Write the equation for and graph the parabola? focus:(-6,7) opens:left contains:(-6,5)

1 Answer
Feb 21, 2018

#x = -1/4(y-7)^2 -5#

Explanation:

The vertex form for the equation of a parabola that opens left or right is:

#x = a(y-k)^2 + h#

In a parabola of this type, the focus has the same y coordinate, k, at the vertex. Substitute the k = 7 into the vertex form:

#x = a(y-7)^2 + h#

The distance from the point #(-6,5)# to the focus #(-6,7)# is:

#d = sqrt((-6- -6)^2+ (7-5)^2)#

#d = 2#

The directrix will be a vertical line the same distance from the focus in the positive x direction:

#x = -6+2#

#x = -4#

The x coordinate of the vertex, h, will be at the midpoint between the focus and the directrix:

#h = (-6+ -4)/2 = -5#

Substitute #h = -5# into the vertex form

#x = a(y-7)^2 -5#

We can use the point #(-6, 5)# to find the value of a:

#-6 = a(5-7)^2 -5#

#-6 = 4a -5#

#a = -1/4#

Substitute #a = -1/4# into the vertex form:

#x = -1/4(y-7)^2 -5#

The graph is:

graph{x=-1/4(y-7)^2-5 [-33.95, 2.09, -1.42, 16.61]}