Vertex form calculator

What is the vertex form calculator?

The vertex form calculator takes a quadratic equation written in standard form and rewrites it in vertex form. Standard form, $y = ax^2 + bx + c$, is convenient for reading off the y-intercept, while vertex form, $y = a(x - h)^2 + k$, immediately reveals the turning point of the parabola. The point $(h, k)$ is the vertex: the lowest point when the parabola opens upward ($a > 0$) and the highest point when it opens downward ($a < 0$).

This tool computes $h$ and $k$ for you, so you can graph the parabola, find its axis of symmetry, or read off its minimum or maximum value without completing the square by hand.

Formula

Given a quadratic in standard form, the vertex coordinates are:

$h = -\frac{b}{2a} \qquad k = c - \frac{b^2}{4a}$

The leading coefficient $a$ is unchanged, so the vertex form is:

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

The axis of symmetry is the vertical line $x = h$.

How to use

1. Enter the coefficient $a$ (it must not be zero, or the equation is not quadratic).
2. Enter the coefficients $b$ and $c$.
3. Read the computed vertex values $h$ and $k$. The vertex form is then $y = a(x - h)^2 + k$.

The results stay blank until all three coefficients are filled in and $a \neq 0$.

Worked example

Convert $y = 2x^2 - 12x + 10$ into vertex form. Here $a = 2$, $b = -12$, and $c = 10$.

Compute $h$:

$h = -\frac{b}{2a} = -\frac{-12}{2 \cdot 2} = \frac{12}{4} = 3$

Compute $k$:

$k = c - \frac{b^2}{4a} = 10 - \frac{(-12)^2}{4 \cdot 2} = 10 - \frac{144}{8} = 10 - 18 = -8$

So the vertex is $(3, -8)$ and the vertex form is:

$y = 2(x - 3)^2 - 8$

FAQ

Why must the coefficient a not be zero?

If $a = 0$, the $x^2$ term disappears and the equation becomes linear, $y = bx + c$, which has no vertex. Both vertex formulas also divide by $a$, so $a = 0$ would make them undefined. To analyse a straight line instead, see the slope calculator.

How does the vertex relate to the rate of change?

At the vertex the parabola’s instantaneous slope is zero, which is why it is the turning point. To measure how a function’s output changes across an interval rather than at a single point, use the average rate of change calculator.