## Equations of Parabolas

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The two most common equations of parabolas are and . In this exploration we will explore the parameters and relationships between the two equations.

We begin our exploration by giving a derivation of the second (commonly known as vertex form) equation from the first (typically known as standard form). We complete this derivation by means of a method known as "completing the square".

Now if we replace the expressions and with h and k respectively we have the vertex form of the equation of a quadratic:

If we were to begin with the vertex form of the parabola and expand the binomial the process would look as follows :

It is easy to see here that the expression for b is equivalent to the expression we derived for h in the first method. Likewise with a little algebra the same can be seen about the expression here for c and the expression for k in the first method.

To explore the parameters and their effects we will look at the two equations along with our expressions from the second derivation for b and c.

- From the standard form it is most easily seen that changing the value of c results in a vertical shift of the graph.

- In vertex form a vertical shift results from a change in k

- Changing the value of a in either equation changes the "openness" of the parabola.

- Changing the value of b in standard form results in a change of vertex

-Changing h in vertex form results in a horizontal shift of the parabola.