Use transformations to graph the function.                                                                                                                          v (x) = 1/3 |x+6| - 5


To graph the function \( v(x) = \frac{1}{3} |x + 6| - 5 \) using transformations, we will start with the basic absolute value function \( f(x) = |x| \) and apply a series of transformations step-by-step. Here's how to do it:

1. **Start with the basic function**: The basic graph is \( f(x) = |x| \), a "V"-shaped graph with its vertex at the origin (0,0) and symmetry along the y-axis.

2. **Horizontal shift**: The expression inside the absolute value, \( |x + 6| \), represents a horizontal shift. The graph of \( |x + 6| \) is the graph of \( |x| \) shifted 6 units to the left. The vertex is now at (-6, 0) instead of (0, 0).

3. **Vertical stretch/compression**: The coefficient \( \frac{1}{3} \) outside the absolute value affects the steepness of the graph. Specifically, it causes a vertical compression by a factor of \( \frac{1}{3} \), making the V-shape wider. Each y-coordinate of the graph is multiplied by \( \frac{1}{3} \).

4. **Vertical shift**: The constant term \( -5 \) at the end of the equation shifts the entire graph down by 5 units. The vertex, which was at (-6, 0), is now moved to (-6, -5).

### Step-by-Step Transformation Summary:
- Start with the basic function \( f(x) = |x| \).
- Shift the graph 6 units to the left.
- Compress the graph vertically by a factor of \( \frac{1}{3} \), making it wider.
- Finally, shift the entire graph 5 units downward.

### Vertex of the transformed function:
The new vertex of the graph is at the point (-6, -5).

Would you like more details or have any questions?

Here are some related questions to deepen understanding:
1. How would the graph change if we replaced \( \frac{1}{3} \) with \( 2 \)?
2. What if the constant outside the absolute value was \( +5 \) instead of \( -5 \)?
3. How do horizontal shifts affect the domain of the function?
4. How can you find the axis of symmetry from the function \( v(x) \)?
5. How would the graph look if the absolute value was negative, e.g., \( -|x+6| \)?

**Tip**: Always identify the vertex first when working with absolute value functions, as it's the key to understanding transformations.

Use transformations to graph the function v(x) = 1/3 |x+6| - 5

Learn how to graph the absolute value function v(x) = 1/3 |x+6| - 5 by applying horizontal and vertical shifts, vertical compression, and identifying the vertex at (-6, -5). This step-by-step guide is suitable for high school students in Grades 9-12.

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