Parabolas - graphing and writing equations (1 of 3)

I introduce the two equations for parabolas: if x is squared then it opens up (positive) or down (negative) and if y is square then it opens right (positive) or left (negative). The distance (p) from the vertex to the focus and the vertex to the directrix are the same, it helps you find the "a" value: a=1/(4p). These parabolas can be shifted off of the origin (0, 0) with a transformation where (h, k) is going to be the vertex.

There are essentially two kinds of questions:
1. Given an equation, graph and/or find certain characteristics (vertex, focus, directrix, and axis of symmetry).
2. Given certain characteristics (vertex, focus, directrix, and axis of symmetry), write the equation of the parabola.

[davidtutorsmath]

@JaeMaJic I'm assuming because you're saying it hasn't been given that you don't have the equation. If you have points or a graph, let's let "p" be the distance between the vertex and focus (it's also the distance between vertex and directrix). a=1/(4p). The equation is y=ax^2 or x=ay^2. depending on which way it opens. So, once you know p, plug it in to a=1/(4p) to find a. Make sure to make "a" negative if it opens to the left for y^2 equations and down for x^2 equations.

[davidtutorsmath]

@tcdude12 You're welcome. I'm glad you found it helpful.

[davidtutorsmath]

@JaeMaJic If you don't know what the focus/directrix is, you can place them anywhere and it will simply adjust how thin or wide the parabola is. Given your focus and directrix, you know the parabola opens up (always toward focus) an that p=2. So the equation is y=1/(4*2)x^2 and simplified y=1/8x^2. Does that answer your question?

[JaeMaJic]

@davidtutorsmath Thank you that was really helpful =]. I have one last question, since my equation is not given can I just choose where I put the directrix and focus, as long as they are both equal distance from the vertex? [ my vertex is (0, 0), directrix y=-2, focus: (0, 2)

[JaeMaJic]

how do you know what a is, if it isn't already given

[JaeMaJic]

@davidtutorsmath that answered my question[ (y=1/8x^2) is the answer i got after you told me that a=1/(4p)] but your answer created another question....I think i'm just going to have to ask my teacher.
Thanks for your help.