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Tranformations of Functions

I'm not so sure about the tranformation of a graph e.g.x^2 or x^3 with y=-f(-x). Do I just add a minus sign in front of the original x and y coordinates of the original graph or what?????? :confused:

For example say a graph is y=x^2 and and one point is (5,25). Therefore would a point with graph y=-f(-x) be (-5,-25)???

Thanks for any help provided. :smile:
(edited 11 years ago)
Reply 1
Avatar for Notnek
Notnek
21
Well what is f(-x) for those functions?

(-x)^2 = x^2
(-x)^3 = -x^3

So for f(x) -> -f(-x) we have x^2 -> -x^2 and x^3 -> x^3

So the graph of x^3 is unchanged and the graph of x^2 is reflected in the x-axis.

Now think again about how the coordinates are affected by these transformations.
(edited 11 years ago)
Reply 2
Avatar for krisshP
krisshP
OP
14
Original post by notnek
Well what is f(-x) for those functions?

(-x)^2 = x^2
(-x)^3 = -x^3

So for f(x) -> -f(-x) we have x^2 -> -x^2 and x^3 -> x^3

So the graph of x^3 is unchanged and the graph of x^2 is reflected in the x-axis.

Now think again about how the coordinates are affected by these transformations.


So does that mean that cubic, exponential and reciprical graphs do not get affected at all by refections whether it's by the x or y axis?
Reply 3
Avatar for krisshP
krisshP
OP
14
(5,25) with f(-x) = (-5,25)
(5,25) with -f(x)= (5,-25)

I can't understand the transformation of -f(-x).
Reply 4
Avatar for Notnek
Notnek
21
Original post by krisshP
So does that mean that cubic, exponential and reciprical graphs do not get affected at all by refections whether it's by the x or y axis?

x^3 is affected by single reflections but f(x) -> -f(-x) is a reflection in both the x-axis and the y-axis which ends up back where you started.

x^3 is known as an odd function which is a function that satisfies

f(x)=-f(-x)

Any odd function remains the same if reflected in both the x and y axes. 1/x is an odd function since

-(1/(-x)) = 1/x

but e^x is not an odd function.
Reply 5
Avatar for krisshP
krisshP
OP
14
so when I'm dealing with a quadratic graph as y=f(x) and I am told to transform it by y=-f(-x), shall I first reflect it by the x-axis and then reflect it by the y-axis?
Reply 6
Avatar for Notnek
Notnek
21
Original post by krisshP
(5,25) with f(-x) = (-5,25)
(5,25) with -f(x)= (5,-25)

I can't understand the transformation of -f(-x).

You were correct earlier when you said the point becomes (-5,-25).

Remember, this doesn't necessarily mean the graph is changed because the coordinates have moved.
Reply 7
Avatar for Notnek
Notnek
21
Original post by krisshP
so when I'm dealing with a quadratic graph as y=f(x) and I am told to transform it by y=-f(-x), shall I first reflect it by the x-axis and then reflect it by the y-axis?

Yes.

But some graphs like x^2 are symmetrical in certain axes so reflecting them doesn't change the graph.
Reply 8
Avatar for krisshP
krisshP
OP
14
Original post by notnek
Yes.

But some graphs like x^2 are symmetrical in certain axes so reflecting them doesn't change the graph.


Okay. I understand now. Thanks a lot for your help which is much appreciated. :smile::smile::smile::smile::smile::smile:

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