Given the following transformation of f(x) = (2)^x

-Vertical stretch by a factor of 3
- Reflection in the y-axis
- Translated 9 units down
- Translated 1 unit left

What is the Equation?

g(x) = 3*2^-(x+1) - 9

Well, if we apply all those transformations to f(x) = (2)^x, it becomes a bit of a circus act. Let's break it down step by step:

1. Vertical stretch by a factor of 3: Now our function becomes f(x) = 3(2)^x. It's like stretching your favorite rubber chicken three times its original size!

2. Reflection in the y-axis: The equation becomes f(x) = 3(2)^(-x). Imagine our graph doing a fancy pirouette!

3. Translated 9 units down: We lower our function by 9 units, so now we have f(x) = 3(2)^(-x) - 9. It's like our graph is now going for a dive into the depths!

4. Translated 1 unit to the left: Finally, we move our function one unit to the left, resulting in f(x) = 3(2)^(-x + 1) - 9. It's like our graph is trying to sneak away!

So, our transformed equation is f(x) = 3(2)^(-x + 1) - 9. It's a real juggling act of transformations!

To find the equation of the transformed function, we need to apply each transformation step by step to the original function f(x) = (2)^x.

1. Vertical Stretch by a factor of 3:
To stretch the function vertically by a factor of 3, we multiply the function by 3.
g(x) = 3f(x) = 3*(2)^x

2. Reflection in the y-axis:
To reflect the function in the y-axis, we change the sign of the x-values.
g(-x) = 3*(2)^(-x)

3. Translation 9 units down:
To translate the function 9 units down, we subtract 9 from the function.
g(-x) - 9 = 3*(2)^(-x) - 9

4. Translation 1 unit left:
To translate the function 1 unit left, we replace x with (x + 1).
g(-(x + 1)) - 9 = 3*(2)^(-(x + 1)) - 9

Simplifying the equation, we get the final transformed function:

g(x) = 3*(2)^(-x - 1) - 9

To determine the equation of the transformed function, let's go step by step through each transformation and adjust the original equation accordingly:

1. Vertical stretch by a factor of 3:
When we vertically stretch a function, we multiply its y-values by the stretch factor. In this case, we have a vertical stretch by a factor of 3. Thus, we multiply the original function f(x) by 3:
f(x) = 3 * (2)^x

2. Reflection in the y-axis:
A reflection in the y-axis occurs when we replace the x-values with their opposites (negatives). Therefore, to reflect the function in the y-axis, we replace x with -x:
f(x) = 3 * (2)^(-x)

3. Translation 9 units down:
To translate a function downward, we subtract the desired amount from the original function. Here, we want to translate the function 9 units down, so we subtract 9 from the function:
f(x) = 3 * (2)^(-x) - 9

4. Translation 1 unit left:
For a translation to the left, we add the desired amount to the x-values. In this case, we want to translate the function 1 unit left, so we add 1 to the x-values:
f(x) = 3 * (2)^(-(x+1)) - 9

Therefore, the equation of the transformed function is f(x) = 3 * (2)^(-(x+1)) - 9.