**Answer each of the following questions about the graph for y=7|x+3|+6**

1. Does the graph open up or down? Select an answer Up Down

2. Give the vertex as an ordered pair.

3. Is this point the maximum or minimum of the graph? Select an answer Maximum Minimum

4. Is the graph wider, narrower, or the same width as the parent function y=|x|? Select an answer Wider Narrower Same Width

5. From the parent fucntion y=|x| this graph has been shift horizontally ? 6 3  unit(s) to the Select an answer Left Up Right Down

6. From the parent function y=|x| this graph has been shifted vertically ? 6 3  unit(s) Select an answer Down Left Right Up  .

Let's break down each question using the given equation \( y = 7|x + 3| + 6 \), which represents a transformation of the parent function \( y = |x| \).

1. **Does the graph open up or down?**  
   - The coefficient of \( |x+3| \) is positive (7), so the graph opens **Up**.

2. **Give the vertex as an ordered pair.**  
   - The vertex form of the absolute value function is \( y = a|x-h| + k \), where \( (h, k) \) is the vertex.
   - In this case, the vertex is \( (-3, 6) \).

3. **Is this point the maximum or minimum of the graph?**  
   - Since the graph opens upwards, the vertex represents the lowest point (minimum).  
   - Therefore, this point is the **Minimum**.

4. **Is the graph wider, narrower, or the same width as the parent function \( y = |x| \)?**  
   - The coefficient of \( |x+3| \) is 7, which stretches the graph vertically, making it **Narrower** than the parent function \( y = |x| \).

5. **From the parent function \( y = |x| \), this graph has been shifted horizontally**:  
   - The graph shifts horizontally by 3 units to the **Left** (since \( x + 3 \) means a shift left by 3 units).

6. **From the parent function \( y = |x| \), this graph has been shifted vertically**:  
   - The graph shifts vertically by 6 units **Up** (since the constant is +6).

### Summary:
1. **Up**
2. **(-3, 6)**
3. **Minimum**
4. **Narrower**
5. **3 units to the Left**
6. **6 units Up**

Would you like more details on any of these transformations or concepts?

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Here are 5 questions that extend this topic:
1. What happens if the coefficient of the absolute value function is negative?
2. How does changing the constant outside the absolute value affect the graph's vertical position?
3. How would the graph shift if the equation was \( y = 7|x - 2| + 6 \)?
4. What is the effect of changing the coefficient inside the absolute value (e.g., \( y = 7|2x + 3| + 6 \))?
5. How does the steepness of the graph change with different coefficients of the absolute value term?

**Tip:** The graph of an absolute value function is symmetric with respect to its vertex, meaning the left and right branches mirror each other.

Answer each of the following questions about the graph for y=7|x+3|+6:
1. Does the graph open up or down?
2. Give the vertex as an ordered pair.
3. Is this point the maximum or minimum of the graph?
4. Is the graph wider, narrower, or the same width as the parent function y=|x|?
5. From the parent function y=|x| this graph has been shifted horizontally by how many units and in which direction?
6. From the parent function y=|x| this graph has been shifted vertically by how many units and in which direction?

Understand how to analyze and graph the absolute value function y=7|x+3|+6. Learn about the vertex, direction of the graph, and how it compares to the parent function y=|x|, including horizontal and vertical shifts. Suitable for Grades 9-11 students.

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