https://file.aiquickdraw.com/mathbot/file/17299885478551u004zei.jpg

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.

Graph the equation y = |2x - 7| and determine the x-intercept, intervals where the graph is increasing and decreasing.

Learn how to graph the absolute value equation y = |2x - 7|. This solution explains finding the x-intercept, identifying intervals of increase and decrease, and selecting the correct graph. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation