Let's analyze each statement in the context of the function $m(x) = 3|x+4|$.

1. Statement A: $m(x)$ has an axis of symmetry at $x = -4$.

   • This is true. The absolute value function $|x + 4|$ has an axis of symmetry at $x = -4$. Since multiplying by 3 only changes the vertical stretch and not the symmetry, the function $m(x) = 3|x + 4|$ also has an axis of symmetry at $x = -4$.

2. Statement B: $m(x)$ has a y-intercept at $(0, -4)$.

   • This is false. To find the y-intercept, we set $x = 0$: $m(0) = 3|0 + 4| = 3|4| = 12$ So, the y-intercept is $(0, 12)$, not $(0, -4)$.

3. Statement C: $m(x)$ is decreasing over the interval $(-\infty, -4)$.

   • This is true. For the function $m(x) = 3|x + 4|$, when $x < -4$, the absolute value function $|x + 4|$ decreases as $x$ approaches $-4$. Therefore, $m(x)$ is decreasing over $(-\infty, -4)$.

4. Statement D: $m(x)$ represents 3 times the distance, $x$, from $-4$.

   • This is true. The function $m(x) = 3|x + 4|$ represents 3 times the absolute value (distance) of $x$ from $-4$.

5. Statement E: $m(x)$ has a minimum at $(0, -4)$.

   • This is false. The function $m(x) = 3|x + 4|$ reaches its minimum at the vertex, which occurs at $x = -4$. The minimum value is $m(-4) = 3| -4 + 4 | = 0$. So, the minimum point is $(-4, 0)$.

Correct Statements:

1. A: $m(x)$ has an axis of symmetry at $x = -4$.
2. C: $m(x)$ is decreasing over the interval $(-\infty, -4)$.
3. D: $m(x)$ represents 3 times the distance, $x$, from $-4$.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. What is the vertex form of the absolute value function?
2. How do transformations affect the graph of an absolute value function?
3. How can you find the range of an absolute value function?
4. What is the significance of the vertex in absolute value functions?
5. How do we determine whether an absolute value function is increasing or decreasing?

Tip: Always check the vertex and axis of symmetry first when analyzing absolute value functions, as they often give you key information about the function's behavior.