SOLUTION: I desperatly need help asap. f(x)=-2x^2+2x+6 i need the x cordinate of the vertex the y coordinate of the vertex the equation of the lines of symmetry the maximum and the mi

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Question 245436: I desperatly need help asap.
f(x)=-2x^2+2x+6
i need the x cordinate of the vertex
the y coordinate of the vertex
the equation of the lines of symmetry
the maximum and the minumum of f(x)
I would also like to learn how to work these problems I have several other the same. Thanks so much.
Found 2 solutions by solver91311, jsmallt9:
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

For any function of the form

the -coordinate of the vertex is given by

The -coordinate of the vertex is the value of the function at the -coordinate of the vertex, namely:

This is a parabola, so there is only one line of symmetry, namely the vertical line that passes through the vertex, so the equation is:

If the lead coefficient is positive, then the parabola opens upward. If the parabola opens upward then quite obviously the -coordinate of the vertex is the minimum of the function. In this case, the function has no maximum.

If the lead coefficient is negative, then the opposite is true: The parabola opens downward, the value of the function at the vertex is a maximum, and there is no minimum.

So, for your particular problem, you need to first calculate the value of , and then calculate the value of the function at that value:

The rest of your answers just fall out from those two calculations.

Here is a graph of your function so that you can check that your calculations make sense:

John

Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
The parabola is the only conic section that can be a function. A form of the equation of a parabola which tells us the most is:

because (h, k) will be the coodinates of the vertex.

To transform your function into this form we will
Use "y" instead of f(x) to save typing.
Move the constant term over to the left side.
Factor out the coefficient of .
And complete the square.

Now we want to complete the square. We want to change into a perfect square. It takes practice to get good at this it is a skill that will help you with conic sections and beyond. (IOW, it is not something that will go away in a few weeks.)

Here are the patterns for perfect squares:

The things to notice here are:
The first and third terms of the right sides are the squares of some monomial (one-term expression), called "a" and "b".
The middle terms are 2 times the "a" times the "b".
So to somplete the square we need to create a trinomial (3-term expression) into one that fits this pattern.

With we have the first term and the middle term. The first term is so our "a" is "x". We do not have a third term. We will have to manufacture one. And to do so we need to know which one we want. We want a "b squared" that will make 2 times "x" times "b" be "x". We need to solve:
2bx = x
for b, Divide both sides by 2x and we get
b = x/2x = 1/2

(There is "shortcut" to finding b if the coefficient of x^2 is 1. Inside the parentheses the coefficient of our x^2 is 1 so we could have used the shortcut on this problem. The shortcut: b = 1/2 of the coefficient of x. For example, if we had , b would be 1/2 of 20 or 10. If we had the b would be 1/2 of 5 or 5/2. Just remember that this only works if the coefficient if x^2 is 1.)

So we want "b" to be 1/2. That makes b squared 1/4. So we want to change into in order to make it fit the perfect square pattern. We want the right side of our equation to be

With the -2 outside the parentheses we are not just adding 1/4 to the right side. We are adding -2*1/4 or -1/2 to the right side. And if we add this to the right side, we must add it to the left side:

becomes

The left side will simplify and we can rewrite the right side with the perfect square we worked so hard to get:

This makes h = 1/2 and k = and the vertex is (1/2, ).
The axis of symmetry is a vertical line thru the vertex: x = 1/2.
Because of the negative number in front of , this is a parabola that opens downward. That makes the vertex a maximum value. (There is no minimum value for parabolas that open downward.)