Question

So I've tried looking for a lot of examples online and couldn't find any to get me jump started on a practice problem that I've no idea to do.
I'm suppose to find the focus,vertex, directrix and axis of symmetry of the parabola with the equation Y = x^2 + 3 but no idea how if anyone could actually walk me through the steps so I can do others I would be very thankful.

Answer 1

@Lyo
I think it would help to look at the relationship of these "parabola parts" which you are asked to find.

I'll attach a diagram.

Here is the link to the diagram if you want to see it in context.
http://hotmath.com/hotmath_help/topics/directrix.html

Answer 2

So, you see what we are asked to find.

Tell me, do you have a general equation of a parabola that you have been using in your class or one that you are supposed to use with this problem?

Answer 3

Thank you, I am just starting on parabolas, I am not sure if it's what you're asking for but I was given two equations: (y - k)^2 = 4p(x - h) and (x -h )^2 = 4p(y - k)
I am currently reading over the link you sent me. The problem doesn't require a specific equation as long as I can show my work and provide accurate answers.

Answer 4

Your given equation y = x^2 + 3 has the x term squared so use the general form that has a linear y and a squared x term. I have posted it below.

I turned it around (symmetric property) to the way I like to look at it.
I also divided both sides by (4p).

I showed the steps below.

(x -h )^2 = 4p(y - k)

4p(y - k) = (x -h )^2

(y - k) = (1/(4p) ) * (x -h )^2

Answer 5

So, we are starting with this:

(y - k) = (1/(4p) ) * (x -h )^2

and comparing y = x^2 + 3 with it.

Answer 6

(y - k) = (1/(4p) ) * (x -h )^2

y - 3 = 1 * (x - 0)^2

(h,k) is the vertex so the vertex of the equation you were given is (0,3).

@Lyo Do you agree with the vertex coordinates? Actually, we did not need this bulky formula to know that.

Answer 7

From what I understand, Yes I agree but I do have a question what are we putting in for 1/4p to get 1, I understand everything else being plugged in just not that.

Answer 8

We are doing all that but because you posted that you had been reading examples and still were not quite sure what to do, I thought we would do this problem one step at a time. So, we will be getting to the value of p.

But, without the vertex, the value of p will not help with the directrix and focus.

Answer 9

One more comment about the vertex.

We could have taken this: Y = x^2 + 3 and noticed that the vertex had to be at (0,3) because the smallest x can be is 0 and then the corresponding y is 3. So, the vertex is a minimum point on the graph and the parabola opens upward.

Answer 10

Going back to this:

(y - k) = (1/(4p) ) * (x -h )^2

y - 3 = 1 * (x - 0)^2

Notice that I placed a "one" in front of the (x-0)^2.
That does not change the given equation.

Answer 11

1/(4p) = 1

So, 4p = 1
p = 1/4

@Lyo It is very important that you understand how the value of p was computed. So, look at it and see if you agree that p = 1/4 or if you have questions.

Answer 12

I understand it, to get P we have to isolate it from everything else right?

Answer 13

Yes, after getting the given equation in the form so that we know the coefficient of the squared x component. That just happened to 1 this time.

Answer 14

Recall the definition of a parabola:

A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix.

Answer 15

The vertex is a point on the parabola.
So, it is equidistant from the directrix and the focus.

Answer 16

The focus and the vertex lie on the same vertical line.
In this case, that line is x = 0. From the vertex (0,3) move up 1/4 (.25) units to get the coordinates of the focus to be (0,3.25).