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Let's analyze the function \( g(x) = 3|x + 1| \) and determine which three statements are correct.

### 1. **Statement: \( g(x) \) is decreasing over the interval \( x < -\frac{1}{3} \).**
- The function \( g(x) = 3|x + 1| \) is a V-shaped graph because it is an absolute value function. The vertex of the function occurs at \( x = -1 \), where the absolute value expression changes. 
- The function decreases to the left of \( x = -1 \) and increases to the right. Thus, this statement is **incorrect** because the function decreases for \( x < -1 \), not for \( x < -\frac{1}{3} \).

### 2. **Statement: \( g(x) \) has a y-intercept at \( (0, 3) \).**
- The y-intercept is found by evaluating the function at \( x = 0 \):
  \[
  g(0) = 3|0 + 1| = 3(1) = 3
  \]
  Therefore, the y-intercept is \( (0, 3) \), and this statement is **correct**.

### 3. **Statement: \( g(x) \) has an axis of symmetry at \( x = -1 \).**
- Since \( g(x) = 3|x + 1| \) is an absolute value function, its graph is symmetric around the point where the expression inside the absolute value equals zero, i.e., at \( x = -1 \). Therefore, the axis of symmetry is \( x = -1 \), making this statement **correct**.

### 4. **Statement: \( g(x) \) has a minimum at \( (-1, 0) \).**
- The minimum value of the absolute value function occurs at the vertex, which is at \( x = -1 \). Substituting \( x = -1 \) into the function:
  \[
  g(-1) = 3|(-1) + 1| = 3(0) = 0
  \]
  Therefore, the minimum occurs at \( (-1, 0) \), and this statement is **correct**.

### Correct Answers:
The three correct statements are:
1. \( g(x) \) has a y-intercept at \( (0, 3) \).
2. \( g(x) \) has an axis of symmetry at \( x = -1 \).
3. \( g(x) \) has a minimum at \( (-1, 0) \).

Would you like more details on any of the steps, or do you have any other questions?

Here are some related questions to explore:
1. How do you determine the vertex of an absolute value function?
2. What is the general shape of an absolute value function's graph?
3. How can you find the y-intercept of a function?
4. How do you check if a function is decreasing or increasing on an interval?
5. What are the properties of even and odd functions, and does symmetry play a role?

**Tip:** For absolute value functions like \( g(x) = a|x - h| + k \), the point \( (h, k) \) represents the vertex, and the function has symmetry about \( x = h \).

Belen draws a graph of g(x) = 3|x + 1| and writes the following statements. Select the THREE statements that are correct.

Learn how to analyze the function g(x) = 3|x + 1|, determine its axis of symmetry, y-intercept, and minimum point. Understand why it decreases over specific intervals and which statements about its graph are correct.

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