The graph of a quadratic function y = ax² + bx + c is a smooth curve called a parabola, identical on both sides of a line of symmetry.
In the most basic form, y = ax², the sign of a decides whether the graph opens upwards or downwards, and the larger the absolute value of a, the narrower it is.
y = a(x − p)² + q is y = ax² moved p in the x-direction and q in the y-direction, so its vertex is (p, q) and its axis of symmetry is x = p.
An expanded equation can be put into this form by completing the square, after which the vertex, the axis and the maximum or minimum value can be read off directly.
The role of a in y = ax²
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The parabola y = ax² narrowing, widening and flipping as a changes, with y = x² for comparison
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📊 What a decides
①If a > 0, the graph opens upwards (it is convex downwards).
②If a < 0, the graph opens downwards (it is convex upwards).
③The larger |a|, the narrower the parabola; the smaller |a|, the wider.
④If a = 0, then y = 0 and the function is not quadratic.
Vertex form y = a(x − p)² + q
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The parabola moving as a, p and q change, with its vertex, axis and expanded form
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🎯 Translations
①Replacing x by x − p in y = ax² gives y = a(x − p)², the graph moved p in the x-direction.
②y = ax² + q is the graph moved q in the y-direction.
③The vertex of y = a(x − p)² + q is (p, q) and its axis is x = p.
④Watch the signs: the vertex of y = (x − 2)² + 3 is (2, 3). The starting values a = 1, p = 2, q = −1 expand to y = x² − 4x + 3, which is also written at the bottom left of the picture.
From expanded form to vertex form
Expanded form
y = ax² + bx + c
The expanded form shows the y-intercept c directly.
Converting to vertex form
y = a(x + b2a)² − b²−4ac4a
Complete the square to find the coordinates of the vertex.
🔄 How to convert
①Take a out of the ax² + bx part.
②Add and subtract the square of half the coefficient of x inside the brackets.
③Write the perfect square and tidy up the remaining constant.
④The vertex comes out as (−b/2a, −(b² − 4ac)/4a).
Properties of the graph together
Information from the graph
vertex (p, q) · axis x = p · y-intercept c · direction from the sign of a
Everything you need to know about the graph is contained in this equation.
💡 Properties of the graph
①The axis of symmetry is x = p, the line through the vertex.
②The y-intercept is c, the value at x = 0.
③The x-intercepts are the roots of the quadratic equation obtained by setting y = 0.
④The maximum or minimum value is the y-coordinate of the vertex.
⑤By symmetry about the axis, f(p + t) = f(p − t).
Worked examples
Example 1
Find the coordinates of the vertex of y=(x−2)²+3.
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In vertex form y = a(x − p)² + q, the vertex is (p, q).
y = (x−2)²+3
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p = 2 and q = 3, so the vertex is (2, 3).
vertex (2, 3)
▸ (2, 3)
(x − 2) has the form x − p, so p = +2. Watch the sign.
Example 2
Rewrite y=x²−4x+1 in vertex form and find the vertex.
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Complete the square for x² − 4x: x² − 4x = (x − 2)² − 4.
y = (x−2)²−4+1
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Simplifying gives y = (x − 2)² − 3, so the vertex is (2, −3).
y = (x−2)²−3 ⇒ vertex (2, −3)
▸ y=(x−2)²−3, vertex (2, −3)
Add and subtract 4, the square of half the coefficient of x (−2).
Exam checklist
Key facts about quadratic functions
y = a(x − p)² + q → vertex (p, q), axis x = p
Most quadratic-function problems start by converting to vertex form.
G9 school-exam type
What is the axis of symmetry of y=x²−6x+5?
①x=−3
②x=−6
③x=3
④x=5
⑤x=6
▸ ③ x=3
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The axis is x = −b/(2a) with a=1, b=−6.
x = −(−6)/(2·1)
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Compute: x=3.
x = 6/2 = 3
🎯 Exam points
①Decide from the sign of a whether the graph opens upwards or downwards.
②The larger |a|, the narrower the graph.
③In y = a(x − p)² + q, (x − 2) means p = +2; this sign is easy to get wrong.
④Practise converting the expanded form to vertex form.
⑤Use the symmetry f(p + t) = f(p − t) about the axis.