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Applications of Exponential Functions

Word problems with variables as exponents

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Exponential Functions

Roberta invested $600 into a mutual fund that paid 4% interest each year compounded annually.

  1. Complete a table showing the value of the mutual fund for the first five years.
  2. Write an exponential function of the form \begin{align*}y=a\cdot b^x\end{align*} to describe the value of the mutual fund.
  3. Use the exponential function to determine the value of the mutual fund in 15 years.

Exponential Functions

An exponential function is a function with a variable in the exponent. Two examples of exponential functions are shown below:

\begin{align*}\boxed{y=2^x}\end{align*}

License: CC BY-NC 3.0

\begin{align*}\boxed{y=\left(\frac{1}{2}\right)^x}\end{align*}

License: CC BY-NC 3.0

Here are some facts to notice about the functions and their graphs:

  • The graph of \begin{align*}y=2^x\end{align*} is an increasing curve. It shows growth.
  • Each y-value of \begin{align*}y=2^x\end{align*} is 2 times the previous y-value (for the integer values of x). For example, the points on the graph go from (0, 1) to (1, 2) to (2, 4). The next point would be (3, 8). The y-values keep being multiplied by 2.
  • The graph of \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*} is a decreasing curve. It shows decay.
  • Each y-value of \begin{align*}y=\left(\frac{1}{2}\right)^x\end{align*} is \begin{align*}\frac{1}{2}\end{align*} the value of the previous y-value (for the integer values of x). For example, the points on the graph go from (0, 1) to \begin{align*}(1, \frac{1}{2})\end{align*} to \begin{align*}(2, \frac{1}{4})\end{align*}. The y-values keep being multiplied by \begin{align*}\frac{1}{2}\end{align*}.
  • Both graphs have a y-intercept of 1. This is because anything to the zero power is equal to 1.
  • The domain of each function is \begin{align*}D=\{x| x \in R\}\end{align*}.
  • The range for each function is \begin{align*}R=\{y | y>0, \ y \in R \}\end{align*}.

Based on the above observations, you can deduce that an exponential function of the form \begin{align*}y=ab^x\end{align*} where \begin{align*}b >0\end{align*} has the following properties:

Properties of an Exponential Function of the form \begin{align*}y=ab^x \ (b>0)\end{align*}

  • \begin{align*}b\end{align*}’ is the value of the common ratio. Within the function, as the x-value increases by 1, the y-value is multiplied by the common ratio.
  • If \begin{align*}b > 1 \end{align*} then the curve will represent exponential growth.
  • If \begin{align*}0 < b < 1\end{align*} then the curve will represent exponential decay.
  • Every exponential function of the form \begin{align*}y=ab^x\end{align*} will pass through the point \begin{align*}(0, a)\end{align*}. \begin{align*}a\end{align*} will always be the y-intercept of the function, or its value at time 0.
  • Every exponential function of the form \begin{align*}y=ab^x\end{align*} will have the domain and range: \begin{align*}D=\{x| x \in R \} \ and \ R=\{y | y>0, \ y \in R \}\end{align*}

Let's determine the common ratio for the following tables of values that represent exponential functions: 

  1. \begin{align*}& x \quad 0 \quad 1 \quad 2 \quad 3 \quad \ 4 \quad \cdots\\ & y \quad 1 \quad 2 \quad 4 \quad 8 \quad 16 \quad \cdots\end{align*}

The common ratio is a constant that is determined by \begin{align*}r=\frac{t_{n+1}}{t_n}\end{align*}.

\begin{align*}& r=\frac{t_{n+1}}{t_n}=\frac{2}{1}=2\\ & r=\frac{t_{n+1}}{t_n}=\frac{4}{2}=2\\ & r=\frac{t_{n+1}}{t_n}=\frac{8}{4}=2\\ & r=\frac{t_{n+1}}{t_n}=\frac{16}{8}=2\\ & \boxed{\text{The common ratio is} \ 2.}\end{align*}

  1. \begin{align*}& x \quad \ 0 \qquad 1 \quad \ 2 \qquad 3 \qquad \ 4 \quad \ \cdots\\ & y \quad 100 \quad 50 \quad 25 \quad 12.5 \quad 6.25 \quad \cdots\end{align*}

