To determine the correct statement regarding the symmetry of the profits made by Megan's clothing boutique, we need to analyze the given data in the table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Years since 2010} & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Profit (dollars)} & 39,000 & 84,000 & 111,000 & 120,000 & 111,000 & 84,000 \\
\hline
\end{array}
\][/tex]
Let's analyze the symmetry about the line [tex]\( x = 3 \)[/tex]. This is essentially the vertical middle of the data set since [tex]\( x = 3 \)[/tex] represents the midpoint in a total span of 6 years (from 0 to 5).
First, look at the data pairs around [tex]\( x = 3 \)[/tex]:
- Profit at [tex]\( x = 1 \)[/tex] and [tex]\( x = 5 \)[/tex]: [tex]\( 84,000 \)[/tex] and [tex]\( 84,000 \)[/tex]
- Profit at [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex]: [tex]\( 111,000 \)[/tex] and [tex]\( 111,000 \)[/tex]
Both pairs are exactly equal, indicating that the profits are symmetric when reflected about the line [tex]\( x = 3 \)[/tex].
With this understanding, we can evaluate the statements:
A. The profit is not symmetric.
- This is incorrect because we have found the profit is symmetric.
B. Since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line [tex]\( x = 3 \)[/tex].
- This statement is true and aligns with our analysis.
C. Since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line [tex]\( y = 120,000 \)[/tex].
- This is incorrect as the symmetry is about [tex]\( x = 3 \)[/tex], not the profit value [tex]\( y = 120,000 \)[/tex].
D. Since the profit made in 2011 is equal to the profit made in 2015, the profit is symmetric about the line [tex]\( x = 4 \)[/tex].
- This is incorrect as well as the symmetry observed is centered on [tex]\( x = 3 \)[/tex].
The correct statement is:
B. Since the profit made in 2012 is equal to the profit made in 2014, the profit is symmetric about the line [tex]\( x = 3 \)[/tex].
