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

• Tyrion101
In summary: Yes, your understanding is incorrect. The correct understanding is that (-2, 2) would be on the graph of y = f(-x), which is the reflection across the y-axis.
Tyrion101
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?

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?

## 1. What is the difference between the graphs of -f(x) and f(-x)?

The main difference between these two graphs is that -f(x) is a reflection of f(x) across the x-axis, while f(-x) is a reflection across the y-axis. This means that the shape of the graph remains the same, but its orientation is flipped.

## 2. How do the x-intercepts of -f(x) and f(-x) compare?

The x-intercepts of -f(x) and f(-x) are the same, since they both occur at the same values of x. This is because the reflection across the x-axis preserves the x-values, while the reflection across the y-axis preserves the y-values.

## 3. Do the y-intercepts of -f(x) and f(-x) differ?

No, the y-intercepts of -f(x) and f(-x) are the same. This is because the y-intercept occurs when x=0, and both -f(0) and f(-0) will give the same value, since a negative sign does not change the value of a number.

## 4. How do the slopes of -f(x) and f(-x) compare?

The slopes of -f(x) and f(-x) are equal, but with opposite signs. This is because the slope of a function is determined by the rate of change between its x and y values, and reflecting a graph across an axis does not change this rate of change.

## 5. What is the overall effect on the graph when comparing -f(x) and f(-x)?

The overall effect is a rotation of the graph by 180 degrees. This is because the reflection across the x-axis followed by the reflection across the y-axis results in a rotation of 180 degrees. This can also be seen by the fact that (-f(-x)) = f(x), meaning that the two graphs are symmetric about the origin.

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