Question

What is the equation of the quadratic graph with a focus of (3, 4) and a directrix of y = 8?

Answer 1

Answer:

The equation of the quadratic graph is f(x)= - (1/8) (x-3)^2 + 6 (second option)

Step-by-step explanation:

Focus: F=(3,4)=(xf, yf)→xf=3, yf=4

Directrix: y=8 (horizontal line), then the axis of the parabola is vertical, and the equation has the form:

f(x)=[1 / (4p)] (x-h)^2+k

where Vertex: V=(h,k)

The directix y=8 must intercept the axis of the parabola at the point (3,8), and the vertex is the midpoint between this point and the focus:

Vertex is the midpoint between (3,8) and (3,4):

h=(3+3)/2→h=6/2→h=3

k=(8+4)/2→k=12/2→k=6

Vertex: V=(h,k)→V=(3,6)

p=yf-k→p=4-6→p=-2

Replacing the values in the equation:

f(x)= [ 1 / (4(-2)) ] (x-3)^2 + 6

f(x)=[ 1 / (-8) ] (x-3)^2 +6

f(x)= - (1/8) (x-3)^2 + 6