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4. Exponential Functions
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United States of America
Grades 9-12

4.02 Exponential growth

Lesson
Practice

Introduction

Exponential functions can be classified as growth or decay based on the value of the constant factor. In the previous lesson, we discussed how the value of the base would determine whether the function is increasing or decreasing. This lesson will focus solely on increasing functions, when {b>1}, also known as exponential growth functions, and will use properties of exponents that were covered in  1.04 Rational exponents  to manipulate expressions with exponents.

Exponential growth

When the y-values get increasingly larger, we call the function an exponential growth function. In an equation, this will occur when the constant factor, b, is greater than 1.

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

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

x
y
Growth factor

The constant factor of an exponential growth 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}
growth rate
Growth rate

The fixed percent by which an exponential function increases

Notice that the base of the exponent is (1+r). This is the constant factor, but more specifically, it is the growth factor because it is greater than 1.

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

\displaystyle b=(1+r)
\bm{b}
growth factor
\bm{r}
growth rate

To solve for the growth rate, we simply need to subtract 1 from both sides.

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.

Examples

Example 1

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

a

f(x)=3\left(1.725\right)^x

Worked Solution
Create a strategy

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

Apply the idea

The growth factor, b, is 1.725.

\begin{aligned} 1.725 &= 1+r \\ r &= 0.725 \end{aligned}

The y-intercept is (0, 3), and the growth rate is 72.5\%.

Reflect and check

In the formula, the growth rate is represented as a decimal. When we are discussing the rate outside of the formula, it is best to state it as a percentage.

b

f(x)=2^{x-1}

Worked Solution
Create a strategy

This is not in either of the forms discussed above because the exponent is not just an x. We need to use the properties of exponents to manipulate the equation until it is in one of the forms mentioned above.

Apply the idea
\displaystyle f(x)\displaystyle =\displaystyle 2^{x-1}Given equation
\displaystyle =\displaystyle 2^{x}\cdot2^{-1}Product property of exponents
\displaystyle =\displaystyle \left(2\right)^x \cdot \dfrac{1}{2}Negative exponent property
\displaystyle =\displaystyle \dfrac{1}{2}\left(2\right)^xCommutative property

Now, it is in the form y=ab^x, so we can identify the y-intercept and growth factor.

a=\dfrac{1}{2},\, b=2

Last, we need to find the growth rate:

\displaystyle b\displaystyle =\displaystyle 1+rEquation for growth factor
\displaystyle 2\displaystyle =\displaystyle 1+rSubstitute b=2
\displaystyle 1\displaystyle =\displaystyle rSubtract 2 from both sides

\begin{aligned} 2 &= 1+r \\ r &= 1 \end{aligned}

The y-intercept is \left(0,\dfrac{1}{2}\right) and the growth rate is 1 or 100\%.

Reflect and check

When something grows by 100\%, it is being increased by its own value. In other words, we are doubling the output each time.

Example 2

Consider the exponential function: f\left(x\right)=\dfrac{2}{5}\left(4\right)^x

a

Explain how we know the function represents exponential growth.

Worked Solution
Create a strategy

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

Apply the idea

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

Reflect and check

We can validate this answer by building a table of values to show that the y-values increase as the x-values increase.

x0123
y\dfrac{2}{5}1\dfrac{3}{5}6\dfrac{2}{5}25\dfrac{3}{5}

The y-values are growing at an increasing rate, so this does represent exponential growth.

b

Identify the growth rate.

Worked Solution
Create a strategy

According to the formula, the growth rate can be found by subtracting 1 from the growth factor.

Apply the idea

The growth factor is 4.

4-1=3

The growth rate is 300\%, so each term is increased by 300\%.

Reflect and check

We can check by finding 300\% of each output. We can start from the y-intercept and find the following points from there.

\dfrac{1}{2}\times \dfrac{300}{100}=\dfrac{3}{2}

Now, we increase \dfrac{1}{2} by \dfrac{3}{2}.

\dfrac{1}{2}+\dfrac{3}{2}=2

This would be the output associated with x=1, so let's check by plugging in x=1.

f(1)=\dfrac{1}{2}(4)^1=2

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

If we know the growth factor, we use the formula b=1+r to solve for the growth rate.

Writing exponential growth equations

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

Exploration

Consider the following exponential growth functions:

Function 1:

1
2
3
4
5
x
1
2
3
4
5
6
7
8
y

Function 2:

x01234
y11.151.32251.52091.7490

Function 3:

Reagan decided to get a part-time job to begin saving for a new car. He was hired by a company that pays \$21\,500 in the first year, and he will receive a 5\% raise each year after that.

  1. Which of the functions would you model using y=ab^x and why?
  2. Which of the functions would you model using y=a\left(1+r\right)^x and why?
  3. In what type of situations would you prefer knowing the growth factor over the growth rate?

In situations where growth is given as a percentage, it is easiest to model the equation in the form y=a\left(1+r\right)^x. When trying to determine the equation from a table or a graph, it is easiest to find the growth factor and convert it to the growth rate if needed.

Examples

Example 3

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

x-3-2-10123
f(x)\dfrac{1}{9}\dfrac{1}{3}1392781
Worked Solution
Create a strategy

To create the equation, we need to identify the y-intercept and growth 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 value of the y-intercept and b is the growth factor when b>1.

