# How do you find the axis of symmetry, and the maximum or minimum value of the function f(x) = 8 - (x + 2) ^2?

Jan 18, 2016

Explanation is given below

#### Explanation:

If you can understand the graph of a parabola opening up or down. You can easily get the solution to the question you have asked.

Let us see how.

The vertex form of the quadratic function is $y = a {\left(x - h\right)}^{2} + k$ where (h,k is the vertex.

$a$ decides vertical shrink or stretch and also which way the parabola is opening.

If $a < 0$ the parabola opens down, and in this case, the graph would have a maximum. The maximum value would be $k$

if $a > 0$ then the parabola opens up, and in this case the graph would have a minimum and the minimum value would be $k$

The axis of symmetry for the graph opening up or down is decided by the $x$ coordinate of the vertex. The equation of axis of symmetry is given by $x = h$

Now let us apply the same with respect to our problem.

$f \left(x\right) = 8 - {\left(x + 2\right)}^{2}$
Let us rewrite in the form $y = a {\left(x - h\right)}^{2} + k$

$f \left(x\right) = - {\left(x + 2\right)}^{2} + 8$

Comparing we can see $a = - 1$, $h = - 2$ and $k = 8$

Vertex is $\left(- 2 , 8\right)$

Since $a < 0$ the graph opens down.
Therefore the graph has a maxmimum.

The maximum value is $8$

The axis of symmetry is $x = - 2$