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4.1 Exponential functions  (Page 6/16)

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Find an equation for the exponential function graphed in [link] .

Graph of an increasing function with a labeled point at (0, sqrt(2)).

f ( x ) = 2 ( 2 ) x .   Answers may vary due to round-off error. The answer should be very close to   1.4142 ( 1.4142 ) x .

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Given two points on the curve of an exponential function, use a graphing calculator to find the equation.

  1. Press [STAT].
  2. Clear any existing entries in columns L1 or L2.
  3. In L1 , enter the x -coordinates given.
  4. In L2 , enter the corresponding y -coordinates.
  5. Press [STAT] again. Cursor right to CALC , scroll down to ExpReg (Exponential Regression) , and press [ENTER].
  6. The screen displays the values of a and b in the exponential equation   y = a ⋅ b x .

Using a graphing calculator to find an exponential function

Use a graphing calculator to find the exponential equation that includes the points   ( 2 , 24.8 )   and   ( 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   a = 6.2   and   b = 2   will be displayed. The exponential equation is   y = 6.2 ⋅ 2 x .

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Use a graphing calculator to find the exponential equation that includes the points (3, 75.98) and (6, 481.07).

y ≈ 12 ⋅ 1.85 x

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Applying the compound-interest formula

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   t , principal   P , APR   r , and number of compounding periods in a year   n :

A ( t ) = P ( 1 + r n ) n t

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

The compound interest formula

Compound interest can be calculated using the formula

A ( t ) = P ( 1 + r n ) n t

where

  • A ( t )   is the account value,
  • t   is measured in years,
  • P   is the starting amount of the account, often called the principal, or more generally present value,
  • r   is the annual percentage rate (APR) expressed as a decimal, and
  • n   is the number of compounding periods in one year.

Calculating compound interest

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,   P = 3000.   Our interest rate is 3%, so   r   =   0.03.   Because we are compounding quarterly, we are compounding 4 times per year, so   n = 4.   We want to know the value of the account in 10 years, so we are looking for   A ( 10 ) , the value when   t   =   10.

A ( t ) = P ( 1 + r n ) n t Use the compound interest formula . A ( 10 ) = 3000 ( 1 + 0.03 4 ) 4⋅10 Substitute using given values . ≈ $ 4045.05 Round to two decimal places .

The account will be worth about $4,045.05 in 10 years.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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