Question

Which functions are symmetric with respect to the y-axis? Check all that apply.

Answer 1

Absolute value functions have a vertex at the origin (0, 0) and create a v shape. if you add or subtract numbers INSIDE your absolute value bars, you change the vertex of the function. this makes them no longer symmetrical with the y-axis. with that in mind, every answer choice that doesn't add or subtract anything inside your absolute value bars will still be symmetrical with respect to your y-axis.

f(x) = |x|
f(x) = |x| + 3
f(x) = |x| + 6

these are all symmetrical. adding on the outside of the absolute value bars does not affect symmetry because of the vertex form equation for an absolute value function: f(x) = |x - h| + k, where (h, k) is your vertex. if you change the y value of your vertex, you only move up or down, but you're haven't changed the x value, which moves you left or right. moving left or right is what affects your symmetry to the y-axis, so adding to the outside (which only moves your graph up or down) won't ruin it.