Exploring Parabolas
Exploring Parabolas in NSW Year 10 Core maths introduces the U-shaped graph of a quadratic relationship (NESA outcome MA5‑NLI‑C‑02, with intercepts of factorised and expanded forms from MA5‑NLI‑P‑01). Every parabola has a vertex, or turning point, an axis of symmetry, a \(y\)-intercept and up to two \(x\)-intercepts, and it is either concave up or concave down.
You will complete tables of values, recognise parabolas, find vertices, intercepts and axes of symmetry for equations such as \(y = (x - 3)(x + 5)\), and explain why some parabolas never cut the \(x\)-axis. You will also sketch parabolas, find \(k\) when \(y = kx^2\) passes through a given point, and model the path of a kicked football. It is part of Year 10 Core Non-Linear Relationships.
Every question with a fully worked solution.
- Exploring Parabolas - Video - Visual introduction to parabolas Watch
Theory
The graph of a quadratic such as \(y = x^2 - 6x + 8\) is a parabola. This Year 10 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. Finding intercepts from factorised and expanded forms comes up if you're heading for the Advanced path.
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.
Intercepts and the axis of symmetry:
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 \((-2) \times (-7) = 14\).
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.
- 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.
(i) complete a table of values for \(x = -3\) to \(3\)
(ii) describe the main features of its graph.
(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 \(4\).
| \(\text{vertex}\) | \(\) | \(\boldsymbol{(0, 4)}\text{, a maximum}\) |
| \(\text{axis}\) | \(\) | \(\boldsymbol{x = 0}\) |
| \(x\text{-intercepts}\) | \(\) | \(\boldsymbol{x = -2,\ x = 2}\) |
| \(\text{shape}\) | \(\) | \(\text{concave down } (a = -1)\) |
The \(x\)-intercepts are where \(y = 0\). Use the null factor law.
| \((x + 3)(x - 1)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(\boldsymbol{-3} \text{ or } \boldsymbol{1}\) |
The \(y\)-intercept is where \(x = 0\):
| \(y\) | \(=\) | \((0 + 3)(0 - 1) = \boldsymbol{-3}\) |
The axis of symmetry is halfway between the \(x\)-intercepts. Substitute it to find the vertex.
| \(x\) | \(=\) | \(\dfrac{-3 + 1}{2} = \boldsymbol{-1}\) |
| \(y\) | \(=\) | \((-1 + 3)(-1 - 1) = -4\) |
| \(\text{vertex}\) | \(\) | \(\boldsymbol{(-1, -4)}\) |
(i) \(y = (x - 3)^2\)
(ii) \(y = -x^2 - 2\)
(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 \(y = 0\).
| \(\text{vertex}\) | \(\) | \((0, -2)\text{, concave down}\) |
| \(\text{answer}\) | \(\) | \(\textbf{no } x\textbf{-intercepts}\) |
(i) How far does it travel before it reaches the ground (\(h = 0\))?
(ii) What is its greatest height?
(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 \(8\).
| \(x\) | \(=\) | \(4\) |
| \(h\) | \(=\) | \(-(4)^2 + 8 \times 4 = -16 + 32 = \mathbf{16}\textbf{ m}\) |
Common pitfalls
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 \(x = 3\).
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.