Topics
6. Exponential & Logarithmic Functions
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United States of AmericaFL
Grade 912

6.02 Exponential models

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Concept summary

Exponential functions can be classified as exponential growth functions or exponential decay functions based on the value of the constant factor.

\displaystyle f\left(x\right)=ab^x
\bm{a}
The initial value of the exponential function
\bm{b}
The constant factor of the exponential function
Exponential growth

Increasing by a constant factor, b, where b>1

Exponential decay

Decreasing by a constant factor, b, where 0<b<1

Growth factor

The constant factor of an exponential growth function.

Decay factor

The constant factor of an exponential decay function.

Exponential functions can also be expressed in terms of their constant percent rate of change.

\displaystyle f\left(x\right)=a(1\pm r)^x
\bm{a}
The initial value of the exponential function
\bm{r}
The growth or decay rate.
Growth rate

The fixed percent by which an exponential function increases.

In the form f\left(x\right)=a(1+r)^x, r is a growth rate, and the constant percent rate of change is positive.

Decay rate

The fixed percent by which an exponential function decreases.

In the form f\left(x\right)=a(1-r)^x, r is a decay rate, and the constant percent rate of change is negative.

Exponential growth and decay models arise in many real-world situations. Here is the base function, its appearance and some real-world examples:

x
y
  • f\left(x\right)=ab^x, b>1
  • f\left(x\right)=a\left(1+r\right)^x, r>0

Growth rate examples

  • Population growth
  • Growth of cells
  • Spread of a disesase in a pandemic
  • Balance of an investment earning compound growth
x
y
  • f\left(x\right)=ab^x, 0<b<1
  • f\left(x\right)=a\left(1-r\right)^x, r>0

Decay rate examples

  • Population decline of a threatened species
  • Cooling of a liquid
  • Depreciation of a car
  • Radioactive decay or half-life
Half-life

How long it takes for a value to halve with exponential decay. Most commonly used to describe how quickly a radioactive element takes to decay.

Euler's number - e

An irrational number approximately equal to 2.71828, that has many applications in natural growth and compound interest. We can evaluate powers of e using the button on our calculator.

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Worked examples

Example 1

Consider the table of values for function f\left(x\right).

x-101234
f(x)\dfrac{5}{3}51545135405
a

Determine whether the function represents an exponential function or not.

Approach

To determine if a set of data represents an exponential function, we want to compare successive y-values, and determine if there is a constant multiplying factor.

Solution

We can see that f\left(-1\right)=\dfrac{5}{3} and f\left(0\right)=5, so we want to calculate the multiplying factor that takes us from \dfrac{5}{3} to 5. \frac{5}{3} \cdot b=5 \longrightarrow b=5 \cdot \frac{3}{5} = 3 So the multiplying factor is 3. We now want to check if this factor works for the remaining y-values: 5 \cdot 3=15 and 15 \cdot 3 = 45, as expected. We can similarly check the other values.

The function represents an exponential function.

b

Write an equation to represent the exponential function.

Approach

The function has an initial value, when x=0, of 5 and a growth factor of 3.

Solution

An equation that represents this exponential function is: f\left(x\right)=5\left(3^x\right)

Example 2

The number of bacterial cells, N\left(t\right), growing in a petri dish in a laboratory, after t minutes can be modeled by the function, N\left(t\right) = 100 \left(1+0.035\right)^{t}.

a

State the initial population of bacterial cells.

Approach

The given function is of the form f\left(x\right)=a\left(1+r\right)^x, where a is the initial value of the function.

Solution

The initial population is a=100 bacterial cells.

b

Determine the constant percent rate of change, and explain what it means in context.

Approach

The given function is of the form f\left(x\right)=a\left(1+r\right)^x, where r is the constant percent rate of change, expressed as a decimal.

Solution

The constant percent rate of change is 3.5\%. This means the population will increase by 3.5\% every minute.

Reflection

In this case the constant percent rate is positive as we have exponential growth. For exponential decay with a function of the form f\left(x\right)=a(1-r)^x, the constant percent rate of change would be a negative percentage.

c

Find the population of bacterial cells after 4 hours, rounding to the nearest integer.

Approach

To find the population after 4 hours, we want to find N\left(240\right), as the function gives us the population after t minutes, not hours, so we need to convert 4 hours into minutes, by multiplying by 60.

