Question

introduction to linear functions
pre-test

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:

⚡ Using what you learned: Rate of Change and Applications

Step 1: Identify the function type

The final cost \( y \) is determined by multiplying the price on the tag \( x \) by \( 75\% \) (or \( 0.75 \)):

$$ y = 0.75x $$

This is a linear function of the form \( y = mx \).

Step 2: Analyze the rate of change

For any linear function, the rate of change (slope) is constant:

$$ \frac{\Delta y}{\Delta x} = 0.75 $$

This means the ratio of the change in the final cost compared to the change in the price tag is constant.

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.