What Are the Key Transformations for Trigonometric Graphs?

[Peter G.]

I am revising my graph transformations and I am curious:

If we graph sin (2x) or sin (x/2) we are able to increase and reduce their cycles.

Is there any transformation for other lines/graphs?

My doubt is we can also do 2 sin (x), which is the stretch parallel to the y-axis as I am familiar.

But I am guessing the cycle increase and decrease is unique to the trigonometric curves?

 

[eumyang]

Sinusoids are written in the form
$$f(x) = a \sin (b(x - h)) + k$$

Altering b affects the period of the graph, as you said. And yes, altering a will affect the amplitude (ie. vertical stretch or shrink). And if a is negative, there would be a reflection across the x-axis as well. Altering h would affect the phase shift (ie. horizontal shift), and altering k would move the graph of the sinusoid up or down.