# Writing the vertex of a parabola

• Aug 10th 2011, 11:23 AM
David Green
Writing the vertex of a parabola
Just like a second opinion please.

I have a parabola y = x^2 - 10x - 8

I have the equation y = a(x - h)^2 + K

where (h, k) are the vertex

am I right to change the signs and write;

h = 10
k = 8

so the vertex of y = x^2 - 10x - 8

= (10, 8)

Thanks
• Aug 10th 2011, 11:26 AM
Siron
Re: Writing the vertex of a parabola
You've to convert \$\displaystyle y=x^2-10x-8\$ into the form \$\displaystyle y=a(x-h)^2+k\$.
In you other topic you used the technique 'completing the square', use this technique for this exercice.

Notice:
The vertex \$\displaystyle V(10,8)\$ is the vertex of the parabola: \$\displaystyle y=(x-10)^2+8=x^2-20x+108\$.
• Aug 10th 2011, 11:38 AM
David Green
Re: Writing the vertex of a parabola
Quote:

Originally Posted by Siron
You've to convert \$\displaystyle y=x^2-10x-8\$ into the form \$\displaystyle y=a(x-h)^2+k\$.
In you other topic you used the technique 'completing the square', use this technique for this exercice.

Notice:
The vertex \$\displaystyle V(10,8)\$ is the vertex of the parabola: \$\displaystyle y=(x-10)^2+8=x^2-20x+108\$.

Thanks for that, so if I already have the equation and the parabola, I can just read straight from the parabola and apply it to the h and k, i.e the vertex (10, 8).
• Aug 10th 2011, 11:41 AM
Siron
Re: Writing the vertex of a parabola
If you've a polynomial in the form:
\$\displaystyle y=a(x-h)^2+k\$
then like you said, you can determine the coordinates of the vertex directly: (h,k)

If you've a polynomial in the form:
\$\displaystyle y=ax^2+bx+c\$
then you've to convert this polynomial into the form \$\displaystyle y=a(x-h)^2+k\$

In case of your exercice, you have given a polynomial in the form \$\displaystyle y=ax^2+bx+c\$ so convert it to the other form (by using 'completing the square, like you did in your previous topic)

So, can you write:
\$\displaystyle y=x^2-10x-8\$ into the form: \$\displaystyle y=a(x-h)^2+k\$?
• Aug 10th 2011, 11:52 AM
David Green
Re: Writing the vertex of a parabola
Quote:

Originally Posted by Siron
If you've a polynomial in the form:
\$\displaystyle y=a(x-h)^2+k\$
then like you said, you can determine the coordinates of the vertex directly: (h,k)

If you've a polynomial in the form:
\$\displaystyle y=ax^2+bx+c\$
then you've to convert this polynomial into the form \$\displaystyle y=a(x-h)^2+k\$

In case of your exercice, you have given a polynomial in the form \$\displaystyle y=ax^2+bx+c\$ so convert it to the other form (by using 'completing the square, like you did in your previous topic)

So, can you write:
\$\displaystyle y=x^2-10x-8\$ into the form: \$\displaystyle y=a(x-h)^2+k\$?

If I am right, then;

y = x^2 - 10x + 8 = x(x - 10)^2 - 8
• Aug 10th 2011, 11:59 AM
Siron
Re: Writing the vertex of a parabola
But \$\displaystyle y=x(x-10)^2-8=x(x^2-20x+100)-8=x^3-20x^2+100x-8\$, so that's not an option.
Do you notice:
\$\displaystyle x^2-10x+25=(x-5)^2\$
?
But in this case you've \$\displaystyle y=x^2-10x-8\$, so you've to subtract a (constant) number of 25 to hold the original equation, so:
\$\displaystyle x^2-10x+25-a=x^2-10x-8\$

Calculate a, then you can wright:
\$\displaystyle x^2-10x-8=(x-5)^2-a\$
• Aug 10th 2011, 12:03 PM
Plato
Re: Writing the vertex of a parabola
Quote:

Originally Posted by David Green
I have a parabola y = x^2 - 10x - 8
I have the equation y = a(x - h)^2 + K
where (h, k) are the vertex

\$\displaystyle y=x^2-10x-8=(x-5)^2-33\$ Completing the square.
So \$\displaystyle h=5~\&~k=-33.\$
• Aug 10th 2011, 12:09 PM
David Green
Re: Writing the vertex of a parabola
Quote:

Originally Posted by Siron
But \$\displaystyle y=x(x-10)^2-8=x(x^2-20x+100)-8=x^3-20x^2+100x-8\$, so that's not an option.
Do you notice:
\$\displaystyle x^2-10x+25=(x-5)^2\$
?
But in this case you've \$\displaystyle y=x^2-10x-8\$, so you've to subtract a (constant) number of 25 to hold the original equation, so:
\$\displaystyle x^2-10x+25-a=x^2-10x-8\$

Calculate a, then you can wright:
\$\displaystyle x^2-10x-8=(x-5)^2-a\$

Thanks Siron, Plato has just beat me to it with the solution, but I have worked it out like this;

y = x^2 - 10x - 8 = (x - 5)^2 - 25 - 8

= (x - 5)^2 - 33

vertex is (5, - 33)

Thanks to you both much appreciated.
• Aug 10th 2011, 12:17 PM
Siron
Re: Writing the vertex of a parabola
Well done and you're welcome! :)
• Aug 10th 2011, 12:23 PM
David Green
Re: Writing the vertex of a parabola
Quote:

Originally Posted by Siron
Well done and you're welcome! :)

Thanks

David