Victoria Has Some Money From Her Birthday

Victoria Has Some Money From Her Birthday And The Amount Is Modeled By

Victoria has some money from her birthday, which is modeled by the function h(x) = 200. She read about a bank with savings accounts that accrue interest according to the function s(x) = (1.05)^x - 1. Victoria is considering depositing her money into this savings account to earn interest. Using complete sentences, explain to Victoria how she can combine her functions to create a new function, and interpret what this new function represents.

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Victoria's initial amount of money from her birthday is represented by the constant function h(x) = 200. This indicates that her initial money does not change with time or any other variable; it remains fixed at $200. She is considering saving this money in a bank account that accumulates interest according to a specific formula, s(x) = (1.05)^x - 1, where x is the number of periods, typically years. This function describes the growth of her savings over time, but it is expressed as the amount of interest earned, not the total amount in her account.

To understand how Victoria's initial money will grow, she can create a new function that combines her initial amount with the interest accrued over time. This process involves adding her initial principal, $200, to the interest generated by her savings. Since the interest function, s(x), represents the interest earned over x periods, the total amount of money in her account after x periods can be modeled by a new function, T(x), which equals her initial amount plus the interest earned.

Specifically, she can define this combined function as T(x) = 200 + 200 s(x). Since s(x) gives the interest earned relative to her initial deposit, multiplying s(x) by 200 translates the interest into absolute dollar amount. Therefore, T(x) = 200 + 200 [(1.05)^x - 1], which simplifies to T(x) = 200 * (1.05)^x. This simplified form reflects the total amount of money in her account at any time x, including her original $200 plus the interest accrued.

This new function, T(x), models the total amount of money Victoria will have in her savings account after x periods of time. It shows exponential growth, highlighting how her initial deposit increases with compound interest. When x = 0, T(0) = 200 (1.05)^0 = 200 1 = 200, which confirms her initial deposit lives consistent with her initial amount. As x increases, the total grows exponentially, illustrating the power of compound interest over time. Using this function, Victoria can estimate how much money she will have after a certain number of years and make informed decisions about how long to leave her savings untouched.

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