The common ratio is a constant that is determined by \begin{align*}r=\frac{t_{n+1}}{t_n}\end{align*}.

\begin{align*}& r=\frac{t_{n+1}}{t_n}\\ & r=\frac{50}{100}=\frac{1}{2}\\ & r=\frac{25}{50}=\frac{1}{2}\\ & r=\frac{12.5}{25}=\frac{1}{2}\\ & r=\frac{6.25}{12}=\frac{1}{2}\\ & \boxed{\text{The common ratio is } \frac{1}{2}.}\end{align*}

Now, for \begin{align*}{f(x)=3^x}\end{align*}, let's determine the value of each of the following:

  1. \begin{align*}f(2)\end{align*}
  2. \begin{align*}f(3)\end{align*}
  3. \begin{align*}f(0)\end{align*}
  4. \begin{align*}f(4)\end{align*}
  5. \begin{align*}f(-2)\end{align*}

Note that \begin{align*}f(x)=3^x\end{align*} is another way to express \begin{align*}y=3^x\end{align*}. To determine the value of the function for the given values, replace the exponent with that value and evaluate the expression.

  1. \begin{align*}& f(x)=3^x\\ & f({\color{red}2})=3^{\color{red}2}\\ & f(2)={\color{red}9}\\ & \boxed{f(2)=9}\end{align*}
  2. \begin{align*}& f(x)=3^x\\ & f({\color{red}3})=3^{\color{red}3}\\ & f(3)={\color{red}27}\\ & \boxed{f(3)=27}\end{align*}
  3. \begin{align*}& f(x)=3^x\\ & f({\color{red}0})=3^{\color{red}0}\\ & f(0)={\color{red}1}\\ & \boxed{f(0)=1}\end{align*}
  4. \begin{align*}& f(x)=3^x\\ & f({\color{red}4})=3^{\color{red}4}\\ & f(4)={\color{red}81}\\ & \boxed{f(4)=81}\end{align*}
  5. \begin{align*}& f(x)=3^x\\ & f({\color{red}-2})=3^{\color{red}-2}\\ & f(-2)={\color{red}\frac{1}{3^2}}\\ & \boxed{f(-2)=\frac{1}{9}}\end{align*}

Examples

Example 1

Earlier, you were told that Roberta invested $600 into a mutual fund that paid 4% interest each year compounded annually. You were asked to do the following:

  1. Complete a table showing the value of the mutual fund for the first five years.
  2. Write an exponential function of the form \begin{align*}y=a\cdot b^x\end{align*} to describe the value of the mutual fund.
  3. Use the exponential function to determine the value of the mutual fund in 15 years.

 

  1. First, do all the calculations for the table:

\begin{align*}& 600(.04)=24 && 624(.04)=24.96 && 648.96(.04)=25.96 \\ &600+24=624 && 624+24.96=648.96 && 648.96+25.96=674.92\\ \\ &674.92(.04)=27.00 && 701.92(.04)=28.08\\ &674.92+27.00=701.92 && 701.92+28.08=730.00\end{align*}

Putting all the information in a table, we get:

\begin{align*}&\text{Time} \ (years) \ \ 0 \qquad 1 \qquad \ \ 2 \qquad \quad \ 3 \qquad \quad \ 4 \qquad \quad \ 5\\ &\text{Value (\$)} \qquad 600 \quad 624 \quad 648.96 \quad 674.92 \quad 701.92 \quad 730.00\end{align*}

  1. To find the function, note that the initial value is $600 and the common ratio is 1.04 which represents the initial investment and the interest rate of 4%. The exponent is the time in years. The exponential function is \begin{align*}y=600(1.04)^x\end{align*} or \begin{align*}v=600(1.04)^t\end{align*}.
  2. To find how much will be in the fund in 15 years, plug in 15 for t. 

\begin{align*}& v=600(1.04)^t\\ & v=600(1.04)^{\color{red}15}\\ & v={\color{red}\$1080.57}\\ &\boxed{v=\$1080.57}\end{align*}

The value of the mutual fund in fifteen years will be $1080.57.

Example 2

On January 1, Juan invested $1.00 at his bank at a rate of 10% interest compounded daily.

