{{ 'ml-label-loading-course' | message }}
{{ toc.name }}
{{ toc.signature }}
{{ tocHeader }} {{ 'ml-btn-view-details' | message }}
{{ tocSubheader }}
{{ 'ml-toc-proceed-mlc' | message }}
{{ 'ml-toc-proceed-tbs' | message }}
Lesson
Exercises
Recommended
Tests
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}. 

{{ article.displayTitle }}

{{ article.intro.summary }}
{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} expand_more
{{ 'ml-heading-abilities-covered' | message }}
{{ ability.description }} {{ ability.displayTitle }}
{{ 'ml-heading-lesson-settings' | message }}
{{ 'ml-lesson-number-slides' | message : article.intro.bblockCount}}
{{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount}}
{{ 'ml-lesson-time-estimation' | message }}
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

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.

Parts of an exponential expression a·b^x: a (coefficient), b (base), x (exponent).
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.
Graph of y=a*2^x where the value of 'a' can be changed from -2.5 to 2.5

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
Graph of y=2*b^x where the value of 'b' can be changed from 0.1 to 2
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.

Coyote.jpg

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.
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.
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.

San Joaquin Kit Fox.jpg

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 intercept.
c How many kit foxes were there in

Answer

a
Graph of the given function
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.
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.

The graph of the given exponential function
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 coordinate is
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.
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
expand_more
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
expand_more

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.

Initial Value of an Exponential Function
3
Use the Constant Multiplier to Find More Points
expand_more
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.
Identifying Points on an Exponential Function

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

Identifying Points on an Exponential Function
4
Draw the Curve
expand_more

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

Drawing the Graph of an Exponential Function
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
Graphing an Exponential Function Using Its Function Rule
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.
Graphing an Exponential Function Using Its Function Rule

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

Graphing an Exponential Function Using Its Function Rule
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.

Identifying Key Features of an Exponential Function
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.
Identifying Key Features of an Exponential Function
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.

Flemish Giant Rabbit.jpg

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.

Exponential Function Showing the Population of Rabbits

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.
Exponential Function Showing the Population of Rabbits
Since the following incomplete function rule can be written.
To determine use another point on the graph.
Exponential Function Showing the Population of Rabbits
The point lies on the graph. To find the value of substitute and into the incomplete rule and solve for
Solve for
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.
Boundary Curve

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

Boundary Curve and Shaded Region

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.

Number of Rabbits
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.