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5. Exponential Functions and Equations
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United States of America
Grades 9-12

5.03 Exponential decay

Lesson
Practice

Introduction

Exponential decay functions will have similar characteristics as the exponential growth functions from the previous lesson. But instead of increasing over the domain, an exponential decay function will decrease over its domain because we are repeatedly multiplying by a number less than one. We will continue using properties of exponents that were covered in  1.04 Rational exponents  to manipulate expressions with exponents.

Exponential decay

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

\displaystyle f\left(x\right)=ab^x
\bm{a}
y-intercept
\bm{b}
constant factor
Exponential decay

The process of decreasing in size by a constant percent rate of change. This occurs when 0<b<1.

x
y
Decay factor

The constant factor of an exponential decay function

Exponential functions can be thought of in terms of their percent change:

\displaystyle f\left(x\right)=a(1-r)^x
\bm{a}
y-intercept
\bm{r}
decay rate
Decay rate

The fixed percent by which an exponential function decreases

Notice that the base of the exponent is (1-r). This is the constant factor, but more specifically, it is the decay factor because it is less than 1.

When we are given the decay factor, we can use it to solve for the decay rate.

\displaystyle b=(1-r)
\bm{b}
decay factor
\bm{r}
decay rate

To solve for the decay rate, we need to add r to both sides and subtract b from both sides. In general, r=1-b.

Sometimes we need to find the key characterstics of an expression that isn't presented in the form ab^x. In which case, we can use properties of exponents to rewrite the expression in a form we recognize.

Exploration

Consider the following exponential functions:

Function 1: A ball is dropped from a ledge. The height of the ball rebounds to 80\% of its previous height after each bounce.

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Function 2
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Function 3

Function 4: f(x)=\dfrac{1}{7}(1.08)^x

Function 5:

x0123
y109.69.2168.847\,36
  1. Which ones can be classified as decay functions?
  2. Explain how you identified the decay functions.

An exponential decay function will have a constant factor than is less than 1, which will cause the y-values to decrease over time.

We can use the key features of an exponential function to build its equation. First, we find the y-intercept, then find the decay factor or decay rate. The specific form of the equation we use will depend on whether we want to highlight the decay factor or the decay rate.

Examples

Example 1

Identify the y-intercept and the decay rate for each of the following exponential functions:

a

f(x)=\dfrac{2}{3}\left(0.625\right)^x

Worked Solution
Create a strategy

This function is in the form y=ab^x. We can identify the y-intercept and the decay factor, but then we need to use r=1-b to solve for the decay rate.

Apply the idea
\displaystyle r\displaystyle =\displaystyle 1-bEquation for decay rate
\displaystyle r\displaystyle =\displaystyle 1-0.625Substitute b=0.625
\displaystyle r\displaystyle =\displaystyle 0.375Evaluate the subtraction

As a percentage, the decay rate is r=37.5\%. Since a=\dfrac{2}{3}, the y-intercept is \left(0,\dfrac{2}{3}\right).

b

f(x)=4^{-x}

Worked Solution
Create a strategy

This is not in one of the function forms expressed above because the exponent in this function is negative. We can use the properties of exponents to change this to the form y=ab^x.

Apply the idea
\displaystyle f\left(x\right)\displaystyle =\displaystyle 4^{-x}Given equation
\displaystyle =\displaystyle \left(\dfrac{1}{4}\right)^xNegative exponent property

Now that it is in the correct form, we see a=1 and b=\dfrac{1}{4}. Next, we need to solve for the decay rate.

\displaystyle r\displaystyle =\displaystyle 1-bEquation for decay rate
\displaystyle r\displaystyle =\displaystyle 1-\dfrac{1}{4}Substitute b=\dfrac{1}{4}
\displaystyle r\displaystyle =\displaystyle \dfrac{3}{4}Evaluate the subtraction

The y-intercept is at (0,1), and the decay rate as a percentage is 75\%.

Reflect and check

We can see how the function decays by creating a table of values:

x-2-1012
y1641\dfrac{1}{4}\dfrac{1}{16}

Each output is reduced by 75\% of its value:

16-16\left(0.75\right)=16-12=4

4-4\left(0.75\right)=4-3=1

1-1\left(0.75\right)=1-0.75=0.25=\dfrac{1}{4}

0.25-0.25\left(0.75\right)=0.25-0.1875=0.0625=\dfrac{1}{16}

Example 2

Consider the exponential function f\left(x\right)=5(1-0.03)^x.

a

Classify the function as either exponential growth or exponential decay.

Worked Solution
Create a strategy

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

Apply the idea

This function represents exponential decay because it is in the form f\left(x\right)=a(1-r)^x.

Reflect and check

We could also simplify what is in parenthesis to determine whether b>1, in which case it would be exponential growth, or if 0<b<1, in which case it would be exponential decay.

The base of the function is b=1-0.03=0.97. Since 0<0.97<1, this will decay exponentially.

b

Identify both the y-intercept and the rate of growth or decay.

Worked Solution
Create a strategy

Since this function is in the the form y=\left(1-r\right)^x, we can easily find a, the y-intercept, and r, the decay rate.

