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Functions

Functions. Exponentials and Logarithms. Overview. In this module you will be learning about two classes of functions called exponentials and logarithