# Parabola question

## Main Question or Discussion Point

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.

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

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

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

...?

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

HallsofIvy
Homework Helper
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?