The problem in the image asks about translating the function $f(x) = 2^{3x - 1}$ by the translation vector $T = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$.

To solve this, we need to understand how translation works on a function. A translation by $T = \begin{pmatrix} a \\ b \end{pmatrix}$ affects the function as follows:

• The horizontal translation $a$ shifts the input $x$ by $-a$, so the new input becomes $x - a$.
• The vertical translation $b$ adds $b$ to the function's output.

Given that $T = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$, the function undergoes the following changes:

• Horizontal shift by $-4$, which modifies $x$ as $x - 4$.
• Vertical shift by $-1$, which subtracts 1 from the function's output.

Step 1: Apply Horizontal Shift

We replace $x$ with $x - 4$ in the function: $f(x - 4) = 2^{3(x - 4) - 1} = 2^{3x - 12 - 1} = 2^{3x - 13}.$

Step 2: Apply Vertical Shift

Now we subtract 1 from the result to account for the vertical translation: $f_T(x) = 2^{3x - 13} - 1.$

Thus, the translated function is: $f_T(x) = 2^{3x - 13} - 1.$

Would you like further details or have any questions?

Related Questions:

1. How do you determine the direction and magnitude of a translation vector?
2. How would the function change if the translation vector was $T = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$?
3. Can you translate other types of functions, like quadratic or trigonometric functions, in a similar way?
4. How does horizontal and vertical translation affect the graph of the function?
5. How would the solution change if the function were $f(x) = 3x^2$ instead of an exponential?

Tip: Always remember that a horizontal translation modifies the input $x$, while a vertical translation affects the output of the function directly.