Linear Functions from Situations Quick Check

Question

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Question
Use the image to answer the question.

A line with an arrow at the bottom is plotted in quadrant 1 of a coordinate plane. The x-axis is labeled Months, and the y-axis is labeled Amount in Dollars.

Barton took out a car loan of $2,625. He makes monthly payments of $175. The graph represents the remaining balance after each payment. Which example models the correct slope?

(1 point)
Responses

m=2,450−2,6250−1
m equals Start Fraction 2,450 minus 2,625 over 0 minus 1 End Fraction

m=1,050−1,2259−8
m equals Start Fraction 1,050 minus 1,225 over 9 minus 8 End Fraction

m=9−81,050−1,225
m equals Start Fraction 9 minus 8 over 1,050 minus 1,225 End Fraction

m=525−70011−12
m equals Start Fraction 525 minus 700 over 11 minus 12 End Fraction
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Answer 1

To find the correct slope that models the situation described, we need to understand how the slope of a linear function is calculated. The slope (m) is defined as the change in the y-value (amount of dollars remaining) divided by the change in the x-value (months).

In Barton’s case, he starts with a car loan of $2,625 and makes monthly payments of $175. Each month, the remaining balance decreases by $175. Therefore, the slope (m) would be negative because the balance decreases over time.

Let's evaluate the slope from the given options, but keep in mind that it should reflect a decrease of $175 for every month (a negative increase in the balance).

1. m = (2,450 - 2,625) / (0 - 1)
   This represents the change in balance from $2,625 to $2,450 after 1 month, which calculates to:
   \( m = (-175) / (-1) = 175 \). This is incorrect as the slope should be -175.

2. m = (1,050 - 1,225) / (9 - 8)
   This represents the change from $1,225 to $1,050 in 1 month (9 to 8). This calculates to:
   \( m = (-175) / (1) = -175 \). This is correct.

3. m = (9 - 8) / (1,050 - 1,225)
   This change doesn't measure what we need. This should be a decrease in balance, so it will lead to an incorrect calculation.

4. m = (525 - 700) / (11 - 12)
   This does represent a change in balance, but it calculates to:
   \( m = (-175) / (-1) = 175 \), again incorrect for the same reason as option 1.

The correct answer representing the slope would be the second option:

m = (1,050 - 1,225) / (9 - 8) indicating a monthly payment reducing the balance by $175.