What is the equation of the parabola that has a vertex at # (-18, -12) # and passes through point # (-3,7) #?

2 Answers
Apr 9, 2017

#y=19/225(x+18)^2-12#

Explanation:

Use the general quadratic formula,

#y=a(x-b)^2+c#

Since the vertex is given #P(-18,-12)#, you know the value of #-b# and #c#,

#y=a(x--18)^2-12#
#y=a(x+18)^2-12#

The only unkown variable left is #a#, which can be solved for using #P(-3,7)# by subbing #y# and #x# into the equation,

#7=a(-3+18)^2-12#
#19=a(15)^2#
#19=225a#
#a=19/225#

Finally, the equation of the quadratic is,

#y=19/225(x+18)^2-12#

graph{19/225(x+18)^2-12 [-58.5, 58.53, -29.26, 29.25]}

There are two equations that represent two parabolas that have the same vertex and pass through the same point. The two equations are:

#y =19/225(x+18)^2-12# and #x = 15/361(y+12)^2-18#

Explanation:

Using the vertex forms:

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

Substitute #-18# for #h# and #-12# for #k# into both:

#y =a(x+18)^2-12# and #x = a(y+12)^2-18#

Substitute #-3# for #x# and 7 for #y# into both:

#7 =a(-3+18)^2-12# and #-3 = a(7+12)^2-18#

Solve for both values of #a#:

#19=a(-3+18)^2# and #15 = a(7+12)^2#

#19=a(15)^2# and #15 = a(19)^2#

#a = 19/225# and #a = 15/361#

The two equations are:

#y =19/225(x+18)^2-12# and #x = 15/361(y+12)^2-18#

Here is a graph of the two points and the two parabolas:

enter image source here