The constant r can then be interpreted as the rate of growth, in decimal form. A value of 0.06, for instance, means that the quantity increases by over every unit of time. As is the case with all exponential functions, a is the y-coordinate of the y-intercept.
The constant r can then be interpreted as the rate of decay, in decimal form. A value of 0.12, for instance, would mean that the quantity decreases by over every unit of time.
In an ideal environment, bacteria populations grow exponentially and can be modeled with an exponential growth function. The bacteria Lactobacillus acidophilus duplicates about once every hour. A single bacteria cell is placed in an ideal environment. State and interpret the constants a and r for the growth that will occur. Then, write a function rule describing this growth.
Thus, the rate of decay is 0.12, or per year. The initial value is 800, and the constant multiplier is 0.88. Using this information, we can graph the exponential decay function by plotting some points that lie on H and connecting them with a smooth curve.
When money is deposited to a savings account, interest is accrued, often yearly — different types of interest work in different ways. Compound interest is when the interest earned is added to the original amount and future interest accrues into a larger amount. To find the balance of an account at any given time, an exponential growth function can be used. When the interest is compounded yearly, the balance can be modeled by a function.
In this context, P stands for the principal, which is the initial amount of money, and r is the interest rate in decimal form. If the interest is not compounded yearly, the function looks a little different.
The constant n is the number of times the interest is compounded per year, while r is still the annual interest rate. For an account with the principal $100 and an annual interest of compounded twice a year, the growth function is:
Notice that this function grows continuously, whereas, in reality, the account balance only increases at the times of compound. Graphing the function together with the actual balance of the account will highlight how it can be used in practice.
Every time the interest is compounded, in this case every half year, the value of B is equal to the account balance. However, at all other times, it is not. To find, for instance, the account balance after 1.75 years, B(1.5) should then be evaluated, since that was the last time interest compounded.
One savings account, with a principal of $100, offers an annual interest rate of compounded twice a year. Find the balance in the account after 5 years. Another savings account with the same principal will have the same balance after 5 years. However, the interest is compounded monthly. Find the interest rate of the second account.
We find an approximate monthly growth factor 1.012, which corresponds to a rate of growth that is 0.012. Thus, the monthly interest rate is roughly