To find the function that represents the graph, we can use the points provided on the graph, $(0, 4)$ and $(1, 8)$, and look for an exponential function of the form:

$f(x) = a \cdot b^x$

1. Using the point $(0, 4)$: $f(0) = a \cdot b^0 = 4$ Since $b^0 = 1$, we get: $a = 4$

So, the function becomes: $f(x) = 4 \cdot b^x$

2. Using the point $(1, 8)$: $f(1) = 4 \cdot b^1 = 8$ Solving for $b$: $4b = 8 \quad \Rightarrow \quad b = 2$

Thus, the function is: $f(x) = 4 \cdot 2^x$

Now, let's match this with the options provided:

• Option A: $f(x) = 4 \cdot (2)^x$, which is the correct form.

Therefore, the correct answer is Option A.

Would you like further details on how this was calculated?

Here are five related questions to expand your understanding:

1. What are the key characteristics of an exponential function?
2. How would you determine if a graph represents an exponential growth or decay?
3. Can you explain how to find the base $b$ from a graph?
4. What happens to an exponential function as $x$ approaches negative infinity?
5. How would the function change if the point $(1, 8)$ was instead $(1, 16)$?

Tip: When identifying exponential functions, always focus on finding the base and initial value first; these will guide you to the correct form.