What's the difference between these 2 graphs?

[mathman100]

I have my original graph f, and I want to make 2 other graphs out of it (each with their own transformations.)
g(x)=2g(1/2x+1)
so it gets stretched vertically by 2, stretched horizontally by 2, goes left 1 and up 1.
But my problem is with h(x)=2[g((x+1)/2)+1)]
What are the transformations made here? Are they the same as g(x)?

 

[Data]

I assume you mean g(x) = 2f(x/2 + 1) and h(x) = 2[g((x+1)/2) + 1]. I don't know why you think that g is vertically shifted (it isn't), but otherwise your description for it is fine.

You can simplify that expression for h(x) to 2g(x/2 + 1/2) + 2. Then substituting in g(x/2 + 1/2) = 2f((x/2 + 1/2)/2 + 1) = 2f(x/4 + 5/4) you get h(x) = 4f(x/4 + 5/4) + 2. That help?

(I am not sure whether I have interpreted your function definitions correctly, so these are just guesses. Are you sure you typed everything out properly? In your post you defined g(x) = 2g(x/2 + 1) which I assume is a typo.)

 

[tacman]

The two graphs g(x) and h(x) have different transformations applied to them. While g(x) has a vertical stretch of 2, a horizontal stretch of 2, and a shift left by 1 and up by 1, h(x) has a vertical stretch of 2, a horizontal stretch of 1/2, and a shift right by 1 and up by 1.

In g(x), the transformations are applied in the order of horizontally stretching first, then vertically stretching, and finally shifting. This results in a wider and taller graph compared to the original graph f.

On the other hand, in h(x), the order of transformations is different. The graph is first shifted to the right by 1, then horizontally stretched by 1/2, and finally vertically stretched by 2. This results in a narrower and taller graph compared to the original graph f.

Therefore, the transformations made in g(x) and h(x) are not the same. They have different effects on the original graph f, resulting in two different graphs.