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.