Using stretching, shrinking, and shifting to transform one graph into anoth

Questions
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1

I am attempting to transform one graph into another, but I am not sure what commands would allow me to do so. I am given two sets of data points to create the two different graphs. I want to be able to pin the two graphs together at a single point, and then I want to be able to transform (using only shifting, shrinking, and stretching) one of the graphs without altering the other.
Forgive the basic blueprint of ideas, I am not really sure of the terminology of wolfram language that would make this question more understandable.

Thanks!

2

These links should be helpful:

Wolfram Alpha is also a good resource:

3

Here is an example of an approach (for two 2-dimensional plots):

graph1 = Plot[Sin[x]/(1 + x^2), {x, 0, 5}]; 

graph2 = Plot[AiryAi[x] (1 + x^2), {x, 0, 5}]; 

Manipulate[
 Graphics[
  {graph1[[1]], Scale[Translate[graph2[[1]], pt], s]},
  Axes -> Automatic, 
  AspectRatio -> 1/GoldenRatio,
  PlotRange -> {{0, 10}, {0, 1}}],
 
 {{pt, {0, 0}}, {0, 0}, {1, 1}},
 {{s, 1}, 1/4, 2}
 ]