Parabola Question: Verifying Solution with Multiple Methods

[sourpatchkid]

mods please move this topic if this is not in the correct section. thanks. :)

the question is:
the vertices of the family of parabolas y = x^2 + bx, b is constant, lie on a single parabola. Find equation for that parabola.

my teacher require me to provide supporting details & background info that back up my answer and have to verify it using a different method. I'm really puzzled and i would greatly appreciate any help that comes my way. thanks in advance.

 

[DavidWhitbeck]

Well why don't you start by showing what you've done on the problem, and we'll take it from there.

 

[HallsofIvy]

You might want to start by completing the square so it's easy to see where the vertices are.

 

[sourpatchkid]

so this is what I've gotten so far, please correct me if I'm wrong.

for any generalized parabola, the equation is given in the standard form: y = ax^2 + bx + c

a = 1 (in the equation y = x^2 + bx)
b = constant
c = (b/2)^2

the equation for the parabola that the question asks for, if written as completing the square, should be: y - k = (x - h)^2.
we need to find the center (h, k) so that y = x^2 + bx.

y - k = (x - h)(x - h)
y - k = x^2 - 2xh + h^2

...?

 

[DavidWhitbeck]

$$x^2+bx=(x+b/2)^2-b^2/4$$ check it and then see if that's helpful for you.

 

[HallsofIvy]

so this is what I've gotten so far, please correct me if I'm wrong.

for any generalized parabola, the equation is given in the standard form: y = ax^2 + bx + c

a = 1 (in the equation y = x^2 + bx)

Yes, but not in this equation! Would it be easier to write it as $y= a(x^2+ (b/a)x)+ c$? How would you complete the square for $a(x^2+ (b/a)x$?

b = constant
c = (b/2)^2

the equation for the parabola that the question asks for, if written as completing the square, should be: y - k = (x - h)^2.
we need to find the center (h, k) so that y = x^2 + bx.

y - k = (x - h)(x - h)
y - k = x^2 - 2xh + h^2

...?

Why, after completing the square, did you then ignore it?