Determine if the graph of the following equation is symmetric?

Question

with respect to the origin, the x-axis or the y-axis. y = x^2 -7

Answer 1

Recall the definitions for even and odd functions.

A function is even if and only if f(x) = f(-x).
A function is odd if and only if f(-x) = -f(x).

Let y = f(x) = x^2 - 7.

Let's first calculate f(-x), since it's required to test for both odd and evenness.

f(-x) = (-x)^2 - 7 = x^2 - 7

Therefore, f(x) = f(-x), so f(x) is even (symmetric across the y-axis).

Now, let's test if it's odd, by calculating -f(x).
-f(x) = -[x^2 - 7] = -x^2 + 7, but
f(-x) = x^2 - 7
So they are not (necessarily) equal, and f(x) is not odd.