Question

Determine whether the following quadratic function has a maximum or a minimum value. Then find the maximum or minimum value and where it occur:

F(x)=-3x²+18x+3
Select the correct choice below and fill in the answer boxes to complete your choice.
(Simplify your answers.)
The function has a maximum value of ___ at x=____.
The function has a minimum value of ____ at x=____.

Answer 1

• maximum: 30
• x = 3

Explanation:

You want to find the maximum or minimum value and where it occurs for the function F(x)=-3x²+18x+3.

Vertex form

The vertex form of a quadratic is ...

f(x) = a(x -h)² +k . . . . . . . vertex at (h, k)

The graph opens downward if a < 0, and opens upward for a > 0. When the graph opens downward, the function has a maximum at the vertex. When the graph opens upward, the function has a minimum at the vertex.

Application

We can put the function in vertex form as follows:

F(x) = -3(x² -6x) +3 . . . . . factor out the leading coefficient

F(x) = -3(x² -6x +9) +3 +3(9) . . . . . . subtract and add 3(9)

F(x) = -3(x -3)² +30 . . . . . . . . . simplify to vertex form

Comparing this equation to the form shown above, we see ...

a = -3 < 0 . . . . . . . . . the function has a maximum

(h, k) = (3, 30) . . . . . the maximum is 30 at x=3

__

Additional comment

The square of the binomial in parentheses is ...

(x -p)² = x² -2px +p²

That is, to "complete the square", we add the square of half the x-coefficient. Here, that coefficient is -2p=-6, so we want to add p²=9 inside parentheses. To keep the expression the same, we need to add the opposite amount outside parentheses. We are effectively adding 0 in the form of -3(9) +3(9) = 0 so the expression can be put in vertex form.

A graphing calculator can quickly show you the minimum or maximum of the function.