Which graph shows the transformation of the function f(x) = e^x where the function is translated three units to the right, vertically compressed by a factor of 1/4, and then translated six units down.

Which graph shows the transformation of the


Original function is f(x) = e^x

For this y intercept = 1. i.e. passes through (0,1)

First it is translated three units to the right.

So y = e^(x-3)

When vertically compressed by a factor of 1/4 we have

y = 4e^(x-3)

Next is translated 6 units down

i.e. new graph would be

y = 4e^(x-3)+6

When x =3, y =6.

When y =-6, x tends to infinity.

i.e. y =-6 is an asymptote

Hence we find that 4th graph is the correct answer.



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Answer:

C is correct graph.

Step-by-step explanation:

Given: f(x)=e^x

Now we do some operation on function f(x)

Step 1: f(x) is translated 3 units to the right.

For a unit right translation, x changes to

x\rightarrow x-a

Therefore, f(x)=e^{x-3}

Step 2: Vertical compressed by a factor of \frac{1}{4}

For vertical compressed. f(x) changes to a f(x)  

f(x)=\frac{1}{4}e^{x-3}

Step 3: Translated 6 unit down

For this translation, y changes to

y\rightarrow y-6

Therefore, f(x)=\frac{1}{4}e^{x-3}-6

Final function after three steps we get,

f(x)=\frac{1}{4}e^{x-3}-6

Please see the attachment for correct graph.

Thus, C is correct graph.


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