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5. Exponential & Piecewise Functions
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United States of AmericaTennessee
Integrated Math 2

5.02 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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x
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y

The graph of e^x appears similar to the graph of 2^x or 3^x since 2<e<3.

Euler's number can be used to model continuous growth and decay:

\displaystyle f\left(x\right)=ae^{kx}
\bm{a}
Initial value
\bm{k}
Growth or decay rate

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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t
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N \text{ (in thousands)}

Solution

30
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240
t
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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

A small town is experiencing a population boom and is currently growing continuously at a rate of 5\% every year. The mayor wants to ensure that the housing needs of the growing population are met and passes a bill to support housing for 1000 additional people every year. The current town population is 5000 residents and the town currently has enough housing to support a population of 8000.

a

Create an algebraic model for the population and amount of housing in this small town after x years.

Approach

The population is growing exponentially 'continuously' so we can use the function f\left(x\right)=ae^{kx} to model the population. The housing is growing linearly so we can use f(x)=mx+b to model the available housing.

Solution

The population can be modeled with f\left(x\right)=5000(e)^{.05x} and the housing can be modeled with f\left(x\right)=1000x+8000.

b

Create a graph of both functions on the same coordinate plane to determine when the town will run out of available housing. Choose an appropriate scale and label the axes of your graph.

Approach

We can use a table of values to get an idea of how the population and the housing grows over time:

Years0102030405060
Population (people)5000824313\,59122\,40836\,94560\,912100\,427
Housing (people)800018\,00028\,00038\,00048\,00058\,00068\,000

This table shows us that the population exceeds the available housing between 40 and 50 years and also gives us an idea of the scale needed to graph the relationship.

Solution

10
20
30
40
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\text{Years}
10000
20000
30000
40000
50000
60000
70000
80000
90000
\text{People}

The population exceeds the available housing between 48 and 50 years.

Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

M2.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.

M2.A.SSE.A.1.B

Interpret complicated expressions by viewing one or more of their parts as a single entity.

M2.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

M2.F.IF.A.1

Use function notation.*

M2.F.IF.A.1.A

Use function notation to evaluate functions for inputs in their domains, including functions of two variables.

M2.F.IF.A.1.B

Interpret statements that use function notation in terms of a context.

M2.F.IF.C.7

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.*

M2.F.IF.C.7.B

Know and use the properties of exponents to interpret expressions for exponential functions in terms of a real-world context.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

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