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.

mark this and return

Explanation:

Step1: Recall linear function definition

A linear function has the form \( y = mx + b \), where \( m \) is the slope (constant rate of change) and \( b = 0 \) here (since no base cost). The rate of change (slope) is \( \frac{\Delta y}{\Delta x} \).

Step2: Calculate rate of change

For \( x = 10 \), \( y = 7.5 \); \( x = 20 \), \( y = 15 \). \( \Delta y = 15 - 7.5 = 7.5 \), \( \Delta x = 20 - 10 = 10 \). Slope \( m=\frac{7.5}{10}=0.75 \).
For \( x = 20 \) to \( x = 30 \), \( \Delta y = 22.5 - 15 = 7.5 \), \( \Delta x = 10 \), slope \( 0.75 \). Constant slope means constant rate of change.

Step3: Analyze options

• Option 1: Correct, as rate of change (ratio of \( \Delta \text{final cost} \) to \( \Delta \text{price tag} \)) is \( 0.75 \) (constant).
• Option 2: Linear functions can be discrete (like this table) or continuous; "continuous" isn't the reason for linearity here.
• Option 3: Multiplying by 0.75 is linear (\( y = 0.75x \)), so this is wrong.
• Option 4: The "price tag" (x) has common difference 10, "final cost" (y) has common difference \( 0.75\times10 = 7.5 \), but linearity depends on slope (ratio), not same common difference. So this is wrong.

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.