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Let's solve the problem step-by-step.

### Problem Analysis

1. **Given Equation**: \( y = |2x - 13| \)
2. **Tasks**:
   - Find the x-intercept.
   - Determine intervals where the graph is increasing and decreasing.
   - Identify the correct graph from options A, B, C, and D.

### Solution Steps

#### Step 1: Finding the X-Intercept
To find the x-intercept, set \( y = 0 \):
\[
0 = |2x - 13|
\]
This implies:
\[
2x - 13 = 0
\]
Solving for \( x \):
\[
x = \frac{13}{2} = 6.5
\]
So, the x-intercept is \( (6.5, 0) \).

#### Step 2: Determine Intervals of Increase and Decrease
Since this is an absolute value function, it has a vertex (turning point) where the expression inside the absolute value equals zero:
\[
2x - 13 = 0 \Rightarrow x = 6.5
\]
- **For \( x < 6.5 \)**: \( y = 13 - 2x \), which is a decreasing function as \( x \) increases.
- **For \( x > 6.5 \)**: \( y = 2x - 13 \), which is an increasing function as \( x \) increases.

Thus:
- The graph is **increasing on the interval \( (6.5, \infty) \)**.
- The graph is **decreasing on the interval \( (-\infty, 6.5) \)**.

#### Step 3: Identify the Correct Graph
From these observations:
- The correct graph should have a vertex at \( (6.5, 0) \), with the graph decreasing to the left of \( x = 6.5 \) and increasing to the right.
- **Option A** seems to match this description, with a V-shape centered at \( (6.5, 0) \), decreasing on the left and increasing on the right.

### Summary of Answers
- **X-intercept**: \( (6.5, 0) \)
- **Increasing interval**: \( (6.5, \infty) \)
- **Decreasing interval**: \( (-\infty, 6.5) \)
- **Correct Graph**: Option A

Would you like further clarification on any of these steps?

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### Related Questions

1. How do you find the x-intercept of an absolute value function in general?
2. Why does the absolute value function create a V-shaped graph?
3. What are other methods to determine increasing and decreasing intervals for more complex functions?
4. How would the solution change if the function were \( y = |3x - 10| \) instead?
5. What transformations affect the shape and position of an absolute value graph?

### Tip
Absolute value graphs form V-shapes because they mirror the linear part of the function across the x-axis when the expression inside becomes negative.

Graph the equation y = |2x - 13| by hand. (a) Find the x-intercept. (b) Determine where the graph is increasing and where it is decreasing.

Learn how to graph the absolute value function y = |2x - 13| and find the x-intercept. This solution includes steps to identify intervals where the graph is increasing and decreasing. Suitable for students in Grades 9-11.

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