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Vertex Form of a Parabola: Completing the Square

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Vertex Form of a Parabola: Completing the Square

In mathematics, a parabola is a symmetrical curve that is shaped like a U. The vertex of a parabola is the point where the curve changes direction. The vertex form of a parabola is a way of writing the equation of a parabola that makes it easy to find the vertex and other important features of the parabola.

What is Vertex Form?

The vertex form of a parabola is given by:

y = a(x - h)^2 + k

where:

  • a is the coefficient of the x^2 term
  • (h, k) is the vertex of the parabola

How to Write a Quadratic Equation in Vertex Form

To write a quadratic equation in vertex form, we need to complete the square. Here are the steps:

  1. Factor out the coefficient of the x^2 term.
  2. Take half of the coefficient of the x term, square it, and add it to both sides of the equation.
  3. Factor the perfect square trinomial.
  4. Move the constant term to the right side of the equation.

Example

Let's write the equation y = x^2 - 6x + 5 in vertex form.

  1. Factor out the coefficient of the x^2 term (which is 1 in this case):

    2. y = (x^2 - 6x) + 5

  2. Take half of the coefficient of the x term (-6), square it ( (-6/2)^2 = 9), and add it to both sides of the equation:

    3. y = (x^2 - 6x + 9) + 5 - 9

  3. Factor the perfect square trinomial:

    4. y = (x - 3)^2 - 4

  4. Move the constant term to the right side of the equation:

    5. y = (x - 3)^2 + (-4)

Therefore, the vertex form of the equation y = x^2 - 6x + 5 is y = (x - 3)^2 - 4.

Why is Vertex Form Important?

The vertex form of a parabola is important because it makes it easy to:

  • Find the vertex of the parabola: The vertex is at the point (h, k).
  • Determine the direction of the parabola: If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards.
  • Find the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex.

Conclusion

Completing the square is a powerful technique that allows us to rewrite quadratic equations in vertex form. This form provides valuable information about the parabola, such as its vertex, direction, and axis of symmetry. Understanding vertex form is essential for solving quadratic equations and analyzing parabolic functions.

Parabolas