9. Catalina and Terri are each saving money to buy a car. Catalina puts \( \$ 1,000 \) in a safe at home and adds \( \$ 25 \) every month. Terri puts \( \$ 500 \) in a savings account at the bank that earns \( 3.2 \% \) interest each month on the total amount of money in the account. For each person, the total amount saved is a function of the number of months. Which of these relationships is a linear function? Construct an argument to justify your response.

To determine which of the two saving relationships is a **linear function**, let's analyze each person's saving strategy and the corresponding mathematical representation. ### **Catalina's Savings Plan** - **Initial Amount:** \$1,000 - **Monthly Addition:** \$25 **Function Representation:** \[ C(m) = 1000 + 25m \] where: - \( C(m) \) is the total amount saved after \( m \) months. - \( 1000 \) is the initial deposit. - \( 25m \) represents the constant addition of \$25 each month. **Characteristics:** - **Linear Function:** The equation is of the form \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y-intercept (initial value). - **Rate of Change:** The savings increase by a fixed amount (\$25) every month. - **Graph:** A straight line, indicating a constant rate of increase. ### **Terri's Savings Plan** - **Initial Amount:** \$500 - **Monthly Interest Rate:** 3.2% **Function Representation:** \[ T(m) = 500 \times (1.032)^m \] where: - \( T(m) \) is the total amount saved after \( m \) months. - \( 500 \) is the initial deposit. - \( 1.032^m \) accounts for the compound interest applied monthly. **Characteristics:** - **Exponential Function:** The equation is of the form \( y = a \times b^x \), where \( a \) is the initial amount and \( b \) is the base representing the growth factor. - **Rate of Change:** The savings grow by a percentage (3.2%) each month, meaning the amount increases by a factor rather than a fixed amount. - **Graph:** A curve that increases at an increasing rate, characteristic of exponential growth. ### **Conclusion:** - **Catalina's Savings (\( C(m) = 1000 + 25m \))** is a **linear function** because it increases by a constant dollar amount each month. - **Terri's Savings (\( T(m) = 500 \times (1.032)^m \))** is an **exponential function** due to the compound interest, which causes the amount to grow by a percentage of the current total each month. **Therefore, the relationship that is a linear function is Catalina's savings plan.**