Solution

\displaystyle N\left(t\right)\displaystyle =\displaystyle 100 \left(1+0.035\right)^{t}State the function
\displaystyle N\left(240\right)\displaystyle =\displaystyle 100 \left(1+0.035\right)^{240}Substitute t=240
\displaystyle =\displaystyle 385\,198Evaluate, rounding to nearest integer

There will be 385\,198 bacterial cells after 4 hours.

d

Graph the bacterial cell population over the domain \left[0, 240\right].

Approach

To graph the population, we can fill out a table of values and then sketch the curve that passes through these points. Finding the population after every 30 minutes will give us enough values to sketch the graph accurately enough. All values will be rounded to the nearest integer.

t0306090120150180210240
N\left(t\right)1002817882211620617\,42048\,895137\,238385\,198

We can now plot the points on a coordinate plane. As the numbers get very large, we will make the y-axis values be multiples of 1000.

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N \text{ (in thousands)}

Solution

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N \text{ (in thousands)}

Reflection

We can see from the table and the graph that the population almost triples every 30 minutes. This makes sense, as \left(1+0.35\right)^{30}\approx 2.8 which is close to 3.

e

Use your graph to determine approximately how many minutes it will take for the population to reach 1 000 times the original population.

Approach

The initial population was 100 bacterial cells, so we want to find how long it takes for the population to reach 100 \cdot 1\,000=100\,000

We can draw a horizontal line from y=100\,000 on the y-axis across to the curve, and then draw a vertical line down to the x-axis to find the value of t.

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t
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N \text{ (in thousands)}

Solution

We can see from the graph that the population reaches 1000 times the original population after approximately 200 minutes or 3 hours and 20 minutes.

Example 3

Kendall is making creme-caramels. They take them out of the oven when they are 150\, \degreeF and then put them in the fridge to cool down. Kendall measured the temperature of the creme-caramels using an infrared thermometer at 3-minute intervals for 18 minutes. The recorded temperature are shown in the table.

\text{Minutes}0369121518
\text{Temperature}\left(\degree\text{F}\right)1701581371251129789
a

Using technology, sketch a graph showing these values and use it to determine if the data suggests an exponential association. Explain your answer.

Approach

We can determine if the data suggests an exponential association by plotting the points on a coordinate plane and determining if the data resembles an exponential curve. We can also use technology to calculate the regression with an exponential. If the value of R^2 is close to 1 then there is likely to be an exponential association.

Solution

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\text{Time (minutes)}
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\text{Temperature} (\degree \text{F})

Yes, an exponential model is appropriate because when drawn on a graph it has an exponential shape which is decreasing less rapidly over time.

Also, if we calculate an exponential regression on this data set using the exponential function, we get R^2=0.9956 which is very close to 1, so there is a strong correlation to the exponential model suggesting an exponential association.

Reflection

The data looks like it may be linear, but this is due to the scale of the graph. We can see that it appears to flatten out as the time increases.

b

Using technology determine an appropriate equation to model the data set to four decimal places.

Solution

y=172.8652\left(0.9636^x\right), where x is the number of minutes and y is temperature.

Reflection

Using technology, students may get the equation for the curve of fit to be y=172.8652\left(e^{-0.0371x}\right) using the laws of exponents to rewrite it as y=172.8652\left(e^{-0.0371}\right)^x and the calculator to simplify the expression involving e to get y=172.8652\left(0.9636\right)^x

c

After 25 minutes, the creme-caramel is the perfect temperature to serve. Find the perfect temperature. Round your answer to the nearest degree.

Approach

We want to substitute x=25 into the equation we found in part (b).

Solution

\displaystyle y\displaystyle =\displaystyle 172.8652\left(0.9636\right)^xState the equation
\displaystyle =\displaystyle 172.8652\left(0.9636\right)^{25}Substitute x=25
\displaystyle \approx\displaystyle 68Evaluate

The perfect temperature is 68\,\degreeF.

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.5.4

Write an exponential function to represent a relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.5.5

Given an expression or equation representing an exponential function, reveal the constant percent rate of change per unit interval using the properties of exponents. Interpret the constant percent rate of change in terms of a real-world context.

MA.912.AR.5.7

Solve and graph mathematical and real-world problems that are modeled with exponential functions. Interpret key features and determine constraints in terms of the context.

MA.912.DP.2.9

Fit an exponential function to bivariate numerical data that suggests an exponential association. Use the model to solve real-world problems in terms of the context of the data.

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

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