What is an equation that model's the trend in #7?

the relationship of the rectangles and the figure number is QUADRATIC.

The graph is also attache below.

Step-by-step explanation:

As we check that

Mapping this information in data set as follows.

x (Figure Number)                      1         2        3

y (Number of Rectangles)          1         3        6        

Here, x represents the Figure number and y represents the number of rectangles.

CHECKING IF THE RELATIONSHIP IS LINEAR

Taking the common difference of y-values

As the common difference between two values is not same. Hence, the relation can not be linear.

CHECKING IF THE RELATIONSHIP IS EXPONENTIAL

Now, taking the common ratio of y- values.

As the common ratio between two values is not same. Hence, the relation can not be exponential.

Since, the relationship is NEITHER linear NOR exponential as we have checked. Lets check if the relationship is quadratic or not.

CHECKING IF THE RELATIONSHIP IS QUADRATIC

Now, let us see if the data set may establish any pattern.

i.e.

As the y-values are showing a pattern such as:

1 + 2 = 3

3 + 3 = 6

so the next term can be obtained by following the same pattern such as

6 + 4 = 10

10+ 5 = 15

15 + 6 = 21

So, the y values become 1, 3, 6, 10, 15, 21,...

In order to check the quadratic relationship between the pattern of y-values, we need to check whether the second common different is constant or not.

So,

3 - 1 = 2

6 - 3 = 3               →      3 - 2 = 1  (second common difference)

10 - 6 = 4              →       4 - 3 = 1  (second common difference)

15 - 10 = 5             →       5 - 4 = 1  (second common difference)

21 - 15 = 6             →        6 - 5 = 1  (second common difference)

From the above calculation we determine that the second common difference remains same. Therefore, the relationship of the rectangles and the figure number is QUADRATIC.

The graph is also attache below.