What is the equation of the parabola that has a vertex at # (0, 0) # and passes through point # (-1,-4) #?

2 Answers
Jim G.
Jul 11, 2018

#y=-4x^2#

Explanation:

#"the equation of a parabola in "color(blue)"vertex form"# is.

#•color(white)(x)y=a(x-h)^2+k#

#"where "(h,k)" are the coordinates of the vertex and a"#
#"is a multiplier"#

#"here "(h,k)=(0,0)" thus"#

#y=ax^2#

#"to find a substitute "(-1,-4)" into the equation"#

#-4=a#

#y=-4x^2larrcolor(blue)"equation of parabola"#
graph{-4x^2 [-10, 10, -5, 5]}

Harish Chandra Rajpoot
Jul 11, 2018

#x^2=-1/4y\quad # or #\quad y^2=-16x#

Explanation:

There are two such parabola satisfying the given conditions as follows

Case 1: Let the vertical parabola with the vertex at #(0, 0)# be

#x^2=ky#

since, above parabola passes through the point #(-1, -4)# then it will satisfy the above equation as follows

#(-1)^2=k(-4)#

#k=-1/4#

hence setting #k=-1/4#, the equation of vertical parabola

#x^2=-1/4y#

Case 2: Let the horizontal parabola with the vertex at #(0, 0)# be

#y^2=kx#

since, above parabola passes through the point #(-1, -4)# then it will satisfy the above equation as follows

#(-4)^2=k(-1)#

#k=-16#

Now, setting #k=-16#, the equation of vertical parabola

#y^2=-16x#

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