Put Function in Vertex Form Calculator

What is Vertex Form?

The vertex form of a quadratic function is written as:

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)

Vertex form is particularly useful because it immediately reveals the vertex coordinates and the parabola's direction. This makes it easier to graph the function and understand its behavior.

How to Convert to Vertex Form

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

1. Start with the standard form: ax² + bx + c
2. Factor out the coefficient of x² from the first two terms: a(x² + (b/a)x) + c
3. Complete the square inside the parentheses:
   • Take half of the coefficient of x: (b/a)/2
   • Square this value: [(b/a)/2]²
   • Add and subtract this squared value inside the parentheses
4. Rewrite the perfect square trinomial as a squared binomial: a[(x + (b/a)/2)² - (b²/a²)/4] + c
5. Distribute the a and simplify: a(x + (b/a)/2)² - (b²/4a) + c
6. Combine the constant terms: a(x + (b/a)/2)² + (c - b²/4a)

Worked Example

Let's convert the quadratic function f(x) = 2x² + 8x + 3 to vertex form.

1. Start with: 2x² + 8x + 3
2. Factor out the coefficient of x²: 2(x² + 4x) + 3
3. Complete the square:
   • Half of 4 is 2
   • Square of 2 is 4
   • Add and subtract 4 inside the parentheses: 2(x² + 4x + 4 - 4) + 3
4. Rewrite as perfect square: 2[(x + 2)² - 4] + 3
5. Distribute the 2: 2(x + 2)² - 8 + 3
6. Combine constants: 2(x + 2)² - 5

The vertex form is f(x) = 2(x + 2)² - 5, with vertex at (-2, -5).