To analyze the given equation $y = |2x - 7|$, let’s go through each part of the question.

Step 1: Understanding the Graph of $y = |2x - 7|$

The equation $y = |2x - 7|$ represents an absolute value function, which typically has a "V" shape. The vertex occurs where the expression inside the absolute value equals zero. Solving $2x - 7 = 0$ gives: $x = \frac{7}{2} = 3.5$ So, the vertex of the graph is at $(3.5, 0)$.

Step 2: Identifying the Correct Graph

To find the correct option among A, B, C, and D, we look for:

• A V-shaped graph with its vertex at $(3.5, 0)$.
• Increasing and decreasing portions corresponding to the behavior of the absolute value function:
  • Decreasing on $(-\infty, 3.5]$
  • Increasing on $[3.5, \infty)$

Looking at the graphs in the options, Option B matches these criteria, as it has a V-shape centered at $x = 3.5$ and appears symmetric about this point.

Step 3: Answering the Questions

1. The x-intercept: The x-intercept is at the vertex, so $(3.5, 0)$.
2. Intervals of Increase and Decrease:
   • Increasing on $[3.5, \infty)$
   • Decreasing on $(-\infty, 3.5]$

Would you like more details on how I analyzed each part?

Here are five related questions to expand your understanding:

1. How do you find the vertex of an absolute value function?
2. Why does the absolute value function always have a V-shape?
3. What changes in the graph if the equation is $y = |2x + 7|$ instead?
4. How do you determine the intervals of increase and decrease for absolute value functions?
5. What would the graph look like if the equation were $y = -|2x - 7|$?

Tip: In absolute value functions, the vertex represents the point where the function changes direction from decreasing to increasing.