anonymous
  • anonymous
How can I find the vertex, focus, directrix and focal width of a parabola with the equation -1/20x^2=y?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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Jack1
  • Jack1
do you know what each of those things are...?
anonymous
  • anonymous
Yes! How do I determine them from the equation though?
Jack1
  • Jack1

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Jack1
  • Jack1
so your vertex is the turning point of your graph, yeah?
anonymous
  • anonymous
Yes.. It would be at the origin in this case
Jack1
  • Jack1
so how did you want to work it out... you could either draw the graph and get your answers from there, or you can do it mathematically
anonymous
  • anonymous
Id like to do it mathematically, so that I can do it without a calculator in case I ever have to. If you could help me figure "P" out then I think I can do it by myself.
Jack1
  • Jack1
and your equation is...? -(1/20) x^2=y or -1/(20x^2) =y ???
anonymous
  • anonymous
The first
Jack1
  • Jack1
k, well that can be re-written as -x^2 / 20 = y so the turning poing of a parabola occurs when the gradient = 0 so what you can do is take the derivative (which give the gradient at any x point) and sub in y=0
Jack1
  • Jack1
so derivative is y = -x/10 so sub in 0 so 0 = -x/10 so x = 0 turning point is a 0,0
anonymous
  • anonymous
So when substitute for that, the answer will be P? In my class our formula for a vertical parabola is x^2=4py
Jack1
  • Jack1
...never heard of P, sorry
Jack1
  • Jack1
what does it let u work out?
anonymous
  • anonymous
Basically everything. For example, if p>0 and the parabola opens downward, then the focus is (h,k+p) and the directrix is y=k-p
anonymous
  • anonymous
Oh! Okay, thank you
Jack1
  • Jack1
so your focus is (0, -4) ...i think
phi
  • phi
you start with \[ -\frac{1}{20} x^2=y \] multiply both sides by -20 : \[ -20 \cdot -\frac{1}{20} x^2= -20y \\ x^2 = -20y\] factor out a 4 from the 20 (i.e. write 20 as 4 * 5): \[ x^2 = 4(-5)y \] as you can see p=-5
anonymous
  • anonymous
Thank you guys!
Jack1
  • Jack1
for the focus, switch the terms so you have x = (something something) y in this case x^2 = -20y so x^2 = 4 *(-5)y P = -5 (sorry, typo before)
anonymous
  • anonymous
No problem!
Jack1
  • Jack1
oooh, shiny!

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