Find equation for a parabola with two x points (3,0) (5,0) vertex (4,-1) with no y intercept known

I'd do this myself but I don't know how.

First, look at what you do know.
First, a parabola is usually of the form of a quadratic.
Next, you know the x-intercepts. These are the roots so y = a(x-3)(x - 5)
You know the vertex (which is on the parabola)
so: -1 = a(4 - 3)(4 -5) = -a so a = 1
so, the equation is
y = (x-3)(x-5) = x^2 -8x +15 = (x -4)^2 -1

or...
You know that the vertex form of a parabola is y = a(x -h)^2 + k

where (h,k) is the vertex
so,
y = a(x -4)^2 -1

so using any of the known points:
0 = a(3 -4)^2 -1
==> a -1 = 0
==> a = 1
so:
y = (x-4)^2 - 1 = x^2 -8x + 15 = (x -3)(x-5)

Use whichever method is easier for you!

y = a(x – 3)(x – 5)

At (4, -1), -1 = a(1)(-1) = -a, so a = 1

y = x² – 8x + 15