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Find an equation for the exponential function graphed in [link] .
$f(x)=\sqrt{2}{\left(\sqrt{2}\right)}^{x}.\text{\hspace{0.17em}}$ Answers may vary due to round-off error. The answer should be very close to $\text{\hspace{0.17em}}1.4142{\left(1.4142\right)}^{x}.$
Given two points on the curve of an exponential function, use a graphing calculator to find the equation.
Use a graphing calculator to find the exponential equation that includes the points $\text{\hspace{0.17em}}(2,24.8)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,198.4).$
Follow the guidelines above. First press [STAT] , [EDIT] , [1: Edit…], and clear the lists L1 and L2 . Next, in the L1 column, enter the x -coordinates, 2 and 5. Do the same in the L2 column for the y -coordinates, 24.8 and 198.4.
Now press [STAT] , [CALC] , [0: ExpReg] and press [ENTER] . The values $\text{\hspace{0.17em}}a=6.2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=2\text{\hspace{0.17em}}$ will be displayed. The exponential equation is $\text{\hspace{0.17em}}y=6.2\cdot {2}^{x}.$
Use a graphing calculator to find the exponential equation that includes the points (3, 75.98) and (6, 481.07).
$y\approx 12\cdot {1.85}^{x}$
Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest . The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate , is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time $\text{\hspace{0.17em}}t,$ principal $\text{\hspace{0.17em}}P,$ APR $\text{\hspace{0.17em}}r,$ and number of compounding periods in a year $\text{\hspace{0.17em}}n:$
For example, observe [link] , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.
Frequency | Value after 1 year |
---|---|
Annually | $1100 |
Semiannually | $1102.50 |
Quarterly | $1103.81 |
Monthly | $1104.71 |
Daily | $1105.16 |
Compound interest can be calculated using the formula
where
If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?
Because we are starting with $3,000, $\text{\hspace{0.17em}}P=3000.\text{\hspace{0.17em}}$ Our interest rate is 3%, so $\text{\hspace{0.17em}}r\text{}=\text{}\mathrm{0.03.}\text{\hspace{0.17em}}$ Because we are compounding quarterly, we are compounding 4 times per year, so $\text{\hspace{0.17em}}n=4.\text{\hspace{0.17em}}$ We want to know the value of the account in 10 years, so we are looking for $\text{\hspace{0.17em}}A\left(10\right),$ the value when $\text{\hspace{0.17em}}t\text{}=\text{}10.$
The account will be worth about $4,045.05 in 10 years.
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