Question

What is the vertex of `y=x^2-x+9-2(x-3)^2`?

Answer 1

`(11/2, 85/4)`

Explanation:

Simplify to `y=ax^2+bx+c` form.

`y=x^2-x+9-2(x-3)^2`
Use FOIL to expand `-2(x-3)^2`
`y=x^2-x+9-2(x^2-6x+9)`
`y=x^2-x+9-2x^2+12x-18`
Combine like terms
`y=-x^2+11x-9`

Now that we have turned the equation to `y=ax^2+bx+c` form,
Let's turn them to `y=a(x-p)^2+q` form which will give the vertex as `(p, q)`.

`y=-(x^2-11x+?)-9+?`

To make perfect square like `(x-p)^2`, We need to find out what `?` is.

We know the formula that when `x^2-ax+b` is factorable by perfect square `(x-a/2)^2`, we get the relationship between `a` and `b`.

`b=(-a/2)^2`

So `b` becomes `?` and `a` becomes `-11`.

Substitute those values and let's find `?`.

`?=(-11/2)^2`
`?=(-11)^2/(2)^2`
`∴?=121/4`

Substitute `?=121/4` to `y=-(x^2-11x+?)-9+?`

`y=-(x^2-11x+121/4)-9+121/4`
`y=-(x-11/2)^2-36/4+121/4`
`y=-(x-11/2)^2+85/4`

`∴y=-(x-11/2)^2+85/4`

Therefore, we have turned the equation to `y=a(x-p)^2+q` form which will give our vertex as `(p, q)`

`∴p=11/2, q=85/4`

`∴Vertex (11/2, 85/4)`

Answer 2

`(5.5, 21.25)`

Explanation:

This equation looks scary, which makes it hard to work with. So, what we're gonna do is simplify it as far as we can and then use a small part of the quadratic formula to find the `x`-value of the vertex, and then plug that into the equation to get out our `y`-value.

Let's start with simplifying this equation:

At the end, there's this part: `-2(x-3)^2`

Which we can factor to `-2(x^2-6x+9)` (remember it isn't just `-2(x^2+9)`)

When we distribute that `-2`, we finally get out `-2x^2+12x-18`.

Put that back into the original equation and we get:

`x^2-x+9-2x^2+12x-18`, which still looks a bit scary.

However, we can simplify it down to something very recognizable:

`-x^2+11x-9` comes together when we combine all the like terms.

Now comes the cool part:

A small piece of the quadratic formula called the vertex equation can tell us the x-value of the vertex. That piece is `(-b)/(2a)`, where `b` and `a` come from the standard quadratic form `f(x)=ax^2+bx+c`.

Our `a` and `b` terms are `-1` and `11`, respectively.

We come out with `(-(11))/(2(-1))`, which comes down to

`(-11)/(-2)`, or `5.5`.

With knowing `5.5` as our vertex's `x`-value, we can plug that into our equation to get the corresponding `y`-value:

`y=-(5.5)^2+11(5.5)-9`

Which goes to:

`y=-30.25+60.5-9`

Which goes to:

`y=21.25`

Pair that with the `x`-value we just plugged in, and you get your final answer of:

`(5.5,21.25)`

Answer 3

Vertex `(11/2, 85/4)`

Explanation:

Given -

`y=x^2-x+9-2(x-3)^2`

`y=x^2-x+9-2(x^2-6x+9)`
`y=x^2-x+9-2x^2+12x-18`
`y=-x^2+11x-9`

Vertex

`x=(-b)/(2a)=(-11)/(2 xx(-1))=11/2`

`y=-(11/2)^2+11((11)/2)-9`

`y=-121/4+121/2-9=(-121+242-36)/4=85/4`

Vertex `(11/2, 85/4)`