Vertex Form of a Quadratic Equation
Quadratic equations written in vertex form provide readily available information about the the parabola it represents on a graph. The variables h and k form a coordinate pair (h, k) and represent the vertex of the parabola. The vertex is the maximum or minimum point to the parabola (depending on if the graph opens upward or downward). Using the vertex form of quadratic equations can be ideal for graphing parabolas, because we can easily identify the vertex, plot points on one side of the vertex, and then reflect them across the axis of symmetry, on which the vertex lies.
Compare Standard Form to Vertex Form
The standard form of quadratic equations expresses the quadratic in expanded terms, containing an x-squared term, an x-term, and a constant term. On the other hand, equations in vertex form resemble factored form of a quadratic, with a linear binomial (x – h) raised to the second power.
We can use a process called completing the square to write an expanded quadratic as a factor squared. In order to do this, our expanded quadratic must be a perfect square trinomial. In other words, half of the x-term coefficient squared is equal to the constant term. Here is an example:
We were able to rewrite x2 + 6x + 9 as (x + 3)2 because we recognized that if we halved the x-term coefficient, and then squared it, the result would be the constant term. Half of the x-term coefficient then becomes the value that accompanies x in the binomial squared. (x + a)2
Convert a Quadratic from Standard Form into Vertex Form
In order to rewrite a quadratic equation from standard form to vertex form, we need be able to recognize the perfect square trinomial relationship. Often times, this isn't readily provided to us, and we need to do some algebraic manipulation, with a process known as completing the square. You may have had some practice completing the square when learning about solving or factoring quadratic equations. To complete the square within this context, we perform the following steps:
There are quite a few steps involved in this process, so we are going to take a look at an example step-by step.
Notice when we added 16 in the 4th step. This quantity was found by dividing 8 by 2, and then squaring it. We added it within the parentheses, with a factor of 2 outside. Due to this outside factor of 2, we actually added 32 to that side of the equation, not 16. This is why we see + 32 on the left side of the equation. When performing this step, it is important to multiply this quantity by the outside factor when adding a quantity on the other side to keep the equation balanced.
Now we have an equation equivalent to the equation given to us in standard from. If we were to graph the two equation, we would get the same parabola. However, you may prefer to work with the equation in vertex form, this form gives the coordinates of the vertex (h, k).