Put Equation in Vertex Form Calculator

Quadratic equations are commonly written in standard form (ax² + bx + c = 0), but vertex form (y = a(x - h)² + k) provides valuable information about the parabola's vertex and symmetry. This calculator helps you convert quadratic equations to vertex form quickly and accurately.

What is Vertex Form?

The vertex form of a quadratic equation is written as y = a(x - h)² + k, where:

(h, k) represents the vertex of the parabola
a determines the parabola's width and direction (upwards if a > 0, downwards if a < 0)
The vertex form makes it easy to identify key features of the parabola

Converting to vertex form is particularly useful for graphing parabolas and solving quadratic equations.

How to Convert to Vertex Form

Converting a quadratic equation from standard form to vertex form involves completing the square. Here's the step-by-step process:

Start with the standard form: y = ax² + bx + c
Factor out the coefficient of x² from the first two terms: y = a(x² + (b/a)x) + c
Complete the square inside the parentheses:
- Take half of the coefficient of x
- Square it
- Add and subtract this squared term inside the parentheses
Rewrite the equation as a perfect square trinomial and simplify
The result will be in vertex form: y = a(x - h)² + k

Note: If the coefficient of x² (a) is not 1, you'll need to factor it out before completing the square.

Vertex Form Formula

The vertex form of a quadratic equation is derived from the standard form through completing the square:

y = ax² + bx + c y = a(x² + (b/a)x) + c y = a[(x + b/(2a))² - (b²/(4a²))] + c y = a(x + b/(2a))² - (ab²)/(4a²) + c y = a(x - h)² + k where h = -b/(2a) and k = c - (ab²)/(4a²)

This formula shows how to transform any quadratic equation into vertex form by completing the square.

Example Conversion

Let's convert the equation y = 2x² + 8x + 5 to vertex form:

Start with: y = 2x² + 8x + 5
Factor out the 2: y = 2(x² + 4x) + 5
Complete the square:
- Half of 4 is 2
- Square of 2 is 4
- Add and subtract 4 inside the parentheses
Rewrite: y = 2(x² + 4x + 4 - 4) + 5 = 2((x + 2)² - 4) + 5
Distribute the 2: y = 2(x + 2)² - 8 + 5 = 2(x + 2)² - 3
Final vertex form: y = 2(x - (-2))² - 3

The vertex of this parabola is at (-2, -3).

FAQ

Why is vertex form important?

Vertex form makes it easy to identify the vertex of a parabola, which is the maximum or minimum point. It also helps with graphing and solving quadratic equations.

Can all quadratic equations be converted to vertex form?

Yes, any quadratic equation in the form ax² + bx + c can be converted to vertex form through completing the square.

What if the coefficient of x² is not 1?

If the coefficient of x² is not 1, you must factor it out before completing the square. The process remains the same but requires an extra step.

How do I know if I've completed the square correctly?

You can verify by expanding your vertex form equation and checking if it matches the original standard form equation.