Use technology to find points and then graph the function y, equals, minus, 2, vertical bar, x, minus, 1, vertical bar, minus, 2, commay=−2∣x−1∣−2, following the instructions below.

To graph the function \( y = -2 |x - 1| - 2 \) and find key points, we can break the process into the following steps:

### Step 1: Rewrite the Equation
The given equation is:
\[
y = -2 |x - 1| - 2
\]
This is a transformation of the absolute value function \( y = |x| \). 

- The term \( |x - 1| \) indicates a **horizontal shift** 1 unit to the right.
- The multiplication by -2 indicates a **vertical stretch** by a factor of 2 and a **reflection** across the x-axis.
- The constant -2 at the end indicates a **vertical shift** 2 units down.

### Step 2: Key Points Calculation
We will find a few key points by plugging in values for \( x \) and solving for \( y \):

1. **Vertex**: The vertex of the absolute value function occurs when \( x = 1 \) because \( |x - 1| = 0 \) at \( x = 1 \).
   - When \( x = 1 \), \( y = -2 |1 - 1| - 2 = -2(0) - 2 = -2 \).
   - So the vertex is \( (1, -2) \).

2. **Other points**: Now, let's calculate \( y \) for a few other values of \( x \):
   
   - When \( x = 0 \), \( y = -2 |0 - 1| - 2 = -2(1) - 2 = -4 \).
   - So the point is \( (0, -4) \).
   
   - When \( x = 2 \), \( y = -2 |2 - 1| - 2 = -2(1) - 2 = -4 \).
   - So the point is \( (2, -4) \).
   
   - When \( x = -1 \), \( y = -2 |-1 - 1| - 2 = -2(2) - 2 = -6 \).
   - So the point is \( (-1, -6) \).
   
   - When \( x = 3 \), \( y = -2 |3 - 1| - 2 = -2(2) - 2 = -6 \).
   - So the point is \( (3, -6) \).

### Step 3: Graphing the Function
The points we have found are:
- Vertex: \( (1, -2) \)
- \( (0, -4) \), \( (2, -4) \)
- \( (-1, -6) \), \( (3, -6) \)

The graph will have a "V" shape, opening **downward** due to the negative coefficient of the absolute value term. The vertex is at \( (1, -2) \), and the function decreases symmetrically away from this point.

### Step 4: Graph Description
To graph this function:
- Plot the vertex at \( (1, -2) \).
- Plot additional points such as \( (0, -4) \), \( (2, -4) \), \( (-1, -6) \), and \( (3, -6) \).
- Draw a V-shaped graph that opens downward, with symmetry about the vertical line \( x = 1 \).

### Graph Example:
```
    y
    ^
  -6|   .       .               .       .
  -5|
  -4|     .   .                   .   .
  -3|
  -2|        V
  -1|
   0|-----------------------------> x
      -3 -2 -1  0   1   2   3
```

Would you like a detailed graph drawn, or any additional information about the steps above?

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Here are 5 related questions to expand your understanding:
1. How does changing the coefficient of the absolute value function affect the graph?
2. What happens if we replace \( |x - 1| \) with \( |x + 1| \)?
3. How do vertical and horizontal shifts impact the symmetry of the graph?
4. Can you graph a similar equation, such as \( y = 3 |x - 2| + 1 \)?
5. How would the graph change if we applied a horizontal stretch to \( |x - 1| \)?

**Tip**: Always identify the vertex of the absolute value function first, as it defines the key point of symmetry.

Use technology to find points and then graph the function y = -2|x - 1| - 2, following the instructions below.

Learn how to graph the function y = -2|x - 1| - 2 by finding key points such as the vertex and additional points on the graph. This tutorial covers algebraic concepts, including absolute value transformations, horizontal and vertical shifts, and reflection across the x-axis. Suitable for students in Grades 9-11.

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