axis is a line. Write the axis of symmetry as an equation, such as \(x = 3\), not just \(3\) or the point \((3, 0)\).

Watch the sign in the vertex. \(y = (x + 4)^2\) has its vertex at \((-4, 0)\), not \((4, 0)\): the bracket is zero when \(x = -4\).

Not every parabola crosses the x-axis. \(y = x^2 + 5\) is always at least \(5\), so it has no \(x\)-intercepts.

Frequently asked questions

What is a parabola?

A parabola is the U-shaped graph of a quadratic relationship such as \(y = x^2\). It is symmetric about a vertical line called the axis of symmetry.

How can you tell if a parabola is concave up or down?

Look at the coefficient of \(x^2\). If it is positive the parabola is concave up and has a minimum. If it is negative it is concave down and has a maximum.

How do you find the axis of symmetry?

It is the vertical line halfway between the two \(x\)-intercepts. If the intercepts are \(2\) and \(4\), the axis is \(x = 3\). From \(y = ax^2 + bx + c\) you can also use \(x = -\dfrac{b}{2a}\).

How many x-intercepts can a parabola have?

Two, one or none. It has one when the vertex sits on the \(x\)-axis, and none when the whole parabola is above or below the \(x\)-axis.

Where are parabolas found in real life?

The path of a thrown ball, the shape of a water jet and the cables of some bridges are close to parabolas. The vertex gives the greatest height.

← Previous subtopic

Applications of Quadratic Equations

Next subtopic →

Sketching y=ax^2 with Dilations and Reflections

More in Introduction to Quadratic Equations and Graphs

The Parabola

📖 Theory

How to sketch a parabola

How to read features from a graph

Common pitfalls

Frequently asked questions

Theory