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Put Something Into Vertex Form Calculator

Reviewed by Calculator Editorial Team

Vertex form is a way to express quadratic equations that makes it easy to identify key features like the vertex and axis of symmetry. This calculator helps you convert standard 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) 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 directly reveals the vertex of the parabola, which is the maximum or minimum point depending on the value of a.

How to Convert to Vertex Form

Converting a quadratic equation from standard form (y = ax² + bx + c) to vertex form involves completing the square. Here's the step-by-step process:

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

Note: If the coefficient of x² (a) is negative, the parabola opens downward. The vertex will be the maximum point of the parabola.

Formula

The vertex form conversion formula is:

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 - (b²/4a)

Worked Example

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

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

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

FAQ

What is the vertex form used for?
Vertex form is used to easily identify the vertex of a parabola, which is the maximum or minimum point, and to understand the parabola's width and direction.
Can all quadratic equations be converted to vertex form?
Yes, any quadratic equation in the form y = ax² + bx + c can be converted to vertex form using the completing the square method.
How do I know if a parabola opens up or down?
If the coefficient 'a' in vertex form is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.