Quadratic Graphs - Roots and Intercepts Revision Notes | GCSE AQA Higher Maths

1. Why Quadratic Graphs Matter

Every quadratic equation tells a story on a graph. The curve reveals exactly where solutions live, where the function reaches its peak or trough, and where it crosses the axes — all without solving a single equation by hand.

At GCSE, you need to read that story fluently in both directions: from a graph to its key features, and from an equation to its roots. These two skills — graphical interpretation and algebraic deduction — are the twin pillars of this topic.

2. The Shape of a Quadratic Graph

A quadratic function has the form $y = ax^2 + bx + c$, where $a eq 0$. Its graph is always a parabola — a smooth, symmetrical curve. The sign of $a$ controls which way it opens.

• If $a > 0$: the parabola opens upwards (U-shape) and has a minimum turning point.
• If $a < 0$: the parabola opens downwards (n-shape) and has a maximum turning point.
• The axis of symmetry runs vertically through the turning point, splitting the curve into two mirror halves.

The blue U-shaped curve above is $y = x^2 - 5x + 6$. Notice it crosses the $x$-axis at two points — those are the roots. The red n-shaped curve is $y = -x^2 + 4x - 1$, which also has two roots but a maximum turning point instead of a minimum.

3. Reading Key Features from a Graph

Three features define a quadratic graph completely: its roots, its $y$-intercept, and its turning point. Each one has a precise graphical location and a precise algebraic meaning.

To read the roots from a graph, locate every point where the curve meets the $x$-axis and read off the $x$-coordinate. The $y$-coordinate at a root is always zero, so you only need to record the $x$-value.

The turning point is the highest or lowest point on the curve. It sits exactly on the axis of symmetry, which is halfway between the two roots. If the roots are at $x = p$ and $x = q$, the axis of symmetry is at $x = \frac{p + q}{2}$.

4. Deducing Roots Algebraically by Factorising

Reading roots from a graph is useful, but graphs are not always drawn accurately enough to trust. The algebraic method — factorising — gives exact roots every time, and it is the method the examiner expects you to demonstrate in questions without a calculator.

The key idea: if $y = (x - p)(x - q)$, then $y = 0$ when $x = p$ or $x = q$. So factorising the quadratic directly reveals the roots. Here is the method step by step.

Worked example: Deducing Roots by Factorising — $y = x^2 - 5x + 6$

Find the roots of $y = x^2 - 5x + 6$ algebraically.

1. $x^2 - 5x + 6 = 0$ — Roots occur where $y = 0$, so we solve the equation $x^2 - 5x + 6 = 0$. [M1 — correct setup]
2. $-2 \times -3 = 6$ and $-2 + (-3) = -5$ ✓ — For a quadratic $x^2 + bx + c$, we need a pair of numbers with product $c$ and sum $b$. Here $c = 6$ and $b = -5$, so the pair is $-2$ and $-3$.
3. $(x - 2)(x - 3) = 0$ — Check by expanding: $(x-2)(x-3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6$ ✓ [M1 — correct factorisation]
4. $x - 2 = 0 \Rightarrow x = 2$ $x - 3 = 0 \Rightarrow x = 3$ — Each bracket set to zero gives one root. [A1 for each correct root]
5. Roots at $(2, 0)$ and $(3, 0)$ — At a root, $y = 0$, so the coordinates are $(x, 0)$. Always write roots as coordinates when the question asks for points on the graph.

Answer: The roots are $x = 2$ and $x = 3$, i.e. the graph crosses the $x$-axis at $(2, 0)$ and $(3, 0)$.

Notice how the factorised form $(x - 2)(x - 3) = 0$ makes the roots visible immediately — they are the values that make each bracket equal to zero. This is the algebraic equivalent of reading the $x$-intercepts from a graph.

5. Worked Example — Full Feature Identification

Now that you can find roots algebraically and read intercepts from a graph, the next step is combining both skills to identify all key features of a quadratic. This is exactly what GCSE Higher exam questions ask you to do.

Worked example: Identify All Key Features of $y = x^2 + 2x - 8$

For the quadratic $y = x^2 + 2x - 8$, find: (a) the $y$-intercept, (b) the roots, (c) the coordinates of the turning point.

1. Substitute $x = 0$: $y = (0)^2 + 2(0) - 8 = -8$ So $y$-intercept is $(0, -8)$. — The $y$-intercept is always at $x = 0$. For $y = ax^2 + bx + c$, this is the constant term $c$. Here $c = -8$. [A1]
2. $x^2 + 2x - 8 = 0$ — Roots occur where the graph crosses the $x$-axis, i.e. where $y = 0$. [M1]
3. Need two numbers: product $= -8$, sum $= +2$ $+4 \times -2 = -8$ and $+4 + (-2) = +2$ ✓ — With a negative product, one number must be positive and one negative. Systematically check pairs: $(1,-8), (2,-4), (4,-2)$. The pair $(4, -2)$ works.
4. $(x + 4)(x - 2) = 0$ $x + 4 = 0 \Rightarrow x = -4$ $x - 2 = 0 \Rightarrow x = 2$ — Roots are $x = -4$ and $x = 2$, giving points $(-4, 0)$ and $(2, 0)$ on the graph. [A1 each root]
5. $x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1$ — The turning point lies halfway between the two roots. The axis of symmetry is at $x = -1$. [M1]
6. $y = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9$ Turning point: $(-1, -9)$ — Substitute $x = -1$ back into the original equation. Since $a = 1 > 0$, this is a minimum. [A1 for correct coordinates]

Answer: $y$-intercept: $(0, -8)$. Roots: $(-4, 0)$ and $(2, 0)$. Turning point (minimum): $(-1, -9)$.

6. Sketching a Quadratic Graph

A sketch is not a precise plot — it is a clear diagram showing the correct shape and all key features labelled. The command word 'sketch' tells you to signal that you should draw the general shape with labelled intercepts and turning point, not plot individual coordinates on squared paper.

• Draw a smooth U-shape (or n-shape if $a < 0$) — never a V-shape.
• Label the roots on the $x$-axis with their exact $x$-values.
• Label the $y$-intercept on the $y$-axis with its exact $y$-value.
• Mark the turning point and state its coordinates.
• Label the axes as $x$ and $y$.

7. Practice — Apply Both Methods

The best way to consolidate these skills is to work through questions that require both graphical reading and algebraic deduction. Try the problems below before checking the solutions.

8. Exam Technique and Common Mistakes

This topic rewards careful reading of the question. GCSE Higher exam questions on quadratic graphs often combine graphical and algebraic skills in the same part, so knowing where each mark comes from is essential.