Exponential relationships include any relations where the outputs increase by a constant factor or decrease by a constant factor for consistent changes in x.

An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an exponential function:

\displaystyle f\left(x\right)=ab^x

\bm{a}

Leading coefficient

\bm{b}

Base where b \gt 0,\, b\neq 1

\bm{x}

Independent variable

\bm{f(x)}

Dependent variable

Exploration

Consider the following equations with a=1:

y=5^x

x	-2	-1	0	1	2
y	\dfrac{1}{25}	\dfrac{1}{5}	1	5	25

y=2^x

x	-2	-1	0	1	2
y	\dfrac{1}{4}	\dfrac{1}{2}	1	2	4

For each of the functions, think about the following questions:

What happens to y as x increases?
Compare y=5^x and y=2^x. How are they similar? How are they different?
What is the y-intercept for each of the functions? How does this relate to the value of a?
Does either function have an x-intercept?
Create a table of values for y=1^x. Does it have the same properties as y=5^x and y=2^x?

We can determine whether a function is exponential by dividing consecutive function values to see if they have a constant factor.

The base, or constant factor, is the number being multiplied repeatedly. It tells us how quickly the output values are growing or shrinking. We can find the base by dividing a term by the previous term, as shown below:

A table with 2 rows titled x and f of x, and with 4 columns. The data is as follows: First row: 0, 1, 2, and 3; Second row: 1, 3, 9, and 27. Below the second row are 3 semi circle arrows, each labeled times 3: from the cell containing 1 to the cell containing 3, from 3 to 9, and from 9 to 27. Below each semi circle arrows is a row of equations titled constant factor: below the left semi circle arrow, 3 divided 1 equals 3; below the middle semi circle arrow, 9 divided by 3 equals 3; below the right semi circle arrow, 27 divided by 3 equals 3.

In this example, we see that the function is growing exponentially. A function grows exponentially when it increases by a constant factor.

Exponential functions change at a faster rate than linear functions. In the table below, we are adding 3 to each term, but the terms do not grow as quickly.

A table with 2 rows titled x and f of x, and with 4 columns. The data is as follows: First row: 0, 1, 2, and 3; Second row: 1, 4, 7, and 10. Below the second row are 3 semi circle arrows, each labeled plus 3: from the cell containing 1 to the cell containing 4, from 4 to 7, and from 7 to 10.

-4

-3

-2

-1

All exponential functions of the form y=ab^x have these features in common:

The domain is all real values of x.
The range is y>0.
For a context, the domain and range may be further restricted.
The y-intercept is at (0,\,a).
The common factor is b.
There is a horizontal asymptote at y=0.

Asymptote

A line that a curve or graph approaches as it heads toward positive or negative infinity.

An exponential function can get infinitely close to an asymptote, but it can never cross it. This means that an exponential function of this form will not have an x-intercept.

Examples

Example 1

Consider the table of values for the function y = 3(2)^{ x }.

x	-3	-2	-1	0	1	2	3	4	5	10
y	\dfrac{3}{8}	\dfrac{2}{4}	\dfrac{3}{2}	3	6	12	24	48	96	3072

Describe the behavior of the function as x increases.

Worked Solution

Create a strategy

We want to identify if the values of y are increasing or decreasing as x increases.

Apply the idea

As x increases, the function increases at a faster and faster rate.

Reflect and check

We can see that the equation has a constant factor that is greater than 1. This is why the function is increasing.

Determine the y-intercept of the function.

Worked Solution

Create a strategy

The y-intercept occurs when x=0. We can read these coordinates from the table.

Apply the idea

\left(0,\,3\right)

Reflect and check

We can see that the equation has a leading coefficient of 3. This is the value of the y-intercept, and the result of substituting x=0 into the equation.

State the domain of the function.

Worked Solution

Create a strategy

The domain is the complete set of possible values for x. For exponential functions, the graph extends indefinitely in both horizontal directions.

Apply the idea

All real values of x.

Reflect and check

All exponential equations of the form y=ab^x have a domain of all real x.

State the range of the function.

Worked Solution

Create a strategy

The range is the complete set of possible values for y. We can see the graph extends indefinitely up towards the left, but it approaches an asymptote at y=0 towards the right.

Apply the idea

y \gt 0

Reflect and check

All exponential equations of the form y=ab^x have a range of y \gt 0 for positive values of a.

Example 2

A population of bacteria can be modeled with the equation p(t)=100\left(2\right)^t, where p(t) is the population after t days. This graph shows the population over time.

200

400

600

800

1000

1200

1400

1600

1800

2000

Identify and interpret the p-intercept.

Worked Solution

Create a strategy

The p-intercept is the vertical intercept and will occur when t=0.

We can read this off the graph or use the equation.

Apply the idea

Using the equation:

\displaystyle p(t)	\displaystyle =	\displaystyle 100(2)^t	Start with the equation
\displaystyle p(0)	\displaystyle =	\displaystyle 100(2)^0	Let t=0
	\displaystyle =	\displaystyle 100(1)	Use the zero power law
	\displaystyle =	\displaystyle 100	Evaluate the product

This means that initially there were 100 bacteria.

Reflect and check

Reading from the graph is not as precise as using the equation, but is still an important approach.

Estimate and interpret when p(t)=1600.

Worked Solution

Create a strategy

We can create a table of values using the equation or read it off the graph.

Apply the idea

200

400

600

800

1000

1200

1400

1600

1800

2000

Starting from p(t)=1600, we can go across to the curve and then down to find the t-value for when p(t)=1600.

This means that after 4 days, the population reaches 1600.

Reflect and check

We can confirm this by creating a table of values:

t	0	1	2	3	4
p(t)	100	200	400	800	1600

Identify and interpret the domain and range in this context.

Worked Solution

Create a strategy

We can read the domain and range from the graph, or think about what is possible given the context.

Apply the idea

Since t represent time, it cannot be negative. This means the domain is x \geq 0.

The initial population is 100, and it is growing from there, so the range is y \geq 100.

Reflect and check

Without the context, the domain and range of y=100(2)^x would be different.

Idea summary

The base of the exponent is the constant factor, or the number being multiplied repeatedly. We can find it by dividing one output by the previous output.

All exponential functions of the form y=ab^x have the following features in common:

The domain is -\infty \lt x \lt \infty.
The range is y>0.
If there is a context, the domain and range may be different based on the realistic constraints.
The y-intercept is at (0,\,a).
There is a horizontal asymptote at y=0.

Outcomes

A.F.2

The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2e

Given an equation or graph of an exponential function in the form y = ab^(x) (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable.

A.F.2g

For any value, x, in the domain of f, determine f(x) of a quadratic or exponential function. Determine x given any value f(x) in the range of f of a quadratic function. Explain the meaning of x and f(x) in context.

5.08 Characteristics of exponential functions

Ideas

Characteristics of exponential functions

Exploration

Examples

Example 1

Example 2

Outcomes

A.F.2

A.F.2e

A.F.2g

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