Question

You are looking to invest $1,200. One savings option follows the

function f(x) = 1200 + 1200(0.055)x, where f(x) is the amount of money
in savings after x years. The second option is represented by the function
0.035 where g(x) is the amount of money after 12 g(x)=12001+ 12 x years.
Which increases faster, f(x) or g(x)?
Use a graph to explain your answer.

Answer 1

Option f(x), which is based on compound interest, increases faster than option g(x), which has a linear growth rate. By graphing these functions, f(x)'s exponential curve overtakes g(x)'s straight line, illustrating the impact of compound interest over time.

Step-by-step explanation:

To determine which investment option, f(x) or g(x), increases faster over time, we can graph the functions and compare their slopes. The function f(x) = 1200 + 1200(0.055)x represents a savings option accruing interest at a 5.5% annual rate on an initial invest $1,200. On the other hand, the function g(x) = 1200(1 + 0.035x) signifies a linear increase with a slope of 0.035 per year over 12 x years.

When graphed, the function f(x) will depict exponential growth due to its compounding nature, where the amount of increase grows larger with every passing year. Whereas g(x) will show a consistent linear increase with no change in the rate of increase over time. Thus, f(x) will increase faster than g(x) the longer the money is invested, due to the effects of compound interest.

As a reference, we can look at the power of compound interest in scenarios with larger amounts and different rates, such as when $3,000 invested at a 7% annual rate of return becomes almost $45,000 after 40 years, showcasing the significant growth potential of compound interest over time.