Finding the vertex at a given point?

**habibixox** · September 21st 2011, 06:31 AM

Find b and c so that the parabola

has vertex

.

What do I do here?

**Siron** · September 21st 2011, 06:34 AM

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

**Sudharaka** · September 21st 2011, 06:39 AM

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.

**habibixox** · September 21st 2011, 03:05 PM

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

**habibixox** · September 21st 2011, 03:07 PM

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

0= -8x + x
0= X(-8+1)
so final answer is 0?

**habibixox** · September 21st 2011, 03:22 PM

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

i understand it...

**Sudharaka** · September 21st 2011, 06:18 PM

Originally Posted by habibixox

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

0= -8x + x
0= X(-8+1)
so final answer is 0?

No.

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

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

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

Hope you can continue.

**Siron** · September 21st 2011, 10:29 PM

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.

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