The function given is \( f(x) = 6|x - 3| \). This is an absolute value function scaled by a factor of 6. Here's how the function behaves:

### Key Characteristics:
1. **Vertex**: The vertex of the function is at the point \( (3, 0) \), since \( |x - 3| \) equals zero when \( x = 3 \).
2. **Slope**: The slope of the lines is determined by the coefficient 6. 
   - For \( x > 3 \), the slope is positive, so the function increases with a slope of 6.
   - For \( x < 3 \), the slope is negative, so the function decreases with a slope of -6.
3. **Domain**: The domain of the function is all real numbers, \( \mathbb{R} \).
4. **Range**: Since the absolute value function is non-negative and is scaled by 6, the range of \( f(x) \) is \( [0, \infty) \).
5. **Piecewise Form**: The function can be written in piecewise form as:
   \[
   f(x) = \begin{cases} 
   6(x - 3) & \text{if } x \geq 3 \\
   6(3 - x) & \text{if } x < 3
   \end{cases}
   \]

### Example Calculations:
- \( f(0) = 6|0 - 3| = 6 \times 3 = 18 \)
- \( f(3) = 6|3 - 3| = 6 \times 0 = 0 \)
- \( f(5) = 6|5 - 3| = 6 \times 2 = 12 \)

Would you like more details or have any questions?

Here are some related questions to consider:
1. How does the graph of \( f(x) = 6|x - 3| \) compare to the graph of \( f(x) = |x - 3| \)?
2. What is the effect of changing the constant 6 to another value?
3. How would you describe the symmetry of the graph of \( f(x) \)?
4. How would the function change if the absolute value was removed?
5. What are the intercepts of this function?

**Tip**: Absolute value functions create V-shaped graphs, with the vertex indicating the point of change from increasing to decreasing or vice versa.

Explore the absolute value function f(x) = 6|x - 3| with a detailed analysis of its key characteristics, including vertex, slope, and piecewise form. Learn how this function behaves and understand the impact of the scaling factor 6. Suitable for students in Grades 9-10.

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