Vertex Form Calculator

What this calculator does

So, a quadratic can be written two different ways, and this tool converts between them in either direction. Standard form is y = ax² + bx + c. Vertex form is y = a(x - h)² + k. Feed it one and it gives you the other, and along the way it tells you the vertex (h, k) and the y-intercept.

A dropdown picks the direction, and you enter three numbers to match the form you are starting from.

How to use it

1. Choose the direction: standard form to vertex form, or vertex form to standard form.
2. Enter the three values. For standard form these are a, b, and c; for vertex form they are a, h, and k.
3. Press Calculate.

The coefficient a cannot be zero, or the expression is not a quadratic and has no parabola to describe.

The two forms, and what each is good for

They describe the same parabola, but each hands you a different fact for free. Standard form, y = ax² + bx + c, shows the y-intercept at a glance: it is simply c, the value when x is zero. Vertex form, y = a(x - h)² + k, shows the vertex at a glance: the turning point sits at exactly (h, k). Which form you want depends on which of those you need, and this tool lets you move between them freely.

Standard form to vertex form

To convert from standard to vertex form, the tool finds the vertex coordinates:

h = -b / (2a) and k = c - b² / (4a)

Those are the coordinates of the turning point, and dropping them into y = a(x - h)² + k gives the vertex form. This is completing the square in disguise: the same rearrangement that solves a quadratic also rewrites it in vertex form, which is why the two topics are really one.

Vertex form to standard form

Going the other way is just expanding the bracket. Given a, h, and k, the tool multiplies out a(x - h)² + k. That produces b = -2ah and c = ah² + k, which are the standard-form coefficients. So vertex form to standard form is nothing more than doing the squaring and tidying the terms.

A worked example, step by step

Convert y = x² - 6x + 5 to vertex form, with a = 1, b = -6, c = 5.

• h = -b / (2a) = 6 / 2 = 3.
• k = c - b² / (4a) = 5 - 36/4 = 5 - 9 = -4.
• So the vertex is (3, -4), and the vertex form is y = (x - 3)² - 4.

The y-intercept is (0, 5), read straight from the c in the standard form.

Why vertex form is worth having

Vertex form tells you the single most useful point on a parabola without any graphing: the vertex, which is the highest point if the curve opens downward or the lowest if it opens upward. That is the answer to a whole class of real questions. The peak height of a thrown ball, the moment a cost bottoms out, the maximum area for a fixed fence, all of these are vertex problems, and vertex form puts the turning point right there in the equation. The parabola calculator takes it further and maps the rest of the curve.

Questions people ask

What is the vertex?

The turning point of the parabola, its highest or lowest point. In vertex form y = a(x - h)² + k, it is the point (h, k).

How are h and k found?

From the standard-form coefficients: h = -b / (2a) and k = c - b² / (4a). Those coordinates are the vertex.

Which direction do I need?

Standard to vertex if you have a, b, c and want the vertex; vertex to standard if you have the vertex form and want it multiplied out into ax² + bx + c.

Is this the same as completing the square?

Yes. The standard-to-vertex step is exactly completing the square, which is why it also connects to solving the quadratic.

Why can't a be zero?

Because with no x² term there is no parabola, just a straight line, and neither form applies.

References

On the two forms and the vertex. Vertex form places the parabola's turning point (h, k) on display, while standard form shows the y-intercept, and converting between them is the completing-the-square rearrangement.

1. Eric W. Weisstein, "Parabola," from MathWorld, a Wolfram resource, on the parabola and the coordinates of its vertex.
2. Eric W. Weisstein, "Quadratic Equation," from MathWorld, a Wolfram resource, on standard form and completing the square.