-The big difference about exponential functions is that the variable is the power rather than the base. For example, rather than looking at f(x)=x^2, expontential functions deal with things such as g(x)=2^x. Evaluating an exponential function is the same as other functions, meaning that we can simply pick values for x, plug them in for x, and simplify to solve. Exponential functions tend to start very small and then grow very quickly. -For example, we can look at g(x)=2^x in FIgure 1 more closely. First, we can make a table....From the table given, you can see that the further negative the x values get, the closer the function approaches zero. Additionally, even though the x values only increase by increments of one, the function increases much more rapidly. -After viewing the first table, one can see that this function only approaches 0 and is never negative. This is not always the case; there are ways to translate the functions so that they can denote negative values. In order to translate the functions they must move down the y-axis vertically, you must either add or subtract values not in the exponent. To translate the function horizontally, the added or subtracted value will be in the exponent with the variable. For example, if you were to make the change the first function to h(x)=(2^X)-2, the new table would look just as Figure 2 does. -As many functions work, there can also be horizontal translations. For horizontal translations, the original function could be changed to k(x)=2^(x+2). Since the addition of 2 is in the parenthesis with the x in the exponent, it therefore directly applies to the x rather than the y, and translates horizontally. Therefore, the new table would look like Figure 3. Back to Top of Page

Graphs

From these tables, if you were to infer how they would be represented graphically, you could imagine they'd look something like this: *(Figure 1a directly correlates to the function and table of Figure 1, as does Figure 2a to Figure 2 and Figure 3a to Figure 3)

*Figure 1a

*Figure 2a

*Figure 3a

Please note that due to the translations, each graph has a different scale, too. The first graph has increments of 1 while the second two increase by 2's. Also, in order to have the best view of each graph, they were manipulated within the viewing box to best fit the function. Back to Top of Page

Asymptotes

Nayland College Mathematics http://maths.nayland.school.nz/

-A quality of these functions that you may not be confidently accustomed to is that they do not have ranges from negative infinity to positive infinity. As you can see there is always a horizontal asymptote. In the original function, the asymptote is at y=0. This makes sense because a positive number raise to ANY number (even negative numbers or fractions) will still get you positive numbers, no matter how close it gets to 0. -In Figure 2a, the function was translated vertically, so the horizontal asymptote is now at y=-2. Then, in Figure 3a, the horizontal asymptote is still at y=0 since the vertical translation was not affected. -According to Wolfram|Alpha, an asymptote is a line or curve that the function approaches but never crosses. An asymptote can be vertical, horizontal, or diagonal along any linear function, but right now we are only dealing with horizontal asymptotes.

A logarithm is another type of exponential function. Many times, the use of a "log" is to manipulate the variable and get it out of the exponent. This is possible because the logarithm is the inverse of exponential function. By applying the log, one is able to work with the variable and also solve for it much easier. Since the log is the inverse of the exponential functions we looked at before, this is an image of how the two functions are graphed side by side:

For more information: http://people.richland.edu/james/lecture/m116/logs/logs.html

*More information on Logarithms can be found later under Real World Applications (and Extended Practice). Back to Top of Page

Michigan Standards Related to Exponential Functions:

Construct and compare linear, quadratic, and exponential models and solve problems:

CCSS.MATH.CONTENT.HSF.LE.A.1 - Distinguish between situations that can be modeled with linear functions and with exponential functions.

B.CCSS.MATH.CONTENT.HSF.LE.A.1.A - Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

CCSS.MATH.CONTENT.HSF.LE.A.1.B - Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

CCSS.MATH.CONTENT.HSF.LE.A.1.C - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

CCSS.MATH.CONTENT.HSF.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).

CCSS.MATH.CONTENT.HSF.LE.A.3 - Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

CCSS.MATH.CONTENT.HSF.LE.A.4 - For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology

Interpret expressions for functions in terms of the situation they model:

CCSS.MATH.CONTENT.HSF.LE.B.5 - Interpret the parameters in a linear or exponential function in terms of a context.

Extend the properties of exponents to rational exponents.

CCSS.MATH.CONTENT.HSN.RN.A.1 - Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

CCSS.MATH.CONTENT.HSN.RN.A.2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents.

It is important to not only know how to solve exponential and logarithmic functions but also be able to apply the knowledge learned to real world examples. The following site does a great job of giving examples that use exponential functions and logarithms to solve problems that incorporate interest rates, mortgage, population, radioactive decay and earthquakes. Also, it gives step by step solutions to each of the given problems along with more examples that you can try on your own. Using real world applications makes a difficult topic such as this one more interesting and gives a reason to understanding exponential functions and logarithms.

Exponential Outbreaks This article from the New York Times displays a lesson plan involving finding an exponential equation for modeling the growth of the Ebola disease following its outbreak. Students are challenged to relate their instrumental learning about exponential equations to form relational understanding on the spread of disease and other phenomena. The activity challenges students to recognize that rumors spread further the longer they have been around per day. The concept of exponential growth is clearly displayed as students model the spread of Ebola with equations. Back to Top of Page

Helpful Videos and Images:

This video from ThinkWell goes over the same example that has been shown above: f(x)=2^x. The professor in this video does a great job of explaining how to set up a table, plot points, and graph this exponential function. He also compares f(x)=2^x to other functions such as g(x)=3^x. He discusses the concept of asymptotes and looks at patterns found in the functions he graphs. He then relates these patterns to the function h(x)=(1/2)^x and shows how h(x) is similar to f(x). (See the ThinkWell webpage or YouTube page for more tutorials from this company.)

