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
Gold Member
Messages
2,825
Reaction score
413
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...
 
Physics news on Phys.org
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.
 
Back
Top