Solving for invariant points on trig transformations

[zeion]

Homework Statement

Hello.
I came across a question that required me to solve for invariant points between a base trig function and the function after horizontal stretch. I can't remember the exact question right now, but I'm just wondering how I would go about solving it if I didn't know any trig identities or if the transformation couldn't be easily simplified with identities.

Homework Equations

The Attempt at a Solution

 

[Mark44]

Your question is a bit on the vague side, so I'll do the best I can with the available information. Suppose the function is y = sin(x), and the new function is y = sin(x/3), which represents a stretch away from the vertical axis by a factor of 3.

For the transformed function the only point that stays the same is the one whose distance is 0 from the vertical axis; namely, (0, 0).

 

[zeion]

What about points where the two graphs intersect?

 

[Mark44]

How are you defining "invariant points"? I am interpreting this to mean points that do not change. If you are really asking about the points of intersection of the two graphs, that's what you should be asking about, I think.

 

[zeion]

But wouldn't the points where the 2 graphs intersect also be points that "don't change"?

 

[Mark44]

I don't think so, not with a horizontal stretch. On the other hand, if you consider vertical stretches, the invariant points, as I would define them, would be all the points that don't get moved. For example, if y = sin(x), the graph of y = 2sin(x) is stretched away from the horizontal axis by a factor of 2. All of the zeroes of sin(x) (e.g., x = 0, π, 2π, -π, -2π, etc.) are also zeroes of 2sin(x), so the zeroes of y = sin(x) are invariant points under this transformation.

 

[zeion]

Yes the vertical case is clear to me.

Since the y point for the zeroes are zero, anything applied them would not change them.

I was just confused about the horizontal stretch/compression.
So does this mean the only invariant point for a horizontal stretch/compression is when x = 0?

 

[Mark44]

Yes. Every other point on the graph of the untransformed graph moves either closer to or farther away from the vertical axis.