SPM Quadratic Functions: Find the Vertex, Axis of Symmetry and Roots

Quadratic functions come up again and again in SPM, and most questions ask for the same few things: the shape of the graph, the axis of symmetry, the maximum or minimum point, and where the graph cuts the axes. Once you have a fixed method, these become very manageable marks.

The 4-step method

For any quadratic function $f(x) = ax^2 + bx + c$:

1. Shape. If $a > 0$ the graph opens upwards and has a minimum point. If $a < 0$ it opens downwards and has a maximum point.
2. Axis of symmetry. $x = \dfrac{-b}{2a}$
3. Vertex (turning point). Substitute that $x$ back into $f(x)$ to get the $y$-coordinate. This is the minimum or maximum value.
4. Intercepts. The $y$-intercept is $c$. The $x$-intercepts (roots) come from solving $f(x) = 0$.

Worked example

Sketch the graph of $f(x) = 2x^2 - 8x + 6$.

• Shape: $a = 2 > 0$, so the graph opens upwards and has a minimum point.
• Axis of symmetry: $b = -8$, so $x = \dfrac{-(-8)}{2 \times 2} = \dfrac{8}{4} = 2$.
• Vertex: $f(2) = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2$, so the minimum point is $(2, -2)$.
• $y$-intercept: $c = 6$, so the graph passes through $(0, 6)$.
• Roots: $2x^2 - 8x + 6 = 2(x^2 - 4x + 3) = 2(x - 1)(x - 3) = 0$, so $x = 1$ or $x = 3$.

Quick check: the roots 1 and 3 sit equally either side of the axis $x = 2$. If your roots aren’t symmetric about your axis, something has gone wrong.

4 mistakes that cost marks

1. Getting the sign of $b$ wrong

In $2x^2 - 8x + 6$, $b$ is $-8$, not 8. Students who use $b = 8$ get $x = -2$ for the axis, which puts the whole graph in the wrong place. Always write down $a$, $b$ and $c$ with their signs before you start.

2. Dropping the common factor

$2x^2 - 8x + 6 = 2(x - 1)(x - 3)$, not $(x - 1)(x - 3)$. The roots are the same either way, but if you later use the factorised form to find the $y$-intercept or the vertex, you’ll be out by a factor of 2.

3. Completing the square without the coefficient of $x^2$

To write the function in vertex form, take the 2 out first:

The vertex form $a(x - h)^2 + k$ gives the vertex $(h, k) = (2, -2)$, which matches our answer above. A common slip is writing $-4 + 6$ instead of $2 \times (-4) + 6$.

4. A sketch with missing labels

In a sketch question, marks are usually given for the shape, the turning point and the intercepts. A neat curve with no coordinates written on it can lose most of the marks. Label every point you’ve calculated.

Practise it

Try these with the 4-step method, then check that your roots are symmetric about the axis:

1. $f(x) = x^2 - 6x + 5$
2. $f(x) = -x^2 + 4x + 5$
3. $f(x) = 3x^2 + 6x - 9$