https://file.aiquickdraw.com/mathbot/file/17264959276391p7tg3ze.jpg

The problem involves the transformation of two exponential functions, \( f(x) \) and \( g(x) \), with a base of 2. Based on the graph and the question, we are asked to determine the value of \( 2a - b \), where the function \( f(x) \) is translated by a vector \( \begin{pmatrix} a \\ b \end{pmatrix} \) to form \( g(x) \).

### Key Observations:
- The graph of \( f(x) \) appears to be \( f(x) = 2^x \).
- The graph of \( g(x) \) appears to be the same function but shifted to the right and upward.
- A translation \( \begin{pmatrix} a \\ b \end{pmatrix} \) means the graph of \( f(x) \) has shifted horizontally by \( a \) units and vertically by \( b \) units to produce \( g(x) \).

### Determining the values of \( a \) and \( b \):
1. From the graph, we see that:
   - The graph of \( f(x) \) passes through the point \( (0, 1) \), which corresponds to \( 2^0 = 1 \).
   - The graph of \( g(x) \) passes through \( (1, 3) \). This suggests that \( g(x) \) is a translation of \( f(x) \) by 1 unit to the right (horizontally) and 2 units up (vertically), because \( f(1) = 2 \), but \( g(1) = 4 \), which means it is shifted upwards by 2.

Thus, the translation vector is \( \begin{pmatrix} 1 \\ 2 \end{pmatrix} \), where:
- \( a = 1 \)
- \( b = 2 \)

### Final Calculation:
Now, we need to find the value of \( 2a - b \):
\[
2a - b = 2(1) - 2 = 2 - 2 = 0
\]

### Answer:
The value of \( 2a - b \) is \( \boxed{0} \).

Let me know if you need any further details or have any questions! 

Here are five related questions you might find helpful:
1. How do translations affect the equation of a function algebraically?
2. What is the general form of an exponential function, and how do shifts modify it?
3. Can you describe the effect of a horizontal and vertical shift on any graph?
4. How can we determine translations from the coordinates of key points on a graph?
5. What other transformations (e.g., scaling or reflection) can be applied to exponential functions?

**Tip:** When solving graph translation problems, always check how specific points, like the intercepts or key points, shift between the original and transformed graphs.

Perhatikan grafik fungsi eksponen dengan bilangan pokok 2 berikut. Grafik f(x) ditranslasi oleh (a, b) menghasilkan peta grafik g(x). Tentukan nilai dari 2a - b.

This problem involves the translation of an exponential function f(x) = 2^x, with the goal of determining the value of 2a - b from the graph of g(x). The problem covers graph translation, exponential functions, and solving for transformations in the function.

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