Key Concepts and Formulas

• Absolute Value Function: Understanding how the absolute value function affects the graph. $y = |x|$ reflects the part of the graph below the x-axis about the x-axis.
• Transformations of Functions: Knowing how to shift a graph vertically and horizontally. $y = f(x) + c$ shifts the graph of $y = f(x)$ vertically by $c$ units. $y = f(x-c)$ shifts the graph of $y = f(x)$ horizontally by $c$ units.
• Area of a Triangle: The area of a triangle is given by $A = \frac{1}{2} \times base \times height$.

Step-by-Step Solution

Step 1: Analyze the given function and sketch the graph.

We have the function $y = ||x-1| - 2|$. To understand this, we build it up in stages:

• Start with $y = x$.
• $y = x-1$ shifts the graph 1 unit to the right.
• $y = |x-1|$ reflects the part of the graph below the x-axis about the x-axis. The vertex is at (1,0).
• $y = |x-1| - 2$ shifts the graph 2 units down. The vertex is at (1, -2).
• $y = ||x-1| - 2|$ reflects the part of the graph below the x-axis about the x-axis.

To find the x-intercepts (where $y=0$), we solve $||x-1| - 2| = 0$. This means $|x-1| - 2 = 0$, so $|x-1| = 2$. This gives us two cases:

Case 1: $x-1 = 2 \implies x = 3$ Case 2: $x-1 = -2 \implies x = -1$

The graph intersects the x-axis at $x = -1$ and $x = 3$. The "vertices" are at $x = 1 \pm 2$, so $x = -1, 3$. The minimum occurs at $x = 1$, where $y = ||1-1| - 2| = |-2| = 2$. The points where $|x-1| - 2$ would be zero are x = -1 and x = 3.

The points where $y=||x-1|-2|$ has sharp corners (i.e. the vertices of the triangles) are at x=-1, x=1, and x=3. The y values at those points are 0, 2, and 0 respectively.

Step 2: Identify the geometric shape formed.

The graph of $y = ||x-1| - 2|$ forms two triangles above the x-axis.

Step 3: Calculate the base and height of each triangle.

• Triangle 1: The base extends from $x = -1$ to $x = 1$, so the base length is $1 - (-1) = 2$. The height is $y = 2$ (at $x = 1$).
• Triangle 2: The base extends from $x = 1$ to $x = 3$, so the base length is $3 - 1 = 2$. The height is $y = 2$ (at $x = 1$).

Step 4: Calculate the area of each triangle.

• Area of Triangle 1: $A_1 = \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 2 = 2$
• Area of Triangle 2: $A_2 = \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 2 = 2$

Step 5: Calculate the total area.

The total area is the sum of the areas of the two triangles: $A = A_1 + A_2 = 2 + 2 = 4$.

Step 6: Re-examine the solution

The solution above is incorrect. The graph intersects the x-axis at -1 and 3. The minimum point is at x=1 where y=2. The region bounded by y=||x-1|-2| is the area enclosed by the two triangular shapes above the x-axis. Let's recalculate the area.

The x intercepts are -1 and 3. The minimum value of y is 2, at x=1. The graph consists of two triangles.

Triangle 1 vertices: (-1,0), (1,2), (3,0). The two triangles are symmetrical. The area of the triangle is given as 1/2 * base * height. The base extends from -1 to 3. The value of the function at x=1 is ||1-1|-2|=2. The vertices of the triangles are (-1,0), (1,2), (3,0).

The area from x=-1 to x=1 is $\int_{-1}^1 ||x-1|-2| dx$. The area from x=1 to x=3 is $\int_{1}^3 ||x-1|-2| dx$.

However, a simpler approach is to recognize two triangles. Each has a base of 2 and a height of 2. The area is (1/2)22 = 2. The total area is 2. The two triangles are congruent.

We must find the area above the x-axis. We need to find where $|x-1|-2$ is negative and reflect it. $|x-1|-2 = 0 \implies |x-1| = 2 \implies x-1 = \pm 2 \implies x = -1, 3$. The minimum is at x=1 where the y value is 2. The graph consists of two triangles. The area of each is (1/2)baseheight. The base is 2, the height is 2. The area of each is 2. The area of the entire region is 2.

The area is the integral from -1 to 3 of ||x-1|-2| dx. The function is symmetric about x=1. So we can say 2* integral from 1 to 3 of ||x-1|-2| dx. This becomes 2* integral from 1 to 3 of |x-1-2| dx or 2* integral from 1 to 3 of |x-3| dx = 2 * integral from 1 to 3 of (3-x) dx = 2*(3x-x^2/2) evaluated from 1 to 3 = 2*[(9-9/2)-(3-1/2)] = 2*[9/2 - 5/2] = 2*[4/2] = 4.

Where did the error occur? The question asks for the area bounded by the lines. This means we want the area enclosed by the graph and the x-axis. The points where ||x-1|-2| = 0 are x=-1 and x=3. The graph is two triangles with vertices at (-1,0) (1,2) (3,0). The area of each triangle is 1/2 * 2 * 2 = 2. The area of both is 4.

The solution provided is wrong. The area is 2, not 4. The area bounded by the function is 2.

Common Mistakes & Tips

• Sign Errors: Be careful with the signs when dealing with absolute values. Remember to consider both positive and negative cases.
• Sketching the Graph: Always sketch the graph to visualize the area you are trying to calculate. This helps in identifying the correct limits of integration or the base and height of geometric shapes.
• Symmetry: Look for symmetry in the function. This can simplify the calculations.

Summary

To find the area bounded by the function $y = ||x-1| - 2|$, we first sketched the graph to visualize the region. The graph consists of two symmetrical triangles above the x-axis. By calculating the base and height of each triangle, we found that the area of each triangle is 2. The total area is therefore 2.

Final Answer

The final answer is \boxed{2}.