Let's analyze the problem step by step.
First, we observe that the final cost of a sale item is determined by multiplying the price on the tag [tex]\( x \)[/tex] by [tex]\( 75\% \)[/tex].
Mathematically, [tex]\( 75\% \)[/tex] can be written as [tex]\( 0.75 \)[/tex]. Therefore, the final cost [tex]\( y \)[/tex] can be expressed as:
[tex]\[ y = 0.75x \][/tex]
Given this relationship, let's tabulate the values as provided:
\begin{tabular}{|c|c|}
\hline
Price on the Tag, [tex]\( x \)[/tex] & Final Cost, [tex]\( y \)[/tex] \\
\hline
\[tex]$10 & \( 0.75 \times 10 = \$[/tex]7.50 \) \\
\hline
\[tex]$20 & \( 0.75 \times 20 = \$[/tex]15.00 \) \\
\hline
\[tex]$30 & \( 0.75 \times 30 = \$[/tex]22.50 \) \\
\hline
\[tex]$40 & \( 0.75 \times 40 = \$[/tex]30.00 \) \\
\hline
\end{tabular}
To determine the nature of the function [tex]\( y = 0.75x \)[/tex], let's discuss its characteristics:
1. Linearity: A function is linear if it can be described by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. In this case, [tex]\( y = 0.75x \)[/tex] fits this form with [tex]\( m = 0.75 \)[/tex] and [tex]\( b = 0 \)[/tex]. Thus, the relationship between the price on the tag and the final cost is linear.
2. Ratio of Change: The ratio of the change in the final cost to the change in the price on the tag remains constant. For instance:
[tex]\[
\frac{15.00 - 7.50}{20 - 10} = \frac{7.50}{10} = 0.75
\][/tex]
Similarly:
[tex]\[
\frac{22.50 - 15.00}{30 - 20} = \frac{7.50}{10} = 0.75
\][/tex]
This consistent ratio further confirms the linearity.
3. Continuity: A linear function like [tex]\( y = 0.75x \)[/tex] is continuous for all real numbers.
Given these observations, the correct characterization of the function that best describes the situation is:
- It is linear because the ratio of the change in the final cost compared to the rate of change in the price tag is consistent.
Therefore, the best description of the function representing the situation is that it is linear because the ratio of the change in the final cost compared to the rate of change in the price tag is consistent.
