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Tranformations of Functions

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    ReputationRep:
    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.
  2. Offline

    ReputationRep:
    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.
  3. Offline

    ReputationRep:
    (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?
  4. Offline

    ReputationRep:
    (5,25) with f(-x) = (-5,25)
    (5,25) with -f(x)= (5,-25)

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

    ReputationRep:
    (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.
  6. Offline

    ReputationRep:
    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?
  7. Offline

    ReputationRep:
    (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.
  8. Offline

    ReputationRep:
    (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.
  9. Offline

    ReputationRep:
    (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.

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