To determine which explicit formula represents the graph, we need to identify the pattern of the sequence shown in the graph. Let's go through the process:

1. Identify the First Term and Common Difference:

   • The first term of the sequence ($a_1$) is visible from the graph. Here, $a_1 = -4$.
   • Calculate the common difference ($d$) by finding the difference between consecutive terms. For example, the difference between $a_2$ and $a_1$ is $3$.

2. Check the Given Formulas: We have four formulas:

   • $a_n = -4 + 3(n - 1)$
   • $a_n = 3 - 4(n - 1)$
   • $a_n = -6 + 3(n - 1)$
   • $a_n = -4 + 2(n - 1)$

3. Verifying Each Option:

   • For Option (a): $a_n = -4 + 3(n - 1)$

     • The formula for the $n$-th term in an arithmetic sequence is given by: $a_n = a_1 + (n - 1)d$.
     • Here, $a_1 = -4$ and $d = 3$.
     • So, $a_n = -4 + 3(n - 1)$. This matches the formula given in option (a).

   • Verifying with graph:

     • $n = 1$: $a_1 = -4$.
     • $n = 2$: $a_2 = -4 + 3(1) = -1$.
     • $n = 3$: $a_3 = -4 + 3(2) = 2$, and so on.
     • This formula fits the graph.

Therefore, the correct formula is:

Option (a): $a_n = -4 + 3(n - 1)$.

Would you like more details on the steps or have any other questions?

Here are some related questions:

1. How do you derive the formula for an arithmetic sequence?
2. What is the general form of an arithmetic sequence?
3. How do you identify the common difference from a graph?
4. How can you check if a sequence is arithmetic or geometric?
5. What are the applications of arithmetic sequences in real-world scenarios?

Tip: Always verify formulas by plugging in values to check if they match the graph or given data points.