The graph of the equation $ A(t) = 3,000(1.14)^t $ represents exponential growth, where the investment amount increases over time as the currency value rises at a constant rate of 14% per year. The initial value of the investment is $3,000, which is the y-intercept of the graph. As time $ t $ increases, the value of $ A(t) $ rises steeply, demonstrating the compounding effect of the growth rate. By interpreting the model, we can see that after each year, the value of the investment not only increases by a percentage of the initial amount but also accumulates on the increased amount from the previous year, illustrating the nature of exponential growth in investments.

Question

The graph of the equation $ A(t) = 3,000(1.14)^t $ represents exponential growth, where the investment amount increases over time as the currency value rises at a constant rate of 14% per year. The initial value of the investment is $3,000, which is the y-intercept of the graph. As time $ t $ increases, the value of $ A(t) $ rises steeply, demonstrating the compounding effect of the growth rate. By interpreting the model, we can see that after each year, the value of the investment not only increases by a percentage of the initial amount but also accumulates on the increased amount from the previous year, illustrating the nature of exponential growth in investments.
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GPT-4o mini · Answer

The equation $ A(t) = 3,000(1.14)^t $ shows how money can grow over time. This means that if you invest $3,000, it grows by 14% every year! So, the more time that passes (that’s what $ t $ represents), the more your money grows. At the start, when $ t = 0 $, you have $3,000—that’s the starting point on the graph. As the years go by, the amount of money goes up really fast because of the way growth works. Each year, you earn money not just on your original $3,000, but also on the extra money you made in the previous years. This is called exponential growth, and it means that your investment gets bigger and bigger more quickly the longer you leave it. So, if you keep your money invested for several years, it can turn into a lot more than you started with!