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Graph transformations - explanation

    • Thread Starter

    Hi i am uncertain about the following, well i have this question which says:

    The original graph is y = sqrt(x) and takes the following transformations y = sqrt(-x)

    Now I have to describe the transformation in words, and to me wouldn't it be a reflection about y-axis? because it is a vertical reflection about the vertical axis right? However my answers say it is a reflection about the the x-axis which i cant seem to understand. If it were the x-axis wouldnt it be a reflection about the x-axis?

    Please any help would be great!

    You are confusing y=sqrt(-x) with y=-sqrt(x). We know that the sqrt fn is always positive (assuming x is real), so a reflection about y=0 (x axis) is impossible as this would make sqrt(x) negative.

    Think about it this way:
    for which x is y=sqrt(x) undefined?
    for which x is y=sqrt (-x) undefined?
    what is the relationship between these two sets of numbers

    is there an x for which x=-x (and so sqrt(x)=sqrt(-x) - this point will have to be on the axis of reflection so that it is not moved).
    • Thread Starter

    ah ok thanks for that. So in terms of describing transformations for graphs how can i tell the difference like when it is a dilation parallel to y-axis or dilation parallel to x-axis, this really confuses me :/. for example y = x^2 --> y = 2x^2, is this a dilation parallel to the y-axis scale factor 2? (parallel to y-axis because it is vertical?)

    And also like y = x^2 --> y = (2x)^2 would that be dilation parallel to x-axis scale factor 1/2 because it is a horizontal dilation?


    In general, if the 'change is being made to x (ie, before the fn is performed on it (usually f(ax) or f(x-a) or f(ax+b) where a and b are constant (but not necessarily positive), then it will be in the x direction

    so a translation in the case of f(x-a) - translate the graph by a units (in the positive x direction). This will result in moving the graph to the left (in the negative x direction) if a is negative. Be careful about signs. if you have f(x+a) think of it as f(x-(-a)) or reverse positive and negative direction in the above.
    'stretch' in the case of f(ax) by a factor of 1/a (so for f(2x), its a stretch by factor 1/2 in x direction ONLY - so eg x^2 to (2x)^2 then the points (-2,4), (-1,1), (0,0) would go to (-1,4), (-1/2,1), (0,0) respectively). If a is negative, then you can think of it as a reflection in y axis (x direction) followed by 'stretch' of factor 1/mod(a) (this was the case in your ex. sqrt(x) -> sqrt(-x) reflection in y axis, followed by 'stretch of factor 1/1 = 1 (this does not change anything).

    If the change is being made AFTER the fn is performed so you have af(x)+b (where a and b are constants, not necessarily positive, one or both could be zero) then the change is in the y direction. Similar rules apply to the ones above, except that the factor for stretches is just a (not 1/a).

    And, this applies to transformations in both directions, always do stretches before translations (unless you have something like f(2(x+3)). But even then its probably easier/safer to just expand it out to f(2x+6) and then do f(x)->f(2x)->f(2x+6).
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