How do you find the greatest value of function?

Question

g: x --> 2 - 6x - x^2

how do i do this??

Answer 1

you could either graph it on a graphing calculator or you can complete the square.

Finding the greatest value of the function just means to find the maximum point.  That means to find the highest y-coordinate which is easy once you know what the vertex is.

I don't know what math class you are in but using the graphing calculator, the vertex is (-3, 11) so the greatest value is 11.

The range of the function is y < = 11

If you are in Algebra 2 or higher, you could complete the square.

2 - 6x - x^2 = y

-x^2 - 6x = y - 2

Factor out -1 on the left side

-1 ( x^2 + 6x .........) = y - 2

Take half of 6, then square the result.  6/2 = (3) ^ 2 = 9

-1 ( x^2 + 6x + 9) = y - 2 - 9  

[hint : you really added -1*9 on the left so you have to add -9 on the right]

Vertex form of parabola : y - k = a ( x - h ) ^2

-1 ( x + 3 ) ^2 = y - 11

The vertex is (-3, 11) and the parabola opens down because of the -x^2 in the beginning so the vertex is a maximum point.

Good luck to you !!

Answer 2

rewriting it as:

g(x) = -x^2 - 6x + 2

this is a parabola that opens downward (because of the -x^2)

the vertex will be a maximum

x-coordinate of the vertex is -b / (2a), where

a = -1, b = -6, and c = 2

-b / (2a) = -(-6) / (2*-1) = 6 / -2 = -3

x-coordinate of the vertex is x = -3

g(-3) = -(-3)^2 - 6(-3) + 2 = -9 + 18 + 2 = 11

vertex is (-3 , 11), and max value of the function is y = 11