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 x equals the number of hexagons. Let y equals the perimeter of the figure.

Question

2
2
2
2
1 hexagon
2 hexagons
3 hexagons
2
2
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Hexagons​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
  
12
  
left parenthesis 1 comma 12 right parenthesis
2
  
20
  
left parenthesis 2 comma 20 right parenthesis
3
  
28
  
left parenthesis 3 comma 28 right parenthesis
​(Simplify your​ answers.)
Part 2
Represent the relationship using words.
Multiply the number of hexagons in each figure by
  
enter your response here​, since all hexagons will have at least
▼
two
four
sides that contribute to the perimeter of the combined figure. Then add
  
enter your response here for the
▼
two
four
unaccounted for sides on the left and right ends of the combined figure to get the entire perimeter.

Answer 1

To establish the relationship between the number of hexagons and the perimeter of the figure they form, let’s analyze it step by step and fill in the provided components.

Part 1: Representing the relationship using a table

| Number of Hexagons (x) | Perimeter (y) | Ordered Pair (x, y) | |-------------------------|----------------|----------------------------| | 1 | 12 | (1, 12) | | 2 | 20 | (2, 20) | | 3 | 28 | (3, 28) |

Part 2: Representing the relationship using words

To derive the perimeter (y) based on the number of hexagons (x), we can say:

Multiply the number of hexagons in each figure by 8, since all hexagons will have at least 4 sides that contribute to the perimeter of the combined figure (4 sides per hexagon, minus the 2 shared sides where hexagons touch). Then add 4 for the unaccounted for sides on the left and right ends of the combined figure to get the entire perimeter.

Part 3: Representing the relationship using an equation

From our observations, we can deduce the equation representing the relationship between the number of hexagons (x) and the perimeter (y):

\[ y = 8x + 4 \]

Part 4: Representing the relationship using a graph

To graph this relationship, we can plot the ordered pairs obtained from our table: (1, 12), (2, 20), (3, 28).

1. Set up the axes:

   • x-axis (Number of Hexagons)
   • y-axis (Perimeter)

2. Plot the points:

   • (1, 12)
   • (2, 20)
   • (3, 28)

3. Draw the line:

   • Connect the points to illustrate that there's a linear relationship.

The slope of the line indicates that for each additional hexagon, the perimeter increases by 8 units, following the pattern established in our relationship definition.

Thus, the table, words, equation, and graph together show the relationship between the number of hexagons and the perimeter of the figure they form.