# How do you find the axis of symmetry, and the maximum or minimum value of the function  f(x)=-x^2+6x+6?

Feb 10, 2016

Complete the square

The line $x = 3$ is a line of symmetry

The maximum is $f \left(3\right) = 15$.

#### Explanation:

$f \left(x\right) = - {\left(x - 3\right)}^{2} + 15$.

Notice that

$f \left(3 + a\right) = f \left(3 - a\right)$

for any $a \in \mathbb{R}$.

Therefore, the line $x = 3$ is a line of symmetry.

The maximum is $f \left(3\right) = 15$.

This is because

$f \left(3\right) = 15 \ge 15 - {\left(x - 3\right)}^{2} = f \left(x\right)$

Feb 10, 2016

Axis of symmetry is $x = 3$.

The maximum$\text{ } \to \left(x , y\right) \to \left(3 , 15\right)$

#### Explanation:

As the coefficient for ${x}^{2} \text{ is } \left(- 1\right)$ the shape type of the quadratic is $\cap$ so it is a maximum.

Finding axis of symmetry: Let us look at the part of the equation that is $+ 6 x$

Compere to the standard form $y = a {x}^{2} + b x + c$

Write standard form as: $y = a \left({x}^{2} + \frac{b}{a} x\right) + c$

From this " "x_("vertex")=(-1/2)xxb/a

So in your case, as " "x_("vertex")=(-1/2)xx6

But $a = \left(- 1\right) \text{ }$so in fact what you have is:

$\text{ "x_("vertex")=(-1/2)xx6/(-1)" "=" } + 3$

Now it is just a matter of substituting $x = 3$ into the original equation to find that $y = 15$