  1. Create a table of values for the first 8 days of the investment.
  2. What is the common ratio?
  3. Determine the equation of the function that would best represent Juan’s investment.
  4. How much money will Juan have in his account on January 31?
  5. If Juan had originally invested $100 instead of $1.00 at 10%, what exponential equation would describe the investment. How much money would he have in his account on January 31?

 

  1. First calculate the values and put them in a table:

 

\begin{align*}&1.00(.10)=0.10 && 1.10(.10)=0.11 && 1.21(.10)=0.12 && 1.33(.10)=0.13\\ &1.00+0.10=1.10 && 1.10+0.11=1.21 && 1.21+0.12=1.33 && 1.33+0.13=1.46\\ \\ &1.46(.10)=0.15 && 1.61(.10)=0.16 && 1.77(.10)=0.18 && 1.95(.10)=0.20\\ &1.46+0.15=1.61 && 1.61+0.16=1.77 && 1.77+0.18=1.95 && 1.95+0.20=2.15\end{align*}

 

\begin{align*}&\# \ \text{ of days} \qquad 0 \quad \ \ 1 \qquad 2 \qquad \ 3 \qquad \ 4 \qquad \ 5 \qquad \ 6 \qquad \ 7 \qquad \ 8\\ &\text{Money } (\$) \qquad 1 \quad 1.10 \quad 1.21 \quad 1.33 \quad 1.46 \quad 1.61 \quad 1.77 \quad 1.95 \quad 2.15\end{align*}

 

  1. Then note that the common ratio is a constant that is determined by \begin{align*}\frac{t_{n+1}}{t_n}\end{align*}. Therefore, the common ratio for this problem is \begin{align*}r=\frac{t_{n+1}}{t_n} \rightarrow \frac{1.10}{1}=1.10 \rightarrow \frac{1.21}{1.10}=1.10 \rightarrow \frac{1.33}{1.21}=1.10\end{align*}.

The common ratio is \begin{align*}{1.10}\end{align*}.

  1. The equation of the function to model Juan’s investment is \begin{align*}y=1.10^x\end{align*}
  2. To find how much Juan will have on January 31, plug in 31 for t.

\begin{align*}y=1.10^x \rightarrow y=1.10^{31} \rightarrow \boxed{y=\$ 19.19}\end{align*}

On January 31, Juan will have $19.19 in his account.

  1. If Juan invested $100 instead of $1.00, the initial value would change.

\begin{align*}y&=100(1.10)^x\\ y&=100(1.10)^{31} \rightarrow y={\color{red}\$1919.43} \rightarrow \boxed{y=\$1919.43}.\end{align*}

On January 31, Juan would have $1919.43 in his account if he had invested $100 instead of $1.00.

Example 2

The graph below shows the change in value of two comic books purchases in the year 2000. Both comics were expected to be good investments, but one of them did not perform as expected. Use the graphs to answer the questions.

License: CC BY-NC 3.0

  1. What was the purchase price of each comic book?
  2. Which comic book shows exponential growth?
  3. Which comic book shows exponential decay?
  4. In what year were both comic books equal in value?
  5. State the domain and range for each comic.

 

  1. The purchase price of each comic book is the \begin{align*}y\end{align*}-intercept. The \begin{align*}y\end{align*}-intercept is the initial value of the books. The Spiderman comic book cost $30.00 and the Superman comic book cost $5.00.
  2. The Superman comic book shows exponential growth.
  3. The Spiderman comic book shows exponential decay.
  4. In 2005 both comic books were equal in value. The graphs intersect at approximately (5, $12.50), where 5 represents five years after the books were purchased.
  5. The domain and range for each comic is \begin{align*}D=\{x| x \in R\}\end{align*} and \begin{align*}R=\{y|y>0, \ y \in R\}\end{align*}

Example 3

Paulette bought a Bobby Orr rookie card for $300. The value of the card appreciates (increases) by 30% each year.

  1. Complete a table of values to show the first five years of the investment.
  2. Determine the common ratio for the successive terms.
  3. Determine the equation of the exponential function that models this investment.

