Financing Frank just bought a motorcycle. After making the...

Financing Frank just bought a motorcycle. After making the down payment, the amount to be financed is $\$ 28,250$. Using a $0 \%$ (or interest-free) loan, the monthly payment is $\$ 387.50$. The amount still owed on the motorcycle is a function of time given by $A(t)=\$ 28,250-\$ 387.50 t$, where $t$ is the number of months after Frank bought the motorcycle. How much is still owed a) 2 months, b) 15 months after Frank bought the motorcycle?

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Key Concepts

Linear Functions

Linear functions represent relationships where one quantity changes at a constant rate relative to another. They are typically expressed in the form y = mx + b, where m is the rate of change (slope) and b is the initial value (y-intercept). This concept is widely used to model financial situations, such as loan repayments, where regular payments consistently decrease the amount owed.

Slope-Intercept Form

The slope-intercept form of a linear equation, y = mx + b, helps in understanding the impact of each component in practical situations. The intercept (b) represents the starting point or initial amount, while the slope (m) indicates how much the output changes for every single unit increase in the input. This understanding is crucial when analyzing functions that represent the balance of a loan over time.

Loan Amortization (Interest-Free)

Loan amortization refers to the process of paying off debt with regular, fixed payments over time. In an interest-free scenario, the situation can be modeled with a linear function where no additional interest accumulates on the outstanding balance. Each payment reduces the principal by a fixed amount, leading to a straightforward, linear decrease in the amount owed.

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