Find the translation and stretch from ##y=2^x## to ##y=2^{x+4}##

AI Thread Summary
The discussion focuses on the transformations of the function from y=2^x to y=2^{x+4}. It is established that the transformation involves a horizontal shift of four units to the left, moving points on the graph accordingly. Additionally, the transformation can be interpreted as a vertical stretch, represented by y=16·2^x. Participants express confusion about the nature of the shifts and transformations, clarifying that while there is a horizontal shift, there is also a vertical aspect to consider. Overall, the key takeaway is the dual nature of the transformation involving both horizontal and vertical changes.
chwala
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Homework Statement
See attached...
Relevant Equations
Translation/Stretch
1655287826660.png


For part (i) i was thinking of the vertical shift from ##(0,1)## to ##(0,16)##, this can be given by;
##y=2^x + 15## but it does not fit onto ##y=2^{x+4}## something wrong here.

For part (ii), =we have a stretch factor of ##a=16>0## (vertical stretch) thus, ##y=16\cdot 2^x##

your thoughts...i do not have solutions...
 
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If f(x) = 2^x then 2^{x+4} = f(x + 4).
 
pasmith said:
If f(x) = 2^x then 2^{x+4} = f(x + 4).
Isn't this a horizontal shift... it's not exactly what we want. Unless I am missing something...
 
chwala said:
For part (i) i was thinking of the vertical shift from (0,1) to (0,16),
As already noted, the transformation is a horizontal translation (or shift). The original graph is translated four units to the left. The point on the original graph that was at (0, 1) is now at (-4, 1). The point that was at (4, 16) is now at (0, 16).
chwala said:
Isn't this a horizontal shift...
Yes.
 
On the other hand, ##2^{x + 4} = 2^x \cdot 2^4 = 16\cdot 2^2##.
Viewed this way there is a vertical stretch away from the x-axis.
 
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