 

  1. First, calculate the values and create the table:

 

\begin{align*}&300(.30)=90 && 390(.30)=117 && 507(.30)=152.10\\ &300+90=390 && 390+117=507 && 507+152.10=659.10\\ \\ &659.10(.30)=197.73 && 856.83(.30)=257.05\\ &659.10+197.73=856.83 && 856.83+257.05=1113.88\end{align*}

 


\begin{align*}&\text{Time} \ (years) \ \ 0 \qquad 1 \qquad 2 \qquad \quad 3 \qquad \quad 4 \qquad \quad \ 5\\ &\text{Value (\$)} \qquad 300 \quad 390 \quad 507 \quad 659.10 \quad 856.83 \quad 1113.88\end{align*}

  1. To find the common ratio, use the formula and simplify:

\begin{align*}& r=\frac{t_{n+1}}{t_n}\\ & r=\frac{390}{300}=1.3\\ & r=\frac{507}{390}=1.3\\ & r=\frac{659.10}{507}=1.3\\ & r=\frac{856.83}{659.10}=1.3\\ & r=\frac{1113.88}{856.83}=1.3\end{align*}

  1. Putting it all together, the exponential function that would model Paulette's investment is:\begin{align*}\boxed{y=300(1.3)^x \ or \ v=300(1.3)^t}\end{align*}

 

Example 4

Due to the closure of the pulp and paper mill, the population of the small town is decreasing at a rate of 12% annually. If there are now 2400 people living in the town, what will the town's projected population be in eight years?

The town's population is decreasing by 12% annually. The simplest way to use this in an exponential function is to use the percent of the population that still exists each year – 88%.

 

Therefore, the exponential function would consist of the present population \begin{align*}(a)\end{align*}, the common ratio is 0.88 \begin{align*}(b)\end{align*} and the time in years would be the exponent \begin{align*}(x)\end{align*}. The function is \begin{align*}p=2400(0.88)^t\end{align*}The population in eight years would be\begin{align*}& p=2400(0.88)^t\\ & p=2400(0.88)^8\\ & p=863.123\\ & p \approx 863 \ people\end{align*}

 

Review

Brandon bought a car for $13,000. The value of the car depreciates by 20% each year.

  1. Complete a table of values to show the car’s values for the first five years.
  2. Determine the exponential function that would model the depreciation of Brandon’s car.

For each of the following exponential functions, identify the common ratio and the \begin{align*}y\end{align*}-intercept, and tell if the function represents a growth or decay curve.

  1. \begin{align*}y=4(5)^x\end{align*}
  2. \begin{align*}y=13(2.3)^x\end{align*}
  3. \begin{align*}y=0.85(0.16)^x\end{align*}
  4. \begin{align*}y=1.6(0.5)^x\end{align*}
  5. \begin{align*}y=0.4(2.1)^x\end{align*}

Match each graph below with its corresponding equation:

License: CC BY-NC 3.0

  1. \begin{align*}y=2^x\end{align*}
  2. \begin{align*}y=3^x\end{align*}
  3. \begin{align*}y=2(3)^x\end{align*}
  4. \begin{align*}y=3(2)^x\end{align*}
  5. Do these graphs represent growth or decay?

Match each graph below with its corresponding equation and identify if the graphs represent growth or  decay.

License: CC BY-NC 3.0

  1. \begin{align*}y=0.5^x\end{align*}
  2. \begin{align*}y=0.2^x\end{align*}
  3. \begin{align*}y=2(0.5)^x\end{align*}
  4. \begin{align*}y=3(0.2)^x\end{align*}
  5. Do these graphs represent growth or decay?
  6. Jolene purchased a summer home for $120,000 in 2002. If the property has consistently increased in value by 11% each year, what will be the value of her summer home in 2012?

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.9. 

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Vocabulary

domain

The domain of a function is the set of x-values for which the function is defined.

Exponential Decay Function

An exponential decay function is a specific type of exponential function that has the form y=ab^x, where a>0 and 0<b<1.

Exponential Form

The exponential form of an expression is b^x=a, where b is the base and x is the exponent.

Exponential Growth Function

An exponential growth function is a specific type of exponential function that has the form y=ab^x, where h=k=0, a>0, and b>1.

Range

The range of a function is the set of y values for which the function is defined.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0

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