MHB Consecutive vertical and horizontal transformations of a function

AI Thread Summary
The discussion revolves around transforming the function f(x) = (2^x) + 1 through a series of specified transformations. The transformations include a vertical stretch by a factor of 8, a translation by the vector (1,4), and a horizontal stretch by a factor of 1/2. The user expresses confusion about the correct order of applying these transformations to arrive at the final function h(x). They initially applied horizontal transformations before vertical ones but did not achieve the textbook solution, which is h(x) = 4^(x+1) + 16x - 4. The thread highlights the importance of the sequence in which transformations are applied to obtain the correct result.
pluto1
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Dear all,
I am stuck on this question:
"If f(x)=(2^x)+1, give in simplest terms the formula for h(x), which is obtained from transforming f(x) by

a vertical stretch, scale factor 8 relative to y=0
a translation by vector (1,4)
a horizontal stretch, scale factor 1/2 relative to x=0"

This is what I understand from the question:

A vertical stretch by scale factor 8 means h(x)= 8 f(x)
A translation by vector (1,4) means h(x)= f(x-1) + 4
A horizontal stretch by scale factor 1/2 means h(x)= f(1/2x)
Where the horizontal shift is applied prior to the horizontal stretch.

This gives me

h(x)= 8 ((2^(1/2 x-1) + 1/2 x - 1))+ 4
h(x)= 2^(2x-2) + 16 x - 4

h(x)=4^(x+1)+16x-4 according to the solutions at the back of my textbook. I would really appreciate some help. Thank you so much in advance.
 
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In what order are you doing these transformations?
 
I applied the horizontal changes before the vertical changes, with the translation by -1 prior to the stretch by 1/2

I just tried the opposite with all the vertical changes first but that does not give me the correct solution either
 
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