The problem is asking for the value of $a$ in a translation of the function $f(x) = 3^{-x}$ to the new function $g(x) = 3^{-(x+2)}$. The transformation is represented by a translation vector $T = \begin{pmatrix} a \\ 0 \end{pmatrix}$.

To solve this, let's analyze how translations affect the function:

1. The original function is $f(x) = 3^{-x}$.
2. The new function is $g(x) = 3^{-(x+2)}$.

The general form of a horizontal translation is: $f(x + a) = g(x)$ This means the function is shifted horizontally by $a$. In this case, we have: $f(x + a) = 3^{-(x + a)} = g(x) = 3^{-(x+2)}$ Equating the exponents: $-(x + a) = -(x + 2)$ Simplifying: $-x - a = -x - 2$ $-a = -2$ $a = 2$

Thus, the value of $a$ is $2$.

The correct answer is $c. \ 2$.

Do you want more details or have any other questions?

Here are 5 related questions to consider:

1. How would the function change if the translation vector was $T = \begin{pmatrix} -2 \\ 0 \end{pmatrix}$?
2. What is the effect of vertical translation on a function like $f(x) = 3^{-x}$?
3. How can you identify the direction of translation based on changes in the function’s exponent?
4. How do translations differ from dilations or reflections in function transformations?
5. What happens if the translation vector is non-zero in both components (horizontal and vertical)?

Tip: A horizontal translation changes the input $x$ by shifting it left or right, while a vertical translation affects the output.