Topics
5. Exponential Functions
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United States of AmericaFL
Grade 912

5.05 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

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}.

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 function shown in the table.

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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 table.

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 3 and the growth factor is \dfrac{9}{3}=3 so the equation for this function is y=3(3)^x.

Example 3

Write an equation 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. To find the growth factor we will take two points, (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

When determining the growth or decay factor be sure to use the first output as the denominator and the second output as the numerator.

Example 4

A sample contains 300 grams of carbon-11, which has a half-life of 20 minutes.

a

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

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 start with the function A=300\left(\dfrac{1}{2}\right)^n. Since the decay factor is happening every 20 minutes we need to adjust the exponent of the function to be A=300\left(\dfrac{1}{2}\right)^\frac{n}{20}.

Reflection

By using \dfrac{n}{20} in the exponent we can see that the function will halve its value once in 20 minutes since n=20 gives us \dfrac{20}{20}=1, twice in 40 minutes since n=40 gives us \dfrac{40}{20}=2, etc.

b

Evaluate the function for n=30 and interpret the meaning in context.

Approach

In this context n represents the time, in minutes, 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)^\frac{30}{20}\approx 106.1 which tells us that there were approximately 106.1 grams of carbon-11 remaining after 30 minutes had passed.

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.5.3

Given a mathematical or real-world context, classify an exponential function as representing growth or decay.

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.F.1.2

Given a function represented in function notation, evaluate the function for an input in its domain. For a real-world context, interpret the output.

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