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6. Exponential Functions
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Integrated Math 1

6.03 Exponential growth and decay

Lesson
Practice
Lesson

Concept summary

Exponential functions can be classified as exponential growth or exponential decay 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

The process of increasing in size by a constant percent rate of change. Sometimes called percent growth. This occurs when b>1.

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Exponential decay

The process of reducing in size by a constant percent rate of change. Sometimes called percent decay. This occurs when 0<b<1.

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Growth factor

The constant factor of an exponential growth function

Decay factor

The constant factor of an exponential decay function

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Geometric sequences with a common ratio greater than 1 can model exponential growth.

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Geometric sequences with a common ratio between 0 and 1, non-inclusive, can model exponential decay.

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Geometric sequences with a negative common ratio cannot be used to model exponential growth or decay as we can see in this graph:

Worked examples

Example 1

Consider the exponential function: f\left(x\right)=\dfrac{1}{2}\left(4\right)^x

a

Classify the function as either exponential growth or exponential decay.

Approach

To classify an exponential function we want to identify the constant factor, b, and determine if b>1 or 0<b<1.

Solution

In this function b=4. Since 4>1, we would classify this function as exponential growth.

b

Identify the initial value.

Approach

In the general form of an exponential function, y=ab^x, the initial value is represented by the variable a, which is the factor, or coefficient, that does not have a variable exponent.

Solution

The initial value is \dfrac{1}{2}.

Reflection

The initial value is where x=0, so this means the y-intercept will be \left(0,\dfrac{1}{2}\right).

c

Identify the growth or decay factor.

Approach

In the general form of an exponential function, y=ab^x, the growth or decay factor is represented by the variable b, which is the factor with a variable exponent.

Solution

The growth factor is 4.

Example 2

Write an equation that models the geometric sequence shown in the table.

n0123
t(n)392781

Approach

To create the equation we need to identify the intial value and growth or decay factor for the sequence modeled in the table.

A geometric seqeunce can be represented with an equation of the form t(n)=a(b)^n, where a is the value of the 0th term and b is the common ratio.

Solution

The 0th term is 3 and the common ratio is \dfrac{9}{3}=3 so the equation for this function is t(n)=3(3)^n.

Reflection

Sometimes we will see this written as t(n)=9(3)^{n-1} where 9 is the 1st term, 3 is the common ratio, and we have (n-1) as the exponent to offset for showing the 1st term instead of the 0th term as the coefficient.

Example 3

Write an equation of the form y=ab^x that models the function shown in the graph.

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Approach

To create the equation we need to identify the intial value and growth or decay factor for the function modeled in the graph.

For functions in the form y=a(b)^x, the y-intercept represents the initial value and the growth or decay factor can be found using the ratio of two successive outputs.

Solution

The initial value is 16 as this is the y-value of the y-intercept.

To find the growth factor we will take two points whose x-values differ by 1, (0,16) and (1,4), and create a ratio of their outputs: \dfrac{4}{16}=\dfrac{1}{4}. The equation for this function is: y=16\left(\dfrac{1}{4}\right)^x

Reflection

We can see from this graph that this model for exponential decay, so the value of b must be between 0 and 1. We have that b=\dfrac{1}{4}, and 0<\dfrac{1}{4}<1, so we do have a model for decay as expected.

Example 4

A sample contains 300 grams of ruthenium-106, which has a half-life of 1 year.

a

Write a geometric sequence to model this exponential relationship.

Approach

We know from the context that we start with 300 and then to get to the next term we multiply by \dfrac{1}{2} (or divide by 2). It does not specify the number of terms, so 5 terms should be sufficient.

Solution

Geometric sequence: 300, \, 150, \, 75, \, 37.5, 18.75, \,\ldots

Reflection

It is important to include the \ldots at the end as this material will continue to decay. At some point the amount will be immeasureable, but we are not expected to go that far.

b

Write a function, A, to represent the amount of the sample remaining after n years.

Approach

We will start with the general form of an exponential function, f\left(x\right)=a(b)^x and input all known values to find the function.

Solution

The initial value of the function is 300 grams and half-life means that the function has a decay factor of \dfrac{1}{2}, so we can write the function A=300\left(\dfrac{1}{2}\right)^n, where A is in grams and n is in years.

Reflection

If the half-life was every 2 years or 6 months, or anything that is not a unit, we would need a more complex model than y=ab^x.

c

Evaluate the function for n=25 and interpret the meaning in context using an appropriate unit.

Approach

In this context n represents the time, in years, and the output represents the amount of the sample remaining. We will evaluate the function and apply these units to interpret the meaning of the solution.

Solution

A=300\left(\dfrac{1}{2}\right)^{25}=0.00000894069 \ldots = 8.94 \times 10^{-6} An appropriate unit would allow us to write the mass without scientific notation. 8.94 \times 10^{-6} \text{\, g} is equivalent to 0.00894 \, \text{mg} or 8.94 \, \text{μg}.

We can then say that there would be approximately 8.94 \, \text{μg} of ruthenium-106 remaining after 25 years had passed.

Reflection

If we were only familiar with milligrams (\text{mg}) and not micrograms (\text{μg}), we could also give the interpretation in milligrams.

Outcomes

M1.N.Q.A.1

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

M1.N.Q.A.1.A

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

M1.N.Q.A.1.C

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

M1.A.SSE.A.1.B

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

M1.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.*

M1.F.LE.A.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.*

M1.F.LE.B.3

Interpret the parameters in a linear or exponential function in terms of a context.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP8

Look for and express regularity in repeated reasoning.

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