Question

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

Answer 1

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(x-h)^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(x) = 8-(x+2)^2`
Let us rewrite in the form `y=a(x-h)^2+k`

`f(x) = -(x+2)^2+8`

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

Vertex is `(-2,8)`

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`