Put A Parabola in Vertex Form Calculator

The vertex form of a parabola is a standard way to represent quadratic equations that makes it easy to identify key features like the vertex, axis of symmetry, and direction of opening. This calculator helps you convert any parabola equation 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) is 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 graph parabolas because you can immediately identify the vertex and the parabola's direction from the equation.

How to Convert to Vertex Form

To convert a quadratic equation from standard form (y = ax² + bx + c) to vertex form, follow these steps:

Factor the coefficient of x² from the first two terms.
Complete the square for the expression inside the parentheses.
Rewrite the equation in vertex form.

Example: Convert y = 2x² + 8x + 5 to vertex form.

Factor: y = 2(x² + 4x) + 5
Complete the square: x² + 4x + 4 - 4 = (x + 2)² - 4
Final form: y = 2[(x + 2)² - 4] = 2(x + 2)² - 8

For more complex equations, the process is similar but may require more steps to complete the square correctly.

Worked Example

Let's convert the equation y = -3x² + 12x - 5 to vertex form:

Factor the coefficient of x²: y = -3(x² - 4x) - 5
Complete the square inside the parentheses:
- Take half of -4: -2
- Square it: 4
- Add and subtract 4 inside the parentheses: y = -3(x² - 4x + 4 - 4) - 5
- This becomes: y = -3[(x - 2)² - 4] - 5
Distribute the -3: y = -3(x - 2)² + 12 - 5
Combine constants: y = -3(x - 2)² + 7

The vertex form is y = -3(x - 2)² + 7, with vertex at (2, 7).

FAQ

What is the vertex form used for?: The vertex form makes it easy to identify the vertex, axis of symmetry, and direction of opening of a parabola, which is useful for graphing and analyzing quadratic functions.
Can all parabolas be written in vertex form?: Yes, any quadratic equation can be rewritten in vertex form by completing the square.
How do I know if a parabola opens up or down?: The sign of the coefficient 'a' in vertex form determines the direction. If a is positive, the parabola opens upwards; if negative, it opens downwards.
What if my equation has a fraction or decimal coefficient?: You can still complete the square, but it may require more steps. The calculator handles these cases automatically.
Can I use this calculator for 3D parabolas?: This calculator works for 2D quadratic equations only. For 3D parabolas, you would need a different tool.