find the equation of the parabola that satisfies the following conditions:?

Question

the vertex is at the origin
it passes through the point (3,1) and the focus lies on the axis

standard form is x=a(y-k)^2+h

sorry the focus lies on the X axis!

Answer 1

If the vertex is at (0,0) and the focus is on the x-axis, then it's a parabola that opens to the left or right, so the general equation you give (with the y being the squared term and not the x) is right: x=a(y-k)^2+h

Using the standard form, the vertex is at (h,k). So those terms are zero, leaving you with. x = ay^2. It includes the point (3,1), so 3 = a(1), and a=3. Thus the equation for the parabola is
x =3y^2.

Answer 2

Because the focus lies on the x axis, it either opens left or right. But if it contains (3,1) with vertex at (0,0), then it opens to the right. So you know ahead of time that your "a" will be positive. (h,k) in your equation is (0,0), while the x and y can be substituted with (3,1). Hence:

3 = a(1-0)^2 + 0
3 = a(1)
a = 3

x = 3(y)^2

If you had to find the coordinates of the focus (a typical question), you use "a" to help you find the distance from vertex to focus. a = 1/(4p) where p is that distance. So

3 = 1/4p --> 12p = 1 ---> p = 1/12
And focus is to the right of the vertex, so it is (1/12, 0)

Answer 3

Silly thing IS lying on its side!! I'll try again... (note to self... read ALL of question before providing answer!)

formula for parabola (lieing on its side!!)
(x - h) = 4a * (y-k)^2

Your vertex is on the origin so h = k = 0 and
x = 4a * y^2

We know that the parabola passes through x = 3, y = 1 so
3 = 4a* 1^2
a = 3/4

Put this back in your original equation:
x = 4a * y^2
x = 4 * 3/4 * y^2
x = 3* y^2

Answer 4

vertex at (a million,2) provides y = a(x - a million) + 2 concentration above vertex tells you a > 0 distance from vertex to concentration, focal length, is two, so a = a million/(4•2) = a million/8, offering you with y = (a million/8)(x - a million) + 2

Answer 5

in terms of y.. the answer is y= sqrt (3x)/ 3