To understand which description best represents the function, we first examine the relationship between the price on the tag and the final cost.
Given data: The final cost is determined by multiplying the price on the tag by 75%.
Let's analyze the table provided:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Price on the Tag, } x & \text{Final Cost} \\
\hline
\$10 & 0.75 \times 10 = \$7.50 \\
\hline
\$20 & 0.75 \times 20 = \$15.00 \\
\hline
\$30 & 0.75 \times 30 = \$22.50 \\
\hline
\$40 & 0.75 \times 40 = \$30.00 \\
\hline
\end{array}
\][/tex]
We can see a linear relationship here because the final cost is directly proportional to the price on the tag. This can be represented by the equation [tex]\( y = 0.75x \)[/tex], where [tex]\( y \)[/tex] is the final cost and [tex]\( x \)[/tex] is the price on the tag.
To show that this function is linear, let's consider the following aspects:
1. Proportionality:
- For each increment of [tex]\(\$10\)[/tex] in the price on the tag, the increment in the final cost is constant at [tex]\(\$7.50\)[/tex].
- Increment from [tex]\(\$10\)[/tex] to [tex]\(\$20: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- Increment from [tex]\(\$20\)[/tex] to [tex]\(\$30: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- Increment from [tex]\(\$30\)[/tex] to [tex]\(\$40: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- The ratio of change [tex]\(\frac{\Delta y}{\Delta x} = \frac{7.50}{10} = 0.75\)[/tex] remains constant, indicating a linear relationship.
2. Linearity:
- A function is linear if it can be described by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In this case, [tex]\( y = 0.75x + 0 \)[/tex], confirming it is a linear function with a slope of 0.75 and an intercept of 0.
3. Continuity:
- The function [tex]\( y = 0.75x \)[/tex] is continuous, and there are no breaks or gaps in the graph.
Based on this analysis, the best description 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 constant.
