The area (in sq. units) bounded by `y = 2 - |x - 2|` and the x-axis is

Answer

To find the area bounded by the curve \( y = 2 - |x - 2| \) and the x-axis, we can follow these steps:

### Step 1: Understand the function
The function \( y = 2 - |x - 2| \) represents a V-shaped graph that opens downwards. The vertex of this graph occurs at the point where \( |x - 2| \) is minimized, which is at \( x = 2 \).

### Step 2: Find the vertex
To find the vertex, substitute \( x = 2 \) into the function:
\[ y = 2 - |2 - 2| = 2 - 0 = 2 \]
Thus, the vertex of the graph is at the point \( (2, 2) \).

### Step 3: Determine the x-intercepts
To find where the graph intersects the x-axis, set \( y = 0 \):
\[ 0 = 2 - |x - 2| \]
This simplifies to:
\[ |x - 2| = 2 \]
This absolute value equation gives us two cases:
1. \( x - 2 = 2 \) → \( x = 4 \)
2. \( x - 2 = -2 \) → \( x = 0 \)

Thus, the x-intercepts are at the points \( (0, 0) \) and \( (4, 0) \).

### Step 4: Sketch the graph
The graph has a vertex at \( (2, 2) \) and intersects the x-axis at \( (0, 0) \) and \( (4, 0) \). The graph is symmetric about the line \( x = 2 \).

### Step 5: Calculate the area
The area bounded by the curve and the x-axis forms a triangle with the following vertices:
- \( O(0, 0) \)
- \( A(2, 2) \)
- \( B(4, 0) \)

To find the area of triangle \( OAB \):
- Base \( OB = 4 - 0 = 4 \) units
- Height from point \( A \) to the x-axis is \( 2 \) units

The area \( A \) of triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 4 \times 2 = 4 \text{ square units} \]

### Final Answer
The area bounded by the curve \( y = 2 - |x - 2| \) and the x-axis is \( \boxed{4} \) square units.

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