Given #-f(x)#, how do you describe the transformation?

Question

Lily Anderson · Accepted Answer

A reflection about the #x#-axis.

For any function #f#, its output at any given input #x# is just #f(x)#. When we take that output #f(x)# and make it negative (i.e. #-f(x)#), we're just flipping the sign of the output—positives become negatives, and vice versa. Let's say that when #x=3,# we have #f(x)=5#. Thus, when #x=3,# we have #"-"f(x)="-"5#. All that happens is the sign of the output value changes—points that were once above the #x#-axis are now below, with no shift left or right. This is simply stated as a reflection where the mirror is the #x#-axis, also called a reflection about the #x#-axis.

Lily Anderson · Answer

undefined To describe the transformation of the function -f(x), you simply need to consider the effects of negating the original function. Specifically:

1. Reflection over the x-axis if f(x) is positive and reflection over the y-axis if f(x) is negative.
2. If f(x) is shifted vertically by a certain amount, -f(x) will be shifted vertically by the same amount in the opposite direction.
3. If f(x) is stretched or compressed vertically, -f(x) will undergo the same vertical stretching or compression.
4. If f(x) is shifted horizontally, -f(x) will be shifted horizontally by the same amount in the opposite direction.
5. If f(x) is stretched or compressed horizontally, -f(x) will undergo the same horizontal stretching or compression.