Question

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.

Answer 1

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.

Answer 2

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.