Difference between graphs of -f(x) and f(-x)

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Discussion Overview

The discussion revolves around the differences in graph transformations represented by the expressions -f(x) and f(-x). Participants explore the implications of these transformations on the coordinates of points on the graph, addressing both conceptual understanding and specific examples.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion, suggesting that -f(x) implies flipping both x and y coordinates, while f(-x) means making x negative.
  • Another participant clarifies that for a point (2,2), the transformation for -f(x) results in (2,-2) and for f(-x), it cannot be determined without knowing f(-2).
  • A specific example is provided where f(x) = 2x - 2, leading to f(2) = 2 and f(-2) = -6, illustrating how f(-x) changes the output based on the function's definition.
  • Further clarification is offered that the graph of y = -f(x) reflects across the x-axis, while y = f(-x) reflects across the y-axis.
  • Another participant reiterates the point about the transformations, emphasizing the reflection across both axes for -f(-x).

Areas of Agreement / Disagreement

Participants exhibit some agreement on the basic transformations but also express differing interpretations of how these transformations apply to specific points, leading to confusion and a lack of consensus on the initial understanding.

Contextual Notes

Some assumptions about the function f and its behavior at specific points are not fully explored, which may affect the understanding of the transformations discussed.

Tyrion101
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It often seems as though it's backwards from what it should be, I had understood -f(x) meant to flip x and y, from positive to negative, or vice versa, and f(-x) to mean you just make x negative. If you had the points (2,2) you'd make them (-2,-2) for -f(x) and (-2,2) for f(-x) Is my understanding wrong?
 
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The rest of the title should be graphing.
 
Tyrion101 said:
If you had the points (2,2) you'd make them (-2,-2) for -f(x) and (-2,2) for f(-x) Is my understanding wrong?

If the point (2,2) is (x,f(x)) then (x, -f(x)) is the point (2,-2) and (x,f(-x)) can't be determined unless you know what f(-2) is. For example if f(x) = 2x -2, f(2) = 2 and f(-2) = -6. So (x,f(-x)) is (2,-6).
 
Ok, that makes more sense.
 
Tyrion101 said:
It often seems as though it's backwards from what it should be, I had understood -f(x) meant to flip x and y, from positive to negative, or vice versa,
No.
Assuming that you have the graph of y = f(x),
1. The graph of y = -f(x) is the reflection across the x-axis of the graph of y = f(x).
2. The graph of y = f(-x) is the reflection across the y-axis of the graph of y = f(x).
3. The graph of y = -f(-x) is the reflection across the origin (that is, across both axes) of the graph of y = f(x).
Tyrion101 said:
and f(-x) to mean you just make x negative. If you had the points (2,2) you'd make them (-2,-2) for -f(x)
If you had the point (2, 2) on the graph of f, the point (-2, -2) would be on the graph of y = -f(-x). Note the two minus signs.
Tyrion101 said:
and (-2,2) for f(-x) Is my understanding wrong?
 

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