To analyze the function \( y = -3|2x - 6| + 4 \), let's break it down step by step.

### 1. Identify the vertex and transformations
The expression \( |2x - 6| \) represents an absolute value function, which is V-shaped. The function \( y \) is vertically stretched by a factor of 3 and reflected over the x-axis because of the negative sign in front of it.

### 2. Find the vertex
To find the vertex of the absolute value function, set the inside of the absolute value to zero:

\[
2x - 6 = 0 \implies 2x = 6 \implies x = 3
\]

Substituting \( x = 3 \) back into the equation to find \( y \):

\[
y = -3|2(3) - 6| + 4 = -3|0| + 4 = 4
\]

Thus, the vertex of the function is \( (3, 4) \).

### 3. Determine the behavior
1. For \( x < 3 \):
   \[
   y = -3(6 - 2x) + 4 = -3(6 - 2x) + 4 = 6x - 18 + 4 = 6x - 14
   \]

2. For \( x \geq 3 \):
   \[
   y = -3(2x - 6) + 4 = -6x + 18 + 4 = -6x + 22
   \]

### 4. Summary of the piecewise function
The function can be expressed as:
\[
y =
\begin{cases}
6x - 14 & \text{if } x < 3 \\
-6x + 22 & \text{if } x \geq 3
\end{cases}
\]

### 5. Key points
- Vertex: \( (3, 4) \)
- The slope is positive (6) for \( x < 3 \) and negative (-6) for \( x \geq 3 \).

Would you like more details or have any questions? Here are some related questions to consider:

1. How does the reflection over the x-axis affect the graph?
2. What would happen to the graph if the coefficient of the absolute value term was positive?
3. How do you find the intercepts of the function?
4. What is the domain and range of this function?
5. How would you graph this piecewise function? 

**Tip:** Always identify the vertex of an absolute value function to understand its transformations clearly.

Explore the analysis of the function y = -3|2x - 6| + 4, including its vertex at (3, 4), piecewise representation, and the impact of transformations. Ideal for students in Grades 9-11 studying absolute value functions.

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