Question

Rewrite the following quadratic function in vertex form. Then, determine if it has a maximum or minimum and say what that value is. y = -x 2 + 6x + 5

Answer 1

The equation y = -x^2+6x+5 is really the equation y = -1x^2+6x+5. It is in the form y = ax^2 + bx + c where

a = -1
b = 6
c = 5

We will use 'a' and 'b' in the formula below

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

The h refers to the x coordinate of the vertex. Since we know the x coordinate of the vertex (is 3), we can use it to find the y coordinate of the vertex

Simply plug x = 3 into the original equation

y = -x^2 + 6x + 5
y = -(3)^2 + 6(3) + 5
y = -(9) + 6(3) + 5
y = -9+18+5
y = 14

This is the k value, so k = 14.

In summary so far, we have a = -1, h = 3 and k = 14. Plug all this into the vertex form below

y = a(x-h)^2 + k
y = -1(x-3)^2 + 14
y = -(x-3)^2 + 14

Therefore the vertex form equation is y = -(x-3)^2 + 14

So when x = 3, the paired y value is y = 14. The point (x,y) = (3,14) is a point on the parabola. This point is either the highest or lowest point.

How can we figure out if it's the highest or lowest point? By looking at the value of 'a'. Notice how a = -1 and this is less than zero. In other words, a < 0

Since a < 0, this means the parabola opens downward forming a "frown" so to speak. That's one way to remember it: negative 'a' leads to sad face. 

Anyways, this parabola opening downward means that the vertex is the highest point. 

So (3,14) is the vertex

The maximum is y = 14.