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

3 Answers
Oct 8, 2017

(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)

Oct 8, 2017

(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)

Oct 8, 2017

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)