Question

Will medal!! How can I find the answer?
What are the coordinates of the vertex for f(x) = 2x^2 + 4x + 9?
(−2, 4)
(−1, 7)
(1, 15)
(2, 9)

Answer 1

You have to complete the square.

Answer 2

To make it easier divide by \(2\) first:\[
\frac 12 f(x) = x^2+2x+\frac 92
\]

Answer 3

The next part is tricky. The way I remember it is by squaring a binomial: \[
(x+a)^2 = x^2+2ax+a^2
\]We want to turn \(x^2+2x\) into a square binomial.
If we let \(2ax = 2x\), and divide by \(2x\), then we find out that \(a=1\).

Answer 4

This means: \[
(x+1)^2 = x^2+2x+1 \implies x^2+2x=(x+1)^2-1
\]We substitute this into our equation: \[
\frac 12 f(x) = [(x+1)^2-1] +\frac 92
\]We can multiply by \(2\) again and get: \[
f(x) = 2(x+1)^2-2+9 = 2(x+1)^2+7
\]

Answer 5

This is our vertex form: \[
f(x) =2(x+1)^2+7
\]Now, typically a parabola with vertex \((h,k)\) will be written as: \[
y=a(x-h)^2+k
\]So if we compare, we get \(-h = 1\) and \(k=7\).

Answer 6

In general, when you have: \[
y=ax^2+bx+c
\]You complete the square: \[
\frac ya=x^2+\frac ba x+\frac ca\\
\frac ya=\left(x+\frac b{2a}\right)^2-\left(\frac b{2a}\right)^2+\frac ca\\
y =a\left(x+\frac b{2a}\right)^2-\frac {b^2}{4a}+c
\]So our vertex will be \((-b/(2a), c-b^2/(4a))\).

Answer 7

Ok I think I was able to get that last one better, lol
thank you! @wio

Answer 8

The last one? Why?

Answer 9

well I guess because it shows the complete setup. in the beginning I was understanding and then I started to get a little confused until the end

Answer 10

The last one makes it easy to do every other problem without having to complete the square every time. You just need to plug the coefficients into the formula.
However, it isn't easy to remember.

Answer 11

Also, you need to understand algebra well to understand how it works.