Question 1116577: Determine whether the graph of y = |x| − 3 is symmetric with respect to the origin, the x-axis, or the y-axis.
a. symmetric with respect to the x-axis only
b.symmetric with respect to the y-axis only
c. symmetric with respect to the origin only
d. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! Symmetry checks:
f(x) = f(-x) ==> symmetric with respect to y-axis
f(x) = -f(x) ==> symmetric with respect to x-axis
f(x) = -f(-x) ==> symmetric with respect to origin
Which of these hold for y = |x| - 3?
If you replace x with -x, will f(x) change? Try a value, say x=7. f(7) = |7|-3 = 4. f(-7) = |-7|-3 = 4.
No, f(x) does not change because of the absolute value operation. So f(x) is symmetric with respect to the y-axis.
Clearly f(x) does not equal -f(x), so no symmetry with respect to the x-axis. (Note that a true "function" can not be symmetrical with respect to the x-axis because it would violate the vertical line test, i.e. it would not be one-to-one).
f(x) clearly does not equal -f(-x) here either, so no symmetry with respect to the origin.
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