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

Select the THREE statements that are correct.
A.g(x)
 has a y-intercept at (0,3).

B.g(x)
 has one x-intercept at (−13, 0).

C.g(x)
 is decreasing over the interval ​​{x∣x<−13}.

D.g(x)
 has an axis of symmetry at  x=−1.

E.g(x)
 has a minimum at (−1,0).


Let's analyze each statement about the function \( g(x) = 3|x+1| \) carefully.

### Step 1: Understanding the structure of the function

The function \( g(x) = 3|x + 1| \) is an absolute value function, which creates a "V"-shaped graph. The graph will have two distinct parts: one part will be decreasing and the other increasing. The vertex occurs where the expression inside the absolute value is zero, i.e., at \( x = -1 \).

### Step 2: Analyze each statement

**A. \( g(x) \) has a y-intercept at \( (0, 3) \).**

To find the y-intercept, we substitute \( x = 0 \) into the function:

\[
g(0) = 3|0 + 1| = 3 \times 1 = 3
\]

Thus, the y-intercept is \( (0, 3) \), so **this statement is correct**.

**B. \( g(x) \) has one x-intercept at \( (-13, 0) \).**

To find the x-intercept, set \( g(x) = 0 \):

\[
3|x + 1| = 0 \quad \Rightarrow \quad |x + 1| = 0 \quad \Rightarrow \quad x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

Thus, the x-intercept is at \( (-1, 0) \), not \( (-13, 0) \). Therefore, **this statement is incorrect**.

**C. \( g(x) \) is decreasing over the interval \( \{x \mid x < -13\} \).**

The function \( g(x) = 3|x+1| \) is decreasing for \( x < -1 \) because for values less than \( -1 \), the slope of the absolute value function is

Belen draws a graph of g(x)=3|x+1| and writes the following statements. Select the THREE statements that are correct. A. g(x) has a y-intercept at (0,3). B. g(x) has one x-intercept at (−13, 0). C. g(x) is decreasing over the interval ​​{x∣x<−13}. D. g(x) has an axis of symmetry at x=−1. E. g(x) has a minimum at (−1,0).

Understand the properties of the absolute value function g(x) = 3|x + 1| with detailed explanations. This problem covers key concepts such as y-intercepts, x-intercepts, intervals of increase and decrease, axis of symmetry, and minimum points. Suitable for students in Grades 9-12.

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