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).
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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