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5. Exponential Functions
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United States of AmericaTennessee
Algebra 1

5.04 Percent growth and decay

Lesson
Practice
Lesson

Concept summary

Exponential functions, both growth and decay, can be thought of in terms of their percent 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 of the exponential function, usually expressed as a decimal value

For functions in the form f\left(x\right)=a(1+r)^x, r is a growth rate, while for functions in the form f\left(x\right)=a(1-r)^x, r is a decay rate.

Growth rate

The fixed percent by which an exponential function increases

Decay rate

The fixed percent by which an exponential function decreases

Worked examples

Example 1

Classify the exponential function f\left(x\right)=5(1-0.03)^x as either exponential growth or exponential decay and identify both the initial value and the rate of growth or decay.

Approach

We know that an exponential function represents growth if the function is in the form f\left(x\right)=a(1+r)^x and decay if it is in the form f\left(x\right)=a(1-r)^x.

Solution

This function represents exponential decay because it is in the form f\left(x\right)=a(1-r)^x. The initial value is 5 and the decay rate is 0.03 or 3\%.

Reflection

Growth and decay rates are represented as percentages.

Example 2

Consider the exponential function modeled by the graph.

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a

Identify the initial value and growth rate.

Approach

The initial value is represented by the y-intercept. To find the growth rate, first find the constant factor by evaluating the ratio of two consecutive outputs, then find \left\vert \text{constant factor}-1\right\vert.

Solution

The initial value of the function is 4. The constant factor is \dfrac{6}{4}=1.5 and the growth rate is \left\vert \text{1.5}-1\right\vert=0.5=50\%.

b

Write the equation of the function in the form y=a(1\pm r)^x.

Solution

y=4(1+0.5)^x

Example 3

Justin purchased a piece of sports memorabilia for \$2900, and it is expected to increase in value by 9\% per year.

a

Write a function, y, to represent the value of the piece of sports memorabilia after v years.

Approach

Since the memorabilia is predicted to increase in value we will use the growth rate form of the function, f\left(x\right)=a(1+r)^x.

Solution

The initial value of the function is \$2900 and the growth rate is 9\% so the function is y=2900(1+0.09)^v

Reflection

Remember to convert the rate as a percentage to a decimal by dividing it by 100.

b

Evaluate the function for v=8 and interpret the meaning in context.

Approach

In this context v represents the time, in years, and the output, y represents the value of Justin's sports memorabilia. We will evaluate the function and apply these units to interpret the meaning of the solution.

Solution

A=2900(1+0.09)^8\approx 5778.43 which tells us that the memorabilia will be worth approximately \$5778.43 after 8 years have passed.

Outcomes

A1.A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

A1.A.SSE.A.1.B

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

A1.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.*

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

A1.F.BF.A.1

Build a function that describes a relationship between two quantities.*

A1.F.BF.A.1.A

Determine steps for calculation, a recursive process, or an explicit expression from a context.

A1.F.LE.A.1.C

Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.

A1.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.*

A1.F.LE.B.3

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

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP5

Use appropriate tools strategically.

A1.MP6

Attend to precision.

A1.MP8

Look for and express regularity in repeated reasoning.

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