Vertex form and parabolas

Every parabola has a turning point, called the vertex. Vertex form shows it to you without any work.

y = (x - 2)^{2} - 3

has its vertex at

(2, - 3)

. Inside the parentheses, the sign looks flipped.

a > 0

opens up,

a < 0

opens down. A bigger

∣ a ∣

means a narrower parabola.

y = x^{2}

in orange opens up.

y = - \frac{1}{2} x^{2}

in navy opens down and is wider.

From standard form

For $y = a x^{2} + b x + c$ , the axis of symmetry is $x = - \frac{b}{2 a}$ . Substitute that $x$ to get the $y$ of the vertex.

Use the formula. $x = - \frac{- 6}{2} = 3$

Substitute to get $y$ . $9 - 18 + 5 = - 4$

So the vertex is here. $(3, - 4)$

The vertex is at $(3, 16)$ , and it opens down. $t = 3$

Its height there is the $k$ value. $h = 16$

Reading $(x + 4)^{2}$ as $h = 4$ .

$(x + 4)$ is $(x - (- 4))$ , so $h = - 4$ .

The short version
In

y = a (x - h)^{2} + k

, the vertex is

(h, k)

. Watch the sign of

h

. From standard form, use

x = - \frac{b}{2 a}

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