# Thread: Finding the vertex at a given point?

1. ## Finding the vertex at a given point?

Find b and c so that the parabola has vertex .

What do I do here?

2. ## Re: Finding the vertex at a given point?

Use completing the square:
Convert $y$ to $y=(x-a)^2+q$
Where $(a,q)$ are the coordinates of the vertex.

3. ## Re: Finding the vertex at a given point?

Originally Posted by habibixox
Find b and c so that the parabola has vertex .

What do I do here?
Hi habibixox,

At the vertex of the parabola the first derivative of y becomes zero. So calculate, $\frac{dy}{dx}$ and equate it to zero. Since the corresponding x value at the vertex is given you can find b. I hope you can do it now.

4. ## Re: Finding the vertex at a given point?

wait so is it
x^2-18x+91?

5. ## Re: Finding the vertex at a given point?

using the second method the derivative is
-8x + x right?
set that = 0

0= -8x + x
0= X(-8+1)

6. ## Re: Finding the vertex at a given point?

nvm b = 72 now im working on c ill get it i understand it...

7. ## Re: Finding the vertex at a given point?

Originally Posted by habibixox
using the second method the derivative is
-8x + x right?
set that = 0

0= -8x + x
0= X(-8+1)
No.

$y=-4x^2+bx+c$

$\frac{dy}{dx}=-4(2x)+b(1)+0$

$\frac{dy}{dx}=-8x+b$

Hope you can continue.

8. ## Re: Finding the vertex at a given point?

The method I was thinking about is the following.
Given is the parabola: $y=-4x^2+bx+c$ and the vertex point with coordinates $(9,10)$
In general we know we can directly read the coordinates of the vertex of a parabola if we convert $y$ to:
$y=a(x-h)^2+q$ where $(h,q)$ are the coordinates of the parabola.

First, divide every term by $-4$ so:
$\frac{y}{-4}=x^2-\frac{b}{4}x-\frac{c}{4}$
Use completing the square in the RHS:
$\frac{y}{-4}=\left(x-\frac{b}{8}\right)^2-\frac{b^2}{64}-\frac{c}{4}$
$\Leftrightarrow y=-4\left(x-\frac{b}{8}\right)^2+\frac{b^2}{16}+c$

That means:
(1) $\frac{b}{8}=9 \Leftrightarrow b=72$ (1)
(2) $\frac{b^2}{16}+c=10$ (2)
Substituting (1) in (2) gives:
$324+c=10 \Leftrightarrow c=-314$

Therefore the parabola with vertex $(9,10)$ is:
$y=-4x^2+72x-314$

But as you see, the method sudharaka wants to use is easier.