Raymond B. answered • 04/14/20
Math, microeconomics or criminal justice
Plot the 2 points. the focus is to the left of the vertex. It's a parabola opening to the left
general form is x-h = a(y-k)^2
where (h,k) is the vertex, so x-5 = a(y-1)^2
all you need now is a value for a, based on the focus
the focus is the same distance from the vertex as the directrix, so the directix is x=8
the focus is 3 to the left of the vertex, so the directrix is 3 to the right, 5+3 = 8, x=8
line of symmetry is y=1
The parabola is the collection of all points equidistant from focus and directrix
(x,y) is equally distant from (2,1) as it is from x=8
take the point on the parabola directly above the focus, it's (2,y). It's also 6 away from the directrix
That point directly above the focus (2,y) is a distance of y-1 = 6
y=7
A point on the parabola is (2,7)
the general form of the parabola was
led to x-5 = a(y-1)^2. Substitute the point (2,7) to get a
2-5 = a(7-1)^2
-3 = a(36)
a = -3/36 = -1/12
the equation for the parabola is now
x-5 = (-1/12)(y-1)^2
To simplify it, maybe multiply by 12 to get
2x-10 = -(y + 1)^2 = -(y^2 +2y + 1) = -y^2 -2y -1
2x = y^2 -2y +9
divide by 2
x = (1/2)y^2 -y + 9/2
or
x = (1/2)(y^2 -y + 9)
x = (1/2)(y-1)^2 +5
where vertex = (5,1)
That's a general method. Mistakes along the way are easy.
