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 $x = h$.