Write an equation for the parabola with focus (1,3) and vertex (0,3).

I cannot find any example in my book that works a problem with the vertex, all of the examples that I have to go by give only the focus and directrix.

Thanks.

Since this parabola has vertex to the left and at the same level as the focus, it opens right and is of form:

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

the vertex is at (h,k)
distance from vertex to focus = a
distance from vertex to directrix = a

here, vertex at (0,3)
so (y-3)^2 = 4 a (x)
since a is distance from vertex to focus
(y-3)^2 = 4*1 x = 4 x
so
x = (1/4)(y^2 - 6 y + 9)

No worries! I can help you with that.

To find the equation of a parabola given the focus and vertex, we can use the formula:

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

Where (h, k) is the vertex, and p is the distance between the focus and the vertex.

In this case, the vertex is (0, 3), which means h = 0 and k = 3.

To find p, we need to calculate the distance between the focus (1, 3) and the vertex (0, 3). The distance formula is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the given values into the distance formula, we have:

p = sqrt((1 - 0)^2 + (3 - 3)^2)
p = sqrt(1 + 0)
p = 1

Now that we know h = 0, k = 3, and p = 1, we can substitute these values into the formula and simplify:

(x - 0)^2 = 4(1)(y - 3)
x^2 = 4y - 12

So, the equation of the parabola with focus (1, 3) and vertex (0, 3) is x^2 = 4y - 12.