The equation of an absolute value function f(x) and the graph of another absolute value function g(x) are shown below.

Listed on the right are four characteristics that may be true for one of the functions both of the functions, or neither of the functions.

Choices

f(x) = |x + 3| - 3

Answers

Minimum value occurs at x = 0
y-intercept is 0
x-intercepts are at 0 and -6
Minimum value is y = -3
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Let’s analyze the function \( f(x) = |x + 3| - 3 \) and how it relates to the listed characteristics.

### Step-by-step Analysis

1. **Form of the function:**
   The equation \( f(x) = |x + 3| - 3 \) is an **absolute value function**. Its general structure is \( f(x) = |x - h| + k \), where \( (h, k) \) is the **vertex** (minimum or maximum point).  
   - Here, \( h = -3 \) and \( k = -3 \), so the vertex is at \( (-3, -3) \). This will be the minimum value because the absolute value function opens upward.

2. **Minimum Value Occurs at \( x = 0 \):**
   - **False**: The minimum value occurs at \( x = -3 \), not at \( x = 0 \).

3. **y-intercept is 0:**
   - To find the \( y \)-intercept, set \( x = 0 \).
     \[
     f(0) = |0 + 3| - 3 = 3 - 3 = 0
     \]
   - **True**: The y-intercept is at \( (0, 0) \).

4. **x-intercepts are at 0 and -6:**
   - To find the \( x \)-intercepts, set \( f(x) = 0 \):
     \[
     |x + 3| - 3 = 0 \implies |x + 3| = 3
     \]
   - This gives two solutions:
     \[
     x + 3 = 3 \quad \text{or} \quad x + 3 = -3
     \]
     \[
     x = 0 \quad \text{or} \quad x = -6
     \]
   - **True**: The x-intercepts are at \( x = 0 \) and \( x = -6 \).

5. **Minimum Value is \( y = -3 \):**
   - As mentioned earlier, the vertex is at \( (-3, -3) \).
   - **True**: The minimum value is \( y = -3 \).

### Summary of Answers

- **Minimum value occurs at \( x = 0 \)**: False  
- **y-intercept is 0**: True  
- **x-intercepts are at 0 and -6**: True  
- **Minimum value is \( y = -3 \)**: True  

Let me know if you have more questions!

---

Here are **five related questions** to deepen your understanding:  
1. How do you find the vertex of any absolute value function?  
2. What transformations occur when adding or subtracting constants inside and outside the absolute value?  
3. How do you find the range of an absolute value function?  
4. How does shifting the graph horizontally or vertically affect the intercepts?  
5. Can an absolute value function ever have a maximum value instead of a minimum?

**Tip:** To confirm intercepts, always substitute 0 for \( x \) and \( f(x) \) carefully!

The equation of an absolute value function f(x) and the graph of another absolute value function g(x) are shown below. Listed on the right are four characteristics that may be true for one of the functions, both of the functions, or neither of the functions.
f(x) = |x + 3| - 3
Choices:
- Minimum value occurs at x = 0
- y-intercept is 0
- x-intercepts are at 0 and -6
- Minimum value is y = -3

Analyze the absolute value function f(x) = |x + 3| - 3 and explore its key features, including its y-intercept at (0, 0), x-intercepts at x = 0 and x = -6, and minimum value at y = -3. Suitable for Grades 9-11 students learning about absolute value functions.

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