Question

the final cost of a sale item is determined by multiplying the price on the tag by 75%. which best describes the function that represents the situation?
item cost

price on the tag, x	final cost
$20	0.75(20)
$30	0.75(30)
$40	0.75(40)

• 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.
• it is linear because the function is continuous.
• it is nonlinear because the final cost is determined by multiplying each price tag by 0.75.
• it is nonlinear because the price tag and final cost columns do not have the same common difference.

Explanation:

Step1: Recall linear function definition

A linear function has a constant rate of change (slope). The general form is \( y = mx + b \), where \( m \) is the slope (constant rate of change) and \( b = 0 \) here (no y - intercept addition).

Step2: Analyze the given function

Let the price on the tag be \( x \) and the final cost be \( y \). From the table, \( y=0.75x \). The rate of change (slope) \( m = 0.75 \), which is constant.

• For the first option: The rate of change (change in final cost over change in price tag) is \( \frac{0.75(20)-0.75(10)}{20 - 10}=\frac{15 - 7.5}{10}=\frac{7.5}{10}=0.75 \), and for other intervals (e.g., from \( x = 20 \) to \( x = 30 \)) \( \frac{0.75(30)-0.75(20)}{30 - 20}=\frac{22.5 - 15}{10}=\frac{7.5}{10}=0.75 \). So the ratio of change in final cost to change in price tag is constant (0.75), which fits the linear function definition.
• For the second option: A function being continuous is not the defining property of a linear function (discrete linear functions like this one with integer - valued \( x \) are still linear). So this is incorrect.
• For the third option: Multiplying by a constant (0.75) gives a linear function (since \( y = 0.75x \) is linear), so this is incorrect.
• For the fourth option: The price tag has a common difference of \( 10 \) (20 - 10 = 10, 30 - 20 = 10, etc.). The final cost: \( 0.75(20)-0.75(10)=15 - 7.5 = 7.5 \), \( 0.75(30)-0.75(20)=22.5 - 15 = 7.5 \), so the common difference of final cost is \( 7.5 \). The fact that the common differences are different (10 vs 7.5) does not make it non - linear. The rate of change (slope) is \( \frac{7.5}{10}=0.75 \), which is constant, so it is linear. So this option is incorrect.

Answer:

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.