elect the correct answer.Which graph is the graph of the function 𝑟⁡(𝑥)=𝑥2+6⁢𝑥+82⁢𝑥+8 ?

To determine which graph corresponds to the function $r(x) = \frac{x^2 + 6x + 8}{2x + 8}$, we need to analyze the function step by step.

Step 1: Simplify the Function

First, let's simplify the function if possible.

$r(x) = \frac{x^2 + 6x + 8}{2x + 8}$

Factor the numerator:

$x^2 + 6x + 8 = (x + 2)(x + 4)$

So the function becomes:

$r(x) = \frac{(x + 2)(x + 4)}{2(x + 4)}$

Step 2: Cancel Common Factors

We can cancel the common factor $(x + 4)$ in the numerator and the denominator, but we must note that $x \neq -4$ because it would make the denominator zero.

$r(x) = \frac{x + 2}{2} \quad \text{for} \quad x \neq -4$

Step 3: Identify the Simplified Function

The simplified function is:

$r(x) = \frac{x + 2}{2}$

This is a linear function with a slope of $\frac{1}{2}$ and a y-intercept of 1.

Step 4: Consider the Domain

The function $r(x)$ is defined for all $x$ except $x = -4$. At $x = -4$, there is a vertical asymptote or a hole in the graph.

Step 5: Sketch the Graph

1. Linear Part: The graph of $r(x) = \frac{x + 2}{2}$ is a straight line with a slope of $\frac{1}{2}$ and a y-intercept of 1.
2. Discontinuity: There is a hole at $x = -4$.

Step 6: Match with Given Graphs

To match the function with the correct graph, look for a graph that:

• Is a straight line with a slope of $\frac{1}{2}$.
• Has a y-intercept at $y = 1$.
• Has a hole at $x = -4$.

Conclusion

The correct graph will be a straight line with the specified slope and y-intercept, and it will have a hole at $x = -4$.

Without the actual graphs to compare, this is the detailed process to identify the correct graph. If you have multiple-choice options, look for the graph that fits these criteria.