Question

What is the axis of symmetry and vertex for the graph `y=x^2-4`?

Answer 1

This function is symmetric in respect to the y axis.
The vertex is (0,-4)

Explanation:

We can define a function as odd,even, or neither when testing for its symmetry.
If a function is odd, then the function is symmetric in respect to the origin.
If a function is even, then the function is symmetric in respect to the y axis.
A function is odd if `-f(x)=f(-x)`
A function is even if `f(-x)=f(x)`

We try each case.
If `x^2-4=f(x)`, then `x^2-4=f(-x)`, and `-x^2+4=-f(x)`

Since `f(x)` and `f(-x)` are equal, we know this function is even.

Therefore, this function is symmetric in respect to the y axis.

To find the vertex, we first try to see what form this function is in.

We see that this is in the form `y=a(x-h)^2+k`

Therefore, we know that the vertex is (0,-4)