To find the area of the given shape, I will break it into simpler geometric shapes (triangles and rectangles), calculate their areas, and sum them. Let's analyze the shape step by step:

Step 1: Break down the shape

From the graph:

• The shape consists of:
  1. A left triangle (base along $x$-axis from $-4$ to $0$).
  2. A right triangle (with the peak at $(1, 4)$ and base along $x$-axis from $0$ to $2$).
  3. A rectangle (base along $x$-axis from $0$ to $2$ and height from $y = 0$ to $y = 2$).

Step 2: Calculate individual areas

1. Area of the left triangle

The left triangle's vertices are: $(-4, 0), (-1, 4), \text{ and } (0, 0).$

• Base: $4$ units (from $-4$ to $0$).
• Height: $4$ units (from $0$ to $4$).

Using the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area} = \frac{1}{2} \times 4 \times 4 = 8.$

2. Area of the right triangle

The right triangle's vertices are: $(0, 0), (1, 4), \text{ and } (2, 2).$

• Base: $2$ units (from $0$ to $2$).
• Height: $4$ units (from $0$ to $4$).

Using the same formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area} = \frac{1}{2} \times 2 \times 4 = 4.$

3. Area of the rectangle

The rectangle's dimensions:

• Base: $2$ units (from $0$ to $2$).
• Height: $2$ units (from $0$ to $2$).

Using the formula for the area of a rectangle: $\text{Area} = \text{base} \times \text{height}$ $\text{Area} = 2 \times 2 = 4.$

Step 3: Sum the areas

Now, summing up all areas: $\text{Total Area} = \text{Area of left triangle} + \text{Area of right triangle} + \text{Area of rectangle}$ $\text{Total Area} = 8 + 4 + 4 = 16.$

Final Answer:

The total area of the shape is 16 square units.

Do you want details about any part of the solution? Let me know!
Here are some follow-up questions to explore:

1. How would the solution change if the dimensions of the rectangle were different?
2. Can you explain why we use $\frac{1}{2} \times \text{base} \times \text{height}$ for the triangle's area?
3. What are some other methods to verify the total area of a composite shape?
4. How can this method be applied to 3D solids (like finding volumes)?
5. Can the shape be divided differently for easier calculation?

Tip: Always double-check your breakdown of a composite shape to avoid missing any sections!