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Earlier in this course, one of the function families called linear functions was covered. However, linear functions are not always the best option for modeling real-life situations. For example, another function family is needed to model populations, carbon date artifacts, compute investments. This lesson will introduce this new function family.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Linear Functions
Making a Table of Values
Translating Between Rational Exponents and Radicals
Extending the Properties of Exponents to Rational Exponents

Explore

Investigating Function Families

Each function family has unique characteristics. By investigating the table of values of a function, several characteristics can be identified. The following tables of values belong to two functions derived from different function families.

Comparing Functions

How do the

values of Function I and Function II change according to their corresponding

values? What do the graph of these functions look like? Which function family does Function I belong? What can be concluded about Function II?

Discussion

The Concept of Exponential Expressions

To understand the functions whose dependent variable is multiplied by a constant factor as its independent variable changes by a constant amount, one algebraic expression needs to be mentioned beforehand.

Concept

Exponential Expression

An exponential expression is a type of algebraic expression which consists of a number, called the base, raised to either a number, a variable, or an expression, called the exponent. Sometimes an exponential expression may be multiplied by another number, called a coefficient.

When both the base and the exponent are real numbers, the expression is a numeric expression. Additionally,

is also referred to as a power.

Now, a function family containing exponential expressions will be introduced.

Discussion

Exponential Function

An exponential function is a nonlinear function that can be written in the following form, where and As the independent variable changes by a constant amount, the dependent variable multiplied by a constant factor. Therefore, consecutive values form a constant ratio.

Here, the coefficient

is the

intercept, which is sometimes referred to as the initial value. The base

can be interpreted as the constant factor. The graph of an exponential function depends on the values of

and

Graph of y=a*b^x for a-values less than and greater than 0, and for b-values less than and greater than 1.

Why

Why

If the coefficient

is

the function becomes a horizontal line.

This is a line along the

axis and, therefore, is a linear relation. This means that if

then the function is not exponential.

Why

Why

and

If the base

is negative, the function gives undefined results for certain

values. For example, since

a negative value for

would yield non-real values for

Hence, a condition on

is needed.

However, if

or

, the function becomes a horizontal line.

Therefore,

cannot be equal to

nor

Pop Quiz

Identifying Functions

Considering the definitions of an exponential function and a linear function, identify the functions given by the following tables of value.

Identifying Functions

Example

Population of Coyotes

Coyotes tend to thrive along the coasts, the deserts, and forests of the United States. A case study has shown that since the population of coyotes in a particular national park triples every years.

This population growth of coyotes can be modeled by an exponential function.

In this function,

is the number of

year periods. Using the function of coyote population growth, answer the following questions.

a How many coyotes were in the national park in

b How many coyotes will be in the national park after

more years?

Hint

a What value of

should be used to find the number of coyotes in

b Identify the value of

corresponding to a

year period.

Solution

a Recall that

is the number of

year periods. In

the value of

was

Therefore, by substituting

for

into the function, the initial number of coyotes can be found.

Identity Property of Multiplication

b Since

is the number of

year periods,

years correspond to

periods. With this information, the number of coyotes in the national park can be found as follows.

Calculate power and product

As a result, there will be

coyotes another

years from now.

Pop Quiz

Evaluating Exponential Functions

The following applet provides several exponential functions. Evaluate the value of the function for the given value of If it is necessary, round the result to two decimal places.

Evaluating Exponential Functions

Example

Population of Foxes

The San Joaquin kit fox was relatively common until the when people began converting grasslands to farms, orchards, and cities.

Since then, the number

of the San Joaquin kit foxes has been decreasing by

every decade. The following exponential function shows the number of foxes since the year

In this function,

is the number of decades. Using the function, answer the following questions.

a Graph the function.

b Find and interpret the

c How many kit foxes were there in

Answer

a

b See solution.

c

coyotes

Hint

a Make a table of values to graph the function.

b The

intercept is a point where the graph intersects the

axis.

x Begin by determining the value of

Solution

a All functions can be graphed by creating a table of values. To do this, arbitrarily chosen

values are used to find their corresponding

values. This method can be used to graph the given exponential function. Begin by finding the

value corresponding to

Remember to follow the order of operations.

Identity Property of Multiplication

Therefore, the point

lies on the given function. Other points can be found in the same way. For

use the whole numbers from

to

The points found above all lie on the function. To graph the function, plot them in a coordinate plane and connect them with a smooth curve. Note that the number of decades cannot be negative, so the function will be restricted by the first quadrant.

b Considering the table and graph from Part A, the

intercept can be identified. It is a point where the curve intersects the

axis. Hence, the

The

coordinate of the point represents the number of decades since

The

coordinate of the point represents the number of kit foxes. Since the function shows the population starting from

the

intercept tells that there were

kit foxes in

c To find the number of kit foxes in

first the number of decades since

needs to be determined.

