The given function is $y = \frac{ax + b}{x + c}$, and you want to find the value of $b$ based on the graph provided.

To determine $b$, we can analyze the behavior of the function. The key properties of rational functions like this one typically involve:

1. Vertical asymptotes (where the denominator equals zero, i.e., $x + c = 0$).
2. Horizontal asymptotes (the behavior of the function as $x \to \infty$).

From the graph, it appears there is a vertical asymptote at $x = -1$ and the graph approaches a horizontal asymptote at $y = 1$ as $x \to \infty$.

Step 1: Analyze the vertical asymptote

The vertical asymptote occurs where the denominator is zero. This gives: $x + c = 0 \quad \Rightarrow \quad c = -1.$

Step 2: Analyze the horizontal asymptote

For the horizontal asymptote, as $x \to \infty$, the function behaves like: $y = \frac{ax + b}{x + c} \approx \frac{ax}{x} = a.$ From the graph, we observe that the horizontal asymptote is at $y = 1$, so $a = 1$.

Step 3: Use the value of $a$ and $c$ to find $b$

Now, substitute $a = 1$ and $c = -1$ into the function: $y = \frac{x + b}{x - 1}.$ We can use the graph's known point, which looks to be $(0, -1)$, to find $b$. Substituting $x = 0$ and $y = -1$ into the equation: $-1 = \frac{0 + b}{0 - 1} \quad \Rightarrow \quad -1 = \frac{b}{-1} \quad \Rightarrow \quad b = 1.$

Thus, the value of $b$ is 1.

Would you like further clarification on any part of this solution or have additional questions? Here are some related questions you might find useful:

1. How do vertical and horizontal asymptotes relate to the structure of rational functions?
2. How do you derive the equation for the horizontal asymptote in a rational function?
3. What would happen if $a$ were different from 1 in this problem?
4. How can you use graphing software to visualize rational functions?
5. What is the role of the numerator and denominator in shaping the graph of a rational function?

Tip: Always check the horizontal asymptote by considering the highest degree terms in both the numerator and denominator for large values of $x$.