Let's analyze the table of values given for Kyle's butterfly collection:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline$x$ & 0 & 2 & 4 \\
\hline$f(x)$ & 5 & 20 & 80 \\
\hline
\end{tabular}
\][/tex]
In this table, [tex]$x$[/tex] represents time in years, and [tex]$f(x)$[/tex] represents the number of butterflies in Kyle's collection. Let's look at each option to determine what the number 5 represents.
1. The number of new butterflies each year
This option implies that 5 could be the number of new butterflies added each year. However, making close observations, such as [tex]$f(2) = 20$[/tex] and [tex]$f(4) = 80$[/tex], would suggest exponential growth rather than a simple addition of new butterflies each year. This option cannot be correct since the number 5 represents a specific quantity at [tex]$x = 0$[/tex].
2. The common ratio of butterfly growth
This option suggests we are looking for a multiplicative rate between the years. Upon inspection, if we were to look for a common ratio, it must be consistent across the intervals:
- From [tex]$x=0$[/tex] to [tex]$x=2$[/tex], [tex]$f(x)$[/tex] grows from 5 to 20.
- From [tex]$x=2$[/tex] to [tex]$x=4$[/tex], [tex]$f(x)$[/tex] grows from 20 to 80.
The ratio from 5 to 20 is [tex]$ \frac{20}{5} = 4$[/tex], and from 20 to 80 is [tex]$\frac{80}{20} = 4$[/tex]. These calculations confirm a ratio of 4 but do not associate directly with the 5 in context.
3. The average rate of change of butterfly growth each year
This option would calculate the change over the given interval (rate of change). To find the average rate of change:
[tex]\[
\frac{\Delta f(x)}{\Delta x} = \frac{80 - 5}{4 - 0} = \frac{75}{4} = 18.75
\][/tex]
While the average provides information per year, 18.75 is different from 5.
4. The number of butterflies Kyle started with
In the table, at [tex]$x = 0$[/tex], the number of butterflies is 5. This initial amount directly indicates the starting point of the collection.
Given all these considerations:
- The number 5 in the table represents the number of butterflies Kyle started with at year [tex]$x=0$[/tex].
The correct choice is:
The number of butterflies Kyle started with.