Apply the idea

The y-intercept is at (0,3), so a=3. We can pick any two points, like (0,3) and (1,9), and divide the outputs to find the growth factor. The growth factor is \dfrac{9}{3}=3, so the equation for this function is f(x)=3(3)^x.

Reflect and check

Using the properties of exponents, we can simplify this function to f(x)=3^{x+1}.

Example 4

Consider the exponential function modeled by the graph.

-4
-3
-2
-1
1
2
3
4
x
2
4
6
8
10
12
14
16
18
y

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

Worked Solution
Create a strategy

The value of a is represented by the y-intercept. To find the growth rate, r, first find the growth factor, b, by evaluating the ratio of two consecutive outputs. Then, use the fact that r=b-1.

Apply the idea

The y-intercept of the function is 4. The growth factor is \dfrac{6}{4}=1.5 and the growth rate is 1.5-1=0.5=50\%.

y=4(1+0.5)^x

Reflect and check

We can validate our result by plugging in values of x to make sure they align with the points on the graph.

y=4(1+0.5)^0=4, and the point (0,4) is on the graph.

y=4(1+0.5)^1=6, and the point (1,6) is on the graph.

y=4(1+0.5)^2=9, and the point (2,9) is on the graph.

Example 5

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, V, to represent the value of the piece of sports memorabilia after t years.

Worked Solution
Create a strategy

Since the memorabilia is predicted to increase in value by a percentage, we will use the growth rate form of the function, y=a(1+r)^x or with the given variables V=a(1+r)^t.

Apply the idea

The initial value of the memorabilia is \$2900. This represents the value when t=0, so this is the y-intercept. The growth rate is 9\% which is 0.09 as a decimal, so the function is V=2900(1+0.09)^t.

Reflect and check

In many exponential growth functions, the independent variable represents time. Since time cannot be negative, the domain will begin from zero as it did in this problem. When this happens, the y-intercept is called the initial value.

b

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

Worked Solution
Create a strategy

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

Apply the idea

V=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.

Example 6

The population of rabbits in Lincoln county can be modeled by an exponential growth function. Conservationists have been measuring the population since 2015. After the first year, there were 46 rabbits. After the third year, there were 66 rabbits.

a

Estimate the growth rate to one decimal place.

Worked Solution
Create a strategy

To estimate the growth rate, we will create data points from the given information. If we let x=0 represent 2015, then x=1 will be the first year. The associated data point is (1,46). The data point for the third year is (3,66).

These are not consecutive outputs, so we need to set up an equation and solve for r.

Apply the idea

To get from the first year to the third year, we multiplied the population by the growth factor twice. The equation would be 46\cdot b \cdot b = 66 which simplifies to 46b^2=66.

\displaystyle 46b^2\displaystyle =\displaystyle 66Given equation
\displaystyle b^2\displaystyle =\displaystyle \dfrac{66}{46}Division property of equality
\displaystyle b\displaystyle \approx\displaystyle 1.198Approximate by taking the square root of both sides

This represents the growth factor, so now we must solve for the growth rate.

\begin{aligned} 1.198 &= 1+r \\ r &= 0.198 \end{aligned}

Next, we can convert to a percentage and round it to one decimal place.

\begin{aligned} r&=19.8\% \\ &= 20\% \end{aligned}

b

Estimate the initial population.

Worked Solution
Create a strategy

The initial population will be the number of rabbits present in 2015. Now that we know the growth rate, we can use the number of rabbits in the first year, 2016, to help us solve for the initial population.

Apply the idea

To set up the equation, we begin by substituting the known values r=0.2 from part (a), x=1, and y=46 into the equation y=a(1+r)^x.

\displaystyle y\displaystyle =\displaystyle a\left(1+r\right)^xExponential growth equation
\displaystyle 46\displaystyle =\displaystyle a\left(1+0.2\right)^1Substitute y=46, r=0.2, and x=1
\displaystyle 46\displaystyle =\displaystyle a\left(1.2\right)Evaluate the addition and exponent
\displaystyle \dfrac{46}{1.2}\displaystyle =\displaystyle aDivision property of equality
\displaystyle 38\displaystyle \approx\displaystyle aApproximate by evaluating the division

The initial population of rabbits in 2015 was approximately 38.

c

Write the equation that models this situation.

Worked Solution
Create a strategy

Since we discussed the percentage growth, we will use the growth rate form of the equation, y=a\left(1+r\right)^x.

Apply the idea

We found r=20\%, a=38 in parts (a) and (b), so we can simply plug them into the formula.

y=38\left(1+0.2\right)^x

Reflect and check

Because we rounded the percentage in part (a), the values will not be exact when we check our answers. But since we are working with rabbits and we cannot have a decimal of a rabbit, it makes sense to round to the nearest whole rabbit.

Year 1: y=38(1+0.2)^1=45.6 \approx 46 rabbits

Year 2: y=38(1+0.2)^2=54.72 \approx 55 rabbits

Year 3: y=38(1+0.2)^3=65.664\approx66 rabbits

These answers match the information given in the problem.

Idea summary

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

If you know the growth 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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