Vertex form calculator

Q: How do I convert vertex form back to standard form?

Expand the square and simplify: a(x − h)² + k = ax² − 2ahx + (ah² + k).

Convert between standard form and vertex form, find the vertex and axis of symmetry, and see the parabola.

$y = ax^2 + bx + c$

a b c

h

=

-\frac{b}{2a} = -\frac{-8}{2(2)} = 2

k

=

c - \frac{b^2}{4a} = 3 - \frac{64}{8} = -5

Vertex form

y = 2\left(x - 2\right)^2 - 5

Vertex: $(2,\ -5)$
Axis of symmetry: $x = 2$
Minimum value: $-5$
y-intercept: $(0,\ 3)$

Standard form vs vertex form

Standard form $y = ax^2 + bx + c$ shows the y-intercept $c$ directly. Vertex form $y = a(x-h)^2 + k$ shows the turning point $(h, k)$ , which makes graphing and max/min problems quick.

Watch the sign of $h$ : in $y = (x+3)^2 - 1$ the vertex is $(-3, -1)$ , because $x + 3 = x - (-3)$ .

Converting by hand is completing the square. The shortcut $h = -\tfrac{b}{2a}$ comes from the same place.

Move a, b, c and track the vertex

b-8

$y = 2x^2 - 8x + 3$

Discriminant b² − 4ac: 40
Real roots: 0.419, 3.581
Vertex (h, k): (2, -5)

Discriminant > 0: the parabola crosses the x-axis twice, so there are two real roots. a > 0 opens it upward.

y = ax² + bx + c

Frequently asked questions

What is vertex form?

y = a(x − h)² + k, where (h, k) is the vertex of the parabola and a is the same leading coefficient as in standard form.

How do I convert standard form to vertex form?

Use h = −b/2a and k = c − b²/4a, or complete the square: factor a out of the x-terms, add and subtract (half the x-coefficient)², and simplify.

What is “a” in vertex form?

It controls the width and direction of the parabola. a > 0 opens upward (the vertex is a minimum), a < 0 opens downward (a maximum). The larger |a|, the narrower the parabola.

How do I convert vertex form back to standard form?

Expand the square and simplify: a(x − h)² + k = ax² − 2ahx + (ah² + k).