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Algebra

Vertex of a parabola

The turning point (h, k) of y=ax2+bx+c: its minimum when a > 0 and its maximum when a < 0.

h=−b2a, k=c−b24a
Vertex formy=a(x−h)2+k
k from hk=ah2+bh+c

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−8−6−4−22468−8−6−4−22468vertex
  • Vertex (2, −5); axis of symmetry x = 2
  • D = 40 > 0: two real roots, x = 0.4189 and x = 3.581

What the symbols mean

h
the x-coordinate of the vertex; the line x=h is the axis of symmetry
k
the y-coordinate of the vertex: the minimum or maximum value of y
a
coefficient of x2 (not 0); its sign says whether the vertex is a minimum or a maximum
b
coefficient of x
c
constant term

When to use it

Use it to find the greatest or least value of a quadratic (maximum height of a ball, minimum cost, largest area for a fixed fence), the axis of symmetry, or to rewrite y=ax2+bx+c in vertex form for sketching.

Worked example

Find the vertex of y = 2x² − 8x + 3 (a = 2, b = −8, c = 3).

  1. Start from the formula.

    h=−b2⁢a

  2. Substitute a = 2, b = −8, c = 3 and work it out.

    h=−(−8)2·2
    =84
    =2

  3. The y-coordinate is the value of the quadratic at x = h:

    k=a⁢h2+b⁢h+c
    =2·22+(−8·2)+3
    =2·4+(−16)+3
    =8+(−16)+3
    =−8+3
    =−5

  4. So the vertex is (h, k) = (2, −5). It is a minimum (a > 0, the parabola opens upward).

Answer h = 2, so the vertex is (2, −5)

Why it works

A parabola is symmetric, so its vertex sits halfway between the roots: the average of the two roots in the quadratic formula is −b/2a. That still holds when there are no real roots. Putting x=h back into the equation gives k.

Common mistakes

  • Losing the minus sign: h=−b/(2a), so b = −8 and a = 2 give h = +2.
  • Writing the vertex form with the wrong sign: y=2(x−2)2−5 has vertex (2, −5), not (−2, −5).
  • Calling the vertex a maximum when a > 0. A positive a opens upwards, so the vertex is the lowest point.