graphs | War and Piecewise Functions - Free Printable

Let’s solve this step by step.

We are given a piecewise graph made of four colored line segments. We need to find the total length of all these segments combined.

Each segment is straight, so we can use the distance formula between two points:

> Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

But since many segments are horizontal or vertical, we can often just count grid units — which is easier and faster.

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Let’s identify each segment by its endpoints and color (as shown in the graph):

1. Red segment: from (-3, -3) to (-1, -1)
- This is diagonal. Let’s calculate:
- Δx = (-1) - (-3) = 2
- Δy = (-1) - (-3) = 2
- Length = √(2² + 2²) = √(4 + 4) = √8 = 2√2

2. Blue segment: from (-1, -1) to (0, -1)
- Horizontal line.
- Δx = 0 - (-1) = 1
- Δy = 0
- Length = 1

3. Green segment: from (0, -1) to (2, 1)
- Diagonal.
- Δx = 2 - 0 = 2
- Δy = 1 - (-1) = 2
- Length = √(2² + 2²) = √8 = 2√2

4. Black segment: from (2, 1) to (3, 0)
- Diagonal.
- Δx = 3 - 2 = 1
- Δy = 0 - 1 = -1 → absolute value 1
- Length = √(1² + 1²) = √2

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Now add them up:

Total length = Red + Blue + Green + Black
= 2√2 + 1 + 2√2 + √2
Combine like terms:
= (2√2 + 2√2 + √2) + 1
= 5√2 + 1

Wait — let me double-check that addition:

2√2 (red)
+ 2√2 (green)
+ √2 (black)
= 5√2
Plus blue = 1
→ Total = 5√2 + 1

That seems correct.

But let me verify if I read the endpoints correctly.

Looking again at the graph:

- Red: starts at (-3, -3), ends at (-1, -1) → yes, slope 1, length √(2²+2²)=√8=2√2 ✔️
- Blue: from (-1,-1) to (0,-1) → horizontal, length 1 ✔️
- Green: from (0,-1) to (2,1) → rise 2, run 2 → same as red → 2√2 ✔️
- Black: from (2,1) to (3,0) → down 1, right 1 → √(1+1)=√2 ✔️

Sum: 2√2 + 1 + 2√2 + √2 = 5√2 + 1

Yes.

Final Answer:
5√2 + 1