What is the equation, in standard form, of a parabola that contains the following points (-2,-20), (0,-4), (4,-20)?

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Answer 1 · 2016-12-08T01:01:24

See below.

A parabola is a conic and have a structure like

#f(x,y)=a x^2+ b x y +c y^2+d#

If this conic obeys the given points, then

#f(-2,-20) =4 a + 40 b + 400 c + d =0#
#f(0,-4)=16 c + d = 0#
#f(4,-20)=16 a - 80 b + 400 c + d = 0#

Solving for #a,b,c# we obtain

#a = 3d, b=3/10d,c=d/16#

Now, fixing a compatible value for #d# we obtain a feasible parabola

Ex. for #d=1# we get #a=3,b=3/10,c=-1/16# or

#f(x,y) = 1 + 3 x^2 + (3 x y)/10 - y^2/16#

but this conic is a hyperbola!

So the sought parabola has a particular structure as for instance

#y=a x^2+bx+c#

Substituting for the previous values we get the conditions

#{(20 + 4 a - 2 b + c = 0 ),(4 + c = 0),(20 + 16 a + 4 b + c =0):}#

Solving we get

#a=-2,b=4,c=-4#

then a possible parabola is

#y-2x^2+4x-4=0#