There are

years from

to

which correspond to

decades. By substituting

for

into the function, the number of kit foxes in

can be found.

Calculate power

MoveLeftFacToNumOne

Calculate quotient

Discussion

Graphing an Exponential Function

The most basic method for graphing a function is making a table of values. This method can be used to graph an exponential function, as well. On the other hand, there is another method to graph an exponential function more effectively.

Method

Graphing an Exponential Function Using the Function Rule

For an exponential function

represents the initial value and

represents the constant multiplier. These values can be used to graph the function. Consider the following exponential function.

The graph of this function will be drawn as an example by using its function rule.

1

Identify

and

The initial value

of an exponential function is the number without an exponent. The constant multiplier

is the number with the exponent. In this case,

and

can be identified as follows.

2

Plot the Initial Value

The initial value is the value when It can also be thought of as the intercept of the function. Here, the initial value is so is intercept of the graph.

3

Use the Constant Multiplier to Find More Points

When the

value increases by

the

value is multiplied by

Since

the

value for

can be calculated as the product of the initial value

and the constant multiplier

Therefore,

also lies on the graph of the function. Similarly, the point

lies on the graph because

These points are shown on the graph.

This process can be repeated until a general form of the graph emerges.

4

Draw the Curve

Lastly, the graph can be drawn by connecting the points with a smooth curve.

Example

Graphing an Exponential Function and Identifying Its Key Features

The graph of an exponential function can be drawn by using its function rule. Then from the graph of a function the key features such as domain, range, and end behavior can be found.

Considering the above function, answer the following questions.

a Graph the function using its function rule.

b What are the key features of the function?

Answer

a

b See solution.

Hint

a Begin by identifying the initial value value and constant multiplier of the function.

b Use the graph of the function to find the domain, range, and end behavior of the function. Recall the definitions of these concepts.

Solution

a The function

has the initial value

and the constant multiplier

Use these values to mark four points on the function's graph.

The function can now be graphed by connecting the points with a smooth curve.

b The function's key features can be described by interpreting its graph.

The graph has a intercept at
The function is greater than for all Although the left-end of the graph approaches the axis, it never intersects it. Therefore, there is no intercept.
As approaches the function also approaches Therefore, the function increases for all

Combining these features, it can be also concluded that the domain of the function is all real numbers and its range is positive real numbers.

Looking at the graph, it can be seen that the left-end approaches

and the right-end extends upward. With this information, the end behavior of

can be written as follows.

This piece of information can be illustrated on the graph.

Example

Population of Flemish Giant Rabbits

By applying reverse engineering on the graph of an exponential function, its function rule can be written as well. In scientists discovered a rare population of Flemish Giant rabbits in a secluded forest.

Since then, they have been monitoring the population. After five years of conducting the study, the number of rabbits could be modeled with the following exponential function.

Use the graph to write the rule for the function. Then, interpret its initial value and constant multiplier.

Answer

Function Rule:
Interpretation: See solution.

Hint

Begin by identifying the initial value of the function.

Solution

To write an exponential function rule, the initial value of the function

and the constant multiplier

are needed.

Notice that the graph starts at

This means that

is the initial value. Since

the following incomplete function rule can be written.

To determine

use another point on the graph. The point

lies on the graph. To find the value of

substitute

and

into the incomplete rule and solve for

,

▼

Solve for

Rearrange equation

Calculate quotient

The constant multiplier is

Therefore, the function rule can be written as follows.

In this context, the initial value means that the initial population when the rabbits were discovered was

Additionally, the constant multiplier means that each year the population is

times more than the previous year.

Closure

Exponential Inequalities

Throughout the lesson, evaluating and graphing an exponential function and interpreting the graph of an exponential function have been covered. Beside these, exponential inequalities can be also mentioned even though they are not frequently used.

They are useful in situations involving repeated multiplication, especially when being compared to a constant value, such as in the case of interest. Their graphs can be drawn similarly to graphs of linear inequalities. For example, consider the following exponential inequality.

Notice that the

variable is already isolated. Since it is an exponential inequality, there will be a boundary curve instead of a boundary line.

The curve can be graphed using the function rule of the exponential function. Note that the inequality is strict, so the curve will be dashed.

Finally, since all of the values are greater than the region above the curve will be shaded.

Consider the example from the collection where scientists modeled the number of Flemish Giant rabbits using They are expecting the number of rabbits to increase along with the curve formed by the exponential function. It can be assumed that the number of rabbits falls below the curve at any given period of time.

In this case, the scientists can interpret this situation as a disease, for example, that is affecting the rabbits. They can then propose some precautionary measures to maintain a balance in the population of the species.

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