SOLUTION: f(x)=-x^2+2x+7 Determine if this has a maximum or a minimum value. Then find the max/min value.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: f(x)=-x^2+2x+7 Determine if this has a maximum or a minimum value. Then find the max/min value.      Log On


   

Question 705562: f(x)=-x^2+2x+7 Determine if this has a maximum or a minimum value. Then find the max/min value.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+-x%5E2%2B2x%2B7
First we should recognize that the graph of a quadratic function, like this one, will have a parabola that opens upward (or downward) as a graph. Such a parabola will have a maximum or minimum value.

Since the coefficient of the squared term is negative (the "a") this parabola will open downward. If you picture such a parabola you will realize that the vertex will be the highest (i.e. maximum) value. So f(x) will have a maximum value at its vertex.

To find this maximum value we just have to find the vertex of the parabola. This can be done by completing the square to put the equation into vertex form. But it is a little easier if you know that the x coordinate of the vertex of any quadratic function will be:
x%5Bv%5D+=+-b%2F2a
Your "a" is -1 and your "b" is 2. So:
x%5Bv%5D+=+-2%2F2%28-1%29
which simplifies to:
x%5Bv%5D+=+1
Now to find the maximum value we find the y coordinate of the vertex, f(1):
f%281%29+=+-%281%29%5E2%2B2%281%29%2B7
Simplifying...
f%281%29+=+-1%2B2%281%29%2B7
f%281%29+=+-1%2B2%2B7
f%281%29+=+8

So the vertex of the parabola is (1, 8) and the maximum value for the function is 8.