Put Equation Into Vertex Form Calculator

Converting a quadratic equation to vertex form is a fundamental algebra skill that helps you understand the graph of the equation and find key features like the vertex and axis of symmetry. This calculator makes the process quick and easy, but understanding the steps behind it will help you use the calculator more effectively.

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. This form makes it easy to identify the vertex and the direction the parabola opens (upwards if a is positive, downwards if a is negative).

Vertex form is particularly useful because it provides immediate information about the graph's highest or lowest point and its direction. This is in contrast to standard form (y = ax² + bx + c) which requires more calculations to find these features.

How to Convert to Vertex Form

Converting from standard form to vertex form involves completing the square, a process that transforms the quadratic equation into a perfect square trinomial. Here are the general steps:

Start with the standard form equation: 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 by adding and subtracting (b/2a)²
Rewrite the perfect square trinomial as a squared binomial
Factor out the coefficient a from the squared binomial and the remaining terms
The equation is now 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. This is why the first step involves factoring out 'a' from the first two terms.

Vertex Form Formula

To convert y = ax² + bx + c to vertex form:

Factor out 'a' from the first two terms: y = a(x² + (b/a)x) + c
Complete the square: x² + (b/a)x + (b/2a)² - (b/2a)²
Rewrite as perfect square: (x + b/2a)² - (b/2a)²
Combine terms: y = a[(x + b/2a)² - (b²/4a²)] + c
Distribute 'a': y = a(x + b/2a)² - ab²/4a² + c
Simplify: y = a(x - (-b/2a))² + (c - b²/4a)

The vertex form is y = a(x - h)² + k, where:

h = -b/2a (x-coordinate of the vertex)
k = c - b²/4a (y-coordinate of the vertex)

Example Conversion

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

Factor out 2: y = 2(x² + 4x) + 5
Complete the square: x² + 4x + 4 - 4
Rewrite: (x + 2)² - 4
Combine: y = 2[(x + 2)² - 4] + 5
Distribute: y = 2(x + 2)² - 8 + 5
Final vertex form: y = 2(x + 2)² - 3

The vertex 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 shows the direction the parabola opens and the axis of symmetry.

What if the coefficient of x² is not 1?

You must factor out the coefficient first, then complete the square. The calculator handles this automatically for any quadratic equation.

Can I convert vertex form back to standard form?

Yes, you can expand the squared binomial and combine like terms to get back to standard form.