Which function represents the situation?
Item Cost
| Price on the Tag, $x$ | Final Cost |
| :--------------------- | :----------- |
| $10 | $0.75(10)$ |
| $20 | $0.75(20)$ |
| $30 | $0.75(30)$ |
| $40 | $0.75(40)$ |
A. 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.
B. It is linear because the function is continuous.
C. It is nonlinear because the final cost is determined by multiplying each price tag by 0.75.
D. It is nonlinear because the price tag and final cost columns do not have the same common difference.
Answer (2)
The final cost is 0.75 times the price on the tag, so the function is f ( x ) = 0.75 x .
Calculate the rate of change between consecutive points to check if it is constant.
The rate of change is constant and equal to 0.75.
The function is linear because the rate of change is constant. The correct answer 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.
Explanation
Understanding the Problem We are given a table that shows the relationship between the price on the tag, denoted as x , and the final cost. Our goal is to determine the function that represents this relationship and to determine whether the function is linear or non-linear.
Finding the Function Let's analyze the data provided in the table:
When the price on the tag is $10 , the final cost is 0.75 ( 10 ) = $7.50 .
When the price on the tag is $20 , the final cost is 0.75 ( 20 ) = $15.00 .
When the price on the tag is $30 , the final cost is 0.75 ( 30 ) = $22.50 .
When the price on the tag is $40 , the final cost is 0.75 ( 40 ) = $30.00 .
From this, we can infer that the final cost is always 0.75 times the price on the tag. Therefore, the function that represents this relationship is f ( x ) = 0.75 x , where x is the price on the tag and f ( x ) is the final cost.
Determining Linearity Now, let's determine if the function is linear or non-linear. A function is linear if the rate of change (slope) is constant. In other words, for every unit increase in x , the value of f ( x ) increases by a constant amount.
Let's calculate the rate of change between consecutive points:
Between x = 10 and x = 20 : The change in x is 20 − 10 = 10 , and the change in f ( x ) is 15 − 7.5 = 7.5 . The rate of change is 10 7.5 = 0.75 .
Between x = 20 and x = 30 : The change in x is 30 − 20 = 10 , and the change in f ( x ) is 22.5 − 15 = 7.5 . The rate of change is 10 7.5 = 0.75 .
Between x = 30 and x = 40 : The change in x is 40 − 30 = 10 , and the change in f ( x ) is 30 − 22.5 = 7.5 . The rate of change is 10 7.5 = 0.75 .
Since the rate of change is constant (0.75), the function is linear.
Conclusion Therefore, the function that represents the situation is f ( x ) = 0.75 x , and it is a linear function because the ratio of the change in the final cost compared to the rate of change in the price tag is constant.
Examples
Understanding linear functions is crucial in everyday scenarios like calculating discounts. In this case, the final cost is 75% of the original price. If you're shopping and see an item with a tag price of $50 , you can quickly calculate the final cost by multiplying $50 by 0.75, which equals $37.50 . This concept extends to various situations, such as calculating sales tax, determining proportions, or understanding simple interest rates, making it a fundamental skill in financial literacy and decision-making.
The function that represents the relationship is f ( x ) = 0.75 x , indicating a linear relationship due to the constant rate of change. The correct answer choice is A, which states that it is linear because the ratio of change in final cost to the rate of change in price tag is constant. Therefore, the final cost is calculated as 75% of the tag price, confirming linearity.
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