Question

How can I find vertices on a parabola if I have the vertex, focus, directrix, and symmetry line?

Answer 1

vertex: (1,-5)
focus: (0,-5)
Directrix: x=2
Symetry: y=-5

Answer 2

\[-4x+4=y ^{2}+10y+25\]

Answer 3

I get standard form: -4(x-1)=(y+5)squared

Answer 4

@Hero

Answer 5

Im trying to graph it but i think I should find some vertices to help graph it?

Answer 6

The focus is \((0,-5)\). The directrix is \((2,y)\) So insert those points into the equation

\((x - x_1)^2 + (y - y_ 1)^2 = (x - x_2)^2 + (y - y_2)^2\) to get:

\((x - 0)^2 + (y - (-5))^2 = (x - 2)^2 + (y - y)^2\)

Which simplifies to:
\(x^2 + (y + 5)^2 = (x - 2)^2\)

And expands to:
\(x^2 + (y + 5)^2 = x^2 - 4x + 4\)

\(x^2\) cancels on both sides to get:
\((y + 5)^2= -4x + 4\)

Now solving for x you get:
\(-\dfrac{(y + 5)^2}{4} + 1 = x\)

Answer 7

Once you graph it, it should look something like this:
https://www.desmos.com/calculator/46lmaqxorx

Answer 8

Basically, if you have the focus and the directrix, you can find the parabola algebraically.

Answer 9

If you look closely, you'll notice that what you have is the same as what I have.

Answer 10

ok i think i see it lol i was just wondering because im drawing the parabola on graphing paper and I just have the focus and vertex...how do i eyeball the two lines?

Answer 11

can I find some points on the actual parabola line?

Answer 12

Of course. To do that, all you have to do is just create a table of values. Begin with
y = {0,1,2...} then find the corresponding x values.

Answer 13

It would be wise to use your calculator for this activity.

Answer 14

i actually dont have a graphing calculator...

Answer 15

There are online calculators you can use.

Answer 16

can I plug points into the equation itself?

Answer 17

You can, but that would be a long arduous process since it seems that many of the y values will be in fraction form.

Answer 18

You need a way to be able to find the x values quickly.

Answer 19

To make things easier, you should begin with the vertex of the parabola.

Answer 20

so i got points like (0,-3) and (-5,0)

Answer 21

I hope those are correct. Also don't forget the vertex.

Answer 22

thanks!

Answer 23

Mostly to graph it, all you need is the general direction of the graph.

Answer 24

If you have three points, (which it seems you do), then that should be enough.

Answer 25

Your (-5,0) seems to be an approximation rather than an exact point on the graph.

Answer 26

I got the points from a graphing calculator, it seems right i dont know

Answer 27

what does it mean when the vertex is at origin? is that (0,0)?

Answer 28

The vertex is already given to you. It's (1,-5) remember?

Answer 29

oh sorry this is a different problem im looking at...

Answer 30

Yes, that's what it means. If it says "vertex at origin" then that corresponds to the point (0,0)

Answer 31

thanks!