This video is based on an activity on eHow, "Cake Cutting". The video was shot by Jerry Mabrito and he was assisted by his three sons, Arnold, Lance, & JJ. What can you get from the video? A demonstration of how cake can make things fun, even exponents!

http://zonalandeducation.com/ - Picture provided twice http://www.flickr.com/photos/ - Provided by k0re
Figures 1, 2, 3, 1a, 2a, 3a - Created and Provided by Amy Burke
Concept Map - Created and Provided by Amy Burke and Emily Groenink

by Amy Burke, Emily Groenink, and Jerry Mabrito

## Sections:

## Tables

-The big difference about exponential functions is that the variable is the power rather than the base. For example, rather than looking at f(x)=x^2, expontential functions deal with things such as g(x)=2^x. Evaluating an exponential function is the same as other functions, meaning that we can simply pick values for x, plug them in for x, and simplify to solve. Exponential functions tend to start very small and then grow very quickly.-For example, we can look at g(x)=2^x in FIgure 1 more closely. First, we can make a table....From the table given, you can see that the further negative the x values get, the closer the function approaches zero. Additionally, even though the x values only increase by increments of one, the function increases much more rapidly.

-After viewing the first table, one can see that this function only approaches 0 and is never negative. This is not always the case; there are ways to translate the functions so that they can denote negative values. In order to translate the functions they must move down the y-axis vertically, you must either add or subtract values

notin the exponent. To translate the function horizontally, the added or subtracted value will be in the exponent with the variable. For example, if you were to make the change the first function to h(x)=(2^X)-2, the new table would look just as Figure 2 does.-As many functions work, there can also be horizontal translations. For horizontal translations, the original function could be changed to k(x)=2^(x+2). Since the addition of 2 is in the parenthesis with the x in the exponent, it therefore directly applies to the x rather than the y, and translates horizontally. Therefore, the new table would look like Figure 3.

Back to Top of Page

## Graphs

From these tables, if you were to infer how they would be represented graphically, you could imagine they'd look something like this:*(Figure 1a directly correlates to the function and table of Figure 1, as does Figure 2a to Figure 2 and Figure 3a to Figure 3)

Please note that due to the translations, each graph has a different scale, too. The first graph has increments of 1 while the second two increase by 2's. Also, in order to have the best view of each graph, they were manipulated within the viewing box to best fit the function.

Back to Top of Page

## Asymptotes

-A quality of these functions that you may not be confidently accustomed to is that they do not have

rangesfrom negative infinity to positive infinity. As you can see there is always a horizontal asymptote. In the original function, the asymptote is at y=0. This makes sense because a positive number raise to ANY number (even negative numbers or fractions) will still get you positive numbers, no matter how close it gets to 0.-In Figure 2a, the function was translated vertically, so the horizontal asymptote is now at y=-2. Then, in Figure 3a, the horizontal asymptote is still at y=0 since the vertical translation was not affected.

-According to Wolfram|Alpha, an asymptote is a line or curve that the function approaches but never crosses. An asymptote can be vertical, horizontal, or diagonal along any linear function, but right now we are only dealing with horizontal asymptotes.

You can also visit Purple Math for more information regarding this example.

Back to Top of Page

## Logarithms

A logarithm is another type of exponential function. Many times, the use of a "log" is to manipulate the variable and get it out of the exponent. This is possible because the logarithm is the inverse of exponential function. By applying the log, one is able to work with the variable and also solve for it much easier. Since the log is the inverse of the exponential functions we looked at before, this is an image of how the two functions are graphed side by side:Extended Practice).Back to Top of Page

Michigan Standards Related to Exponential Functions:Common Core Standards - http://www.corestandards.org/Math/Content/HSF/LE/

Standard A3: Families of FunctionsMichigan Merit Curriculum Standards:the graphs of exponential and logarithmic

functions.

logarithmic functions.

Back to Top of Page

## Real World Applications (and Extended Practice)

It is important to not only know how to solve exponential and logarithmic functions but also be able to apply the knowledge learned to real world examples. The following site does a great job of giving examples that use exponential functions and logarithms to solve problems that incorporate interest rates, mortgage, population, radioactive decay and earthquakes. Also, it gives step by step solutions to each of the given problems along with more examples that you can try on your own. Using real world applications makes a difficult topic such as this one more interesting and gives a reason to understanding exponential functions and logarithms.Exponential Word Problem Practice

Logarithmic Problem Practice

Teaching Resource for LessonExponential OutbreaksThis article from the New York Times displays a lesson plan involving finding an exponential equation for modeling the growth of the Ebola disease following its outbreak. Students are challenged to relate their instrumental learning about exponential equations to form relational understanding on the spread of disease and other phenomena. The activity challenges students to recognize that rumors spread further the longer they have been around per day. The concept of exponential growth is clearly displayed as students model the spread of Ebola with equations.Back to Top of Page

## Helpful Videos and Images:

This video from ThinkWell goes over the same example that has been shown above: f(x)=2^x. The professor in this video does a great job of explaining how to set up a table, plot points, and graph this exponential function. He also compares f(x)=2^x to other functions such as g(x)=3^x. He discusses the concept of asymptotes and looks at patterns found in the functions he graphs. He then relates these patterns to the function h(x)=(1/2)^x and shows how h(x) is similar to f(x). (See the ThinkWell webpage or YouTube page for more tutorials from this company.)

Exponential Function Concept Map

Place Link HereThis video is based on an activity on eHow, "Cake Cutting". The video was shot by Jerry Mabrito and he was assisted by his three sons, Arnold, Lance, & JJ. What can you get from the video? A demonstration of how cake can make things fun, even exponents!

Back to Top of Page

## Citations for Pictures

http://zonalandeducation.com/ - Picture provided twicehttp://www.flickr.com/photos/ - Provided by k0re

Figures 1, 2, 3, 1a, 2a, 3a - Created and Provided by Amy Burke

Concept Map - Created and Provided by Amy Burke and Emily Groenink

Back to Top of Page