A quadratic function has an x2 term — its graph is a smooth U-shape called a parabola. Every quadratic has a few key features (vertex, roots, y-intercept, axis of symmetry) and can be written in three different forms — each one giving you instant info about a different feature.
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What you need to know
A quadratic has the form y = ax2 + bx + c, where a ≠ 0
The shape depends on the sign of a: positive = ∪ (min), negative = ∩ (max)
The y-intercept is at (0, c)
The roots (a.k.a. zeros, x-intercepts) come from solving ax2 + bx + c = 0
The axis of symmetry is at x = −b2a (in the formula booklet)
The vertex sits on the axis of symmetry — it’s the maximum or minimum point
A quadratic can be written in three forms — each one reveals different features instantly
Shape: Concave Up or Down?
The sign of a tells you everything about the shape:
a > 0
Concave up (∪) — graph has a minimum point
a < 0
Concave down (∩) — graph has a maximum point
Anatomy of a Quadratic Graph
Here’s everything you need to identify on a parabola:
The Key Features
How many x-intercepts?
A parabola can cross the x-axis 0, 1, or 2 times — this is set by something called the discriminant (covered in its own note). For now, just remember:
2 roots → the parabola crosses the x-axis at two points
1 root → the parabola touches the x-axis once (tangent)
0 roots → the parabola never touches the x-axis
Quick fact: if a parabola has two roots, the axis of symmetry passes through their midpoint. So if the roots are at x = 2 and x = 8, the axis of symmetry is at x = 5.
The Three Forms of a Quadratic
Each form is best for a different situation:
General Form
f(x) = ax2 + bx + c
Shows the y-intercept(0, c) directly. Axis of symmetry: x = −b2a
Factorised Form
f(x) = a(x − p)(x − q)
Shows the roots(p, 0) and (q, 0). Axis of symmetry: x = p + q2
Vertex Form
f(x) = a(x − h)2 + k
Shows the vertex(h, k). Axis of symmetry: x = h
The number a stays the same in all three forms — it’s the same scale factor everywhere. Pick the form that matches what you’ve been given.
How to Sketch a Quadratic Graph
Determine the shape from the sign of a — concave up (∪) or down (∩).
Find the axis of symmetry using x = −b2a.
Find the vertex by substituting that x-value back into the equation to get y.
Find the roots by setting the equation equal to zero and solving.
Mark the y-intercept at (0, c).
Draw a smooth parabola through all the labelled points.
How to Find the Equation of a Quadratic
Match the form to whatever info you’ve been given:
Given the rootsx = p, x = q: use factorised formy = a(x − p)(x − q). You’ll need a third point to find a.
Given the vertex(h, k): use vertex formy = a(x − h)2 + k. You’ll need a second point to find a.
Given three random points: use general formy = ax2 + bx + c and set up a system of three equations for a, b, c.
GDC shortcut: on Paper 2, just plot the function and read off the roots, vertex, and intercepts directly. Sketch the GDC graph as part of your working — it’s good exam technique.
Worked Examples
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Example 1 — Identify all key features
For the quadratic y = x2 − 6x + 5, find: (a) the shape, (b) the y-intercept, (c) the axis of symmetry, (d) the vertex, (e) the roots.
Answer:
(a) Shape: a = 1 > 0, so concave up ∪ (minimum)(b) y-intercept: read off c.(0, 5)(c) Axis of symmetry: x = −b/(2a)x = −(−6) / (2 × 1) = 3x = 3(d) Vertex: substitute x = 3 into the equation.y = (3)² − 6(3) + 5 = 9 − 18 + 5 = −4vertex: (3, −4)(e) Roots: factorise.x² − 6x + 5 = (x − 1)(x − 5) = 0x = 1 or x = 5
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Example 2 — Sketch a quadratic
Sketch the graph of y = −x2 + 4x + 5, labelling all key features.
Answer:
Shape: a = −1 < 0, so concave down ∩ (max)y-intercept: c = 5 → (0, 5)Axis of symmetry: x = −4/(2 × −1) = 2Vertex: y = −(2)² + 4(2) + 5 = −4 + 8 + 5 = 9vertex: (2, 9) — maximum pointRoots: −x² + 4x + 5 = 0 → x² − 4x − 5 = 0(x − 5)(x + 1) = 0x = 5 or x = −1Sketch: ∩ shape through (−1, 0), (5, 0), with max at (2, 9) and y-intercept (0, 5).
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Example 3 — Equation from vertex and a point
A quadratic graph has its vertex at (−1, 8) and passes through the y-intercept (0, 6). Find an expression for y = f(x).
Answer:
Step 1: vertex given → use vertex form.y = a(x − h)² + ky = a(x − (−1))² + 8y = a(x + 1)² + 8Step 2: substitute the second point (0, 6) to find a.6 = a(0 + 1)² + 86 = a + 8a = −2y = −2(x + 1)² + 8
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Example 4 — Equation from roots and a point
A quadratic has roots at x = 2 and x = −3, and passes through the point (1, 8). Find its equation.
Answer:
Step 1: roots given → use factorised form.y = a(x − p)(x − q)y = a(x − 2)(x − (−3))y = a(x − 2)(x + 3)Step 2: substitute (1, 8) to find a.8 = a(1 − 2)(1 + 3)8 = a(−1)(4)8 = −4aa = −2y = −2(x − 2)(x + 3)Negative a means the parabola opens downwards. Check: at x=1, y = −2(−1)(4) = 8 ✓
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Example 5 — Reading features from vertex form
A quadratic is given by y = (x − 3)2 − 4. Find: (a) the vertex, (b) the y-intercept, (c) the roots.
Answer:
(a) Vertex: read directly from y = a(x − h)² + k.h = 3, k = −4vertex: (3, −4)(b) y-intercept: substitute x = 0.y = (0 − 3)² − 4 = 9 − 4 = 5y-intercept: (0, 5)(c) Roots: set y = 0 and solve.(x − 3)² − 4 = 0(x − 3)² = 4x − 3 = ± 2x = 3 + 2 = 5 or x = 3 − 2 = 1x = 1 or x = 5
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Tips
Match the form to the info given. Roots → factorised. Vertex → vertex form. Random points → general form.
The number a never disappears. Don’t assume a = 1 just because you can’t see it. You always need a second/third point to pin it down.
Use your GDC on Paper 2 to find roots, vertex, and intercepts instantly. You don’t need to factorise or complete the square — let the calculator do the work.
Sketch the GDC graph as part of your working. Examiners give credit for clear visual reasoning.
Always label all key points on a sketch — vertex, x-intercepts, y-intercept. Unlabelled sketches lose marks.
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Common mistakes
Forgetting the negative in the axis of symmetry formula. It’s x = −b2a, not b2a. The minus sign is essential.
Wrong sign of h in vertex form. If you see y = (x + 1)2 + 8, the vertex is at h = −1, not +1 (because x + 1 = x − (−1)).
Confusing min and max. Always check the sign of a first. Positive a = minimum vertex; negative a = maximum vertex.
Assuming a = 1. Writing y = (x − 2)(x + 3) when only roots are given is wrong — you must find a using a third point.
Substituting into the wrong formula. To find the y-coord of the vertex, substitute the x-coord into the original equation, not into the axis-of-symmetry formula.
Reporting y-intercept as just a number instead of a coordinate. The y-intercept is at the point(0, c), not just “c“.
Final word: Master the three forms — general, factorised, and vertex — and you can answer any quadratic question by picking whichever form makes the work easiest. Each form gives you a feature for free.
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