Apply the idea

a=5, r=0.03

The y-intercept is at (0,5), and the decay rate is 3\%.

Example 3

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

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Worked Solution
Create a strategy

To create the equation, we need to identify the y-intercept 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 leading coefficient, and the growth or decay factor can be found using the ratio of two successive outputs.

Apply the idea

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

To find the constant 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

Reflect and check

We can see from this graph that this is a 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

Write an equation that models the exponential function shown in the table.

x-2-10123
f(x)7.81256.25543.22.56
Worked Solution
Create a strategy

To create the equation, we need to identify the y-intercept and decay factor for the function modeled in the table.

An exponential function can be represented with an equation of the form f(x)=a(b)^x, where a is the y-value of the y-intercept and b is the decay factor when 0<b<1.

Apply the idea

The y-intercept is (0,5) and the decay factor is \dfrac{4}{5} or 0.8, so the equation for this function is f(x)=5(0.8)^x.

Reflect and check

To check our answer, we can substitute different values of x in the function and see if they match the values in the table.

f(-2)=5(0.8)^{-2}=7.8125

f(-1)=5(0.8)^{-1}=6.25

f(1)=5(0.8)^1=4

f(3)=5(0.8)^3=2.56

These are the same values for f(x) as we see in the table, so this is the correct equation.

Example 5

Raya purchased a car for \$21\,500, and it is expected to depreciate by 18\% per year.

a

Write a function, y, to represent the value of the car after t years.

Worked Solution
Create a strategy

Since the car is predicted to decrease in value, we will use the decay rate form of the function, y=a(1-r)^x.

Apply the idea

The initial value of the function is \$21\,500 and the decay rate is 18\%, so the function is y=21\,500\left(1-0.18\right)^t.

Reflect and check

Since negative values for time do not make sense, the y-intercept will be the leftmost point on the graph. In this case, we refer to it as the initial value.

b

Evaluate the function for t=3 and interpret the meaning in context.

Worked Solution
Create a strategy

In this context, t represents the time in years, and the output, y, represents the value of Raya's car. We will evaluate the function and apply these units to interpret the meaning of the solution.

Apply the idea

y=21\,500\left(1-0.18\right)^3=11\,854.412 which tells us that the car will be worth approximately \$11\,854 after 3 years have passed.

Example 6

Find the equation of the exponential decay function passing through the points \left(-1, \dfrac{8}{3}\right) and \left(2,\dfrac{9}{64}\right).

Worked Solution
Create a strategy

Since these are not consecutive points, we will need to determine how many times the leftmost value was multiplied by the decay factor, then set up an equation to find the decay factor.

The leftmost point is at x=-1, and we would need to multiply by the decay factor 3 times to get to the next point at x=2. This can be represented by the equation \dfrac{8}{3}b^3=\dfrac{9}{64}.

Apply the idea
\displaystyle \dfrac{8}{3}b^3\displaystyle =\displaystyle \dfrac{9}{64}Given equation
\displaystyle b^3\displaystyle =\displaystyle \dfrac{27}{512}Multiply both sides by \dfrac{3}{8}
\displaystyle b\displaystyle =\displaystyle \dfrac{3}{8}Take the cube root of both sides

Now that we have the decay factor, we can solve for the y-intercept, or the leading coefficient in the formula. To do this, we can use the form y=ab^x with the decay factor we found and one of the given points.

\displaystyle y\displaystyle =\displaystyle ab^x
\displaystyle \dfrac{8}{3}\displaystyle =\displaystyle a\left(\dfrac{3}{8}\right)^{-1}Substitute b=\dfrac{3}{8} and first point \left(-1,\dfrac{8}{3}\right)
\displaystyle \dfrac{8}{3}\displaystyle =\displaystyle \dfrac{8}{3}aNegative exponent property
\displaystyle a\displaystyle =\displaystyle 1Division property of equality

Therefore, the equation that passes through the given points is y=\left(\dfrac{3}{8}\right)^x.

Reflect and check

We can check our equation with the other given point, \left(2,\dfrac{9}{64}\right). We will substitue x=2 into our equation to see if y=\dfrac{9}{64}.

y=\left(\dfrac{3}{8}\right)^{2}=\dfrac{9}{64}

This verifies that our equation is correct.

Idea summary
\displaystyle y=a\left(1-r\right)^x
\bm{a}
y-intercept
\bm{\left(1-r\right)}
decay factor
\bm{r}
decay rate

If we know the decay factor, we can use the formula r=1-b to solve for the decay rate.

If you know the decay factor, use the form y=ab^x to write the equation.

If you know the decay rate, use the form y=a(1-r)^x to write the equation.

The y-intercept is also known as the initial value, a, in some cases.

Outcomes

A.SSE.A.1.B

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

A.SSE.A.2

Use the structure of an expression to identify ways to rewrite it.

A.SSE.B.3.C

Use the properties of exponents to transform expressions for exponential functions.

F.IF.C.8

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

F.IF.C.8.B

Use the properties of exponents to interpret expressions for exponential functions.

F.LE.A.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

F.LE.B.5

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

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