What is the vertex form of a parabola with a focus of (-2,-3) and a directrix of y=5?

Answer 1

the equation of the parabola is y-1=(-1/16)(x+2)² where the vertex is (-2,1), a=-1/16

the vertex form of a parabola is: y-k=a(x-h)² where (h,k) is the vertex

The vertex is the midpoint between the focus and directrix. Just find the midpoin.

The y-coordinate of the focus is -3. The directrix is 8 above it.

5 - (-3) = 5+3=8

Half of 8 is 4, for the midpoint. Subtract 4 from 5

5-4=1

Or add 4 to -3

4-3=1

1 is the y-coordinate of the vertex.

The vertex has the same x-coordinate as the focus when the parabola opens up or downward as this one does. the x-coordinate of the focus and vertex is -2.

The vertex is (-2,1)

The parabola is y-k=a(x-h)² with (h,k)=(-2,1)

or y-1 = a(x+2)² All you need now is the value of a, the coefficient of the quadratic term

any point on the parabola is equidistant from the focus and the directrix line

take the point (x,-3) which is horizontal from the focus to the right, and a distance from the focus equal to x-(-2) = x+2

x+2 also equals the distance from (x,-3) to the directrix line, which is 5-(-3) = 8

x+2=8

x+2= distance from (x,-3) to the focus

8 = distance from (x-3) to y=5 the directrix line

x+2=8

x=8-2=6

(6,-3) is a point on the parabola.

Plug it into y-1=a(x+2)² and solve for a

-3-1 = a(6+2)² = a(8)² or 64a = -4

a=-4/64=-1/16

the parabola is

y-1=(-1/16)(x+2)² in vertex form

where the vertex is (-2,1)

a<0 when the parabola opens downward. When the directrix is above the focus, the parabola opens downward.

a=-1/16 has an absolute value relatively close to zero. The smaller the absolute value, the flatter the parabola. the larger the absolute value of a, the more narrow the parabola. If a is very large, the parabola looks as if it is approaching a vertical line in shape. If the directrix line is very close to the focus, or approaches it, the parabola approaches a vertical line.

Pick another point on the parabola, (x,y) = (-10,-3)

plug that into the equation, to get -3-1=(-1/16)(-10+2)²=-64/16=-4

The parabola is symmetric about the vertical line through the focus or vertex, x=-2

since (6,3) was on the parabola, we know (-10,3) is also, as it's distance 8 from x=-2, just on the left side of the line of symmetry, while (6,3) is 8 to the right of x=-2.