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

• chwala
In summary, the conversation discusses the vertical shift from (0,1) to (0,16) in a graph, which can be represented by the equation y=2^x + 15. However, this does not fit onto y=2^{x+4}. For part (ii), the stretch factor is determined to be a=16>0, leading to the equation y=16*2^x. The expert notes that this is a horizontal shift, but also points out that it can be viewed as a vertical stretch away from the x-axis.

#### chwala

Gold Member
Homework Statement
See attached...
Relevant Equations
Translation/Stretch

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...

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.

chwala
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.

chwala

## 1. What is the translation and stretch in the equation y = 2^x to y = 2^(x+4)?

The translation in this equation is 4 units to the left, and the stretch is a factor of 2 in the y-direction.

## 2. How do you find the translation and stretch in this equation?

To find the translation, you need to look at the exponent of the variable x. In this case, the exponent has changed from x to x+4, indicating a translation of 4 units to the left. To find the stretch, you need to look at the coefficient of the variable x. In this case, the coefficient is 2, indicating a stretch in the y-direction by a factor of 2.

## 3. What does the translation and stretch represent in this equation?

The translation represents a shift in the graph of the original function, moving it 4 units to the left. The stretch represents a change in the steepness of the graph, making it twice as steep in the y-direction.

## 4. Can you provide an example of how to graph this equation?

First, plot the points for the original function y = 2^x. Then, for the translated and stretched function y = 2^(x+4), shift all the points 4 units to the left and stretch them by a factor of 2 in the y-direction. Connect the points to create the new graph.

## 5. How does the translation and stretch affect the domain and range of the function?

The translation does not affect the domain and range of the function, as it only shifts the graph horizontally. The stretch, however, affects the range by making it twice as large as the original function. The domain remains the same.