Question

In the diagram​ below, what is the relationship between the number of hexagons and the perimeter of the figure they​ form? Represent this relationship using a​ table, words, an​ equation, and a graph. Let xequalsthe number of hexagons. Let yequalsthe perimeter of the figure.
2
2
2
2
1 hexagon
2 hexagons
3 hexagons
2
2
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Part 1
Represent the relationship using a table. Complete the table below.
Number of Hexagons​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
  
enter your response here
  
enter your response here
2
  
enter your response here
  
enter your response here
3
  
enter your response here
  
enter your response here
​(Simplify your​ answers.)

Answer 1

To find the relationship between the number of hexagons and the perimeter of the figure they form, we can analyze the shape created by the hexagons.

### Analysis:
When we have hexagons sharing sides, adding more hexagons will not lead to a linear increase in perimeter. Each hexagon has 6 sides, but when they share sides, we lose some perimeter.

1. **1 hexagon:** The perimeter is simply the perimeter of one hexagon.
- Perimeter = 6 (since each hexagon has 6 sides).

2. **2 hexagons:** When two hexagons are placed next to each other, they share one side.
- Perimeter = 6 (hexagon 1) + 6 (hexagon 2) - 2 (the shared side) = 10.

3. **3 hexagons:** When three hexagons are arranged in a line, they share two sides.
- Perimeter = 6 (hexagon 1) + 6 (hexagon 2) + 6 (hexagon 3) - 4 (the two shared sides) = 14.

### Table Representation:
Now, we can create a table based on this analysis.

| Number of Hexagons (x) | Perimeter (y) | Ordered Pair (x, y) |
|------------------------|----------------|----------------------|
| 1 | 6 | (1, 6) |
| 2 | 10 | (2, 10) |
| 3 | 14 | (3, 14) |

### Words:
- The perimeter increases as the number of hexagons increases, but not linearly. Each hexagon contributes 6 sides, but shared sides reduce the total perimeter.

### Equation:
From the values in our table, we can identify a pattern in the perimeter for each number of hexagons. The relationship can be derived as:
\[ y = 2x + 4 \]
Where:
- \( x \) is the number of hexagons.
- \( y \) is the perimeter.

### Graph:
To graph this relationship, you would plot the ordered pairs \( (1, 6) \), \( (2, 10) \), and \( (3, 14) \) on a coordinate system with \( x \) on the horizontal axis and \( y \) on the vertical.
You will likely see a line that starts from the point (1, 6) and rises, indicating the increasing perimeter with more hexagons.

### Summary:
The relationship shows a pattern where each additional hexagon adds approximately 4 units to the perimeter after the first. This is due to the sharing of edges between hexagons. The simplest equation representing this would be \( y = 2x + 4 \), and visualizing it on a graph would show a linear trend, reflecting the incremental growth in perimeter.