Answer: Even functions
Symmetric about the yy-axis
When you plug -x−x into the function, it will simplify to be the same as the original function. This means that it doesn’t matter whether you plug in xx or -x−x, your output will be the same. So
f(-x)=f(x)f(−x)=f(x)
Below are graphs that are even and symmetric about the yy-axis.
even functions are symmetric about the y-axis
Odd functions
Symmetric about the origin
When you plug -x−x into the function, it will simplify to be negative of the original function, or the original function multiplied by -1−1. This means that when you plug -x−x into the function, you’ll get the same output as you do when you plug in xx, except it will be negative (or have the opposite sign as the original output). So
f(-x)=-f(x)f(−x)=−f(x)
Below are graphs that are odd and symmetric about the origin. Be sure to visually compare quadrants that are diagonal from each other (quadrants 1 and 3, and quadrants 2 and 4).
odd functions are symmetric about the origin
Neither even nor odd
Not symmetric about the yy-axis, and not symmetric about the origin
The function has no symmetry. It’s possible that a graph could be symmetric to the xx-axis, but then it wouldn’t pass the Vertical Line Test and therefore wouldn’t be a function.
functions that are neither even nor odd
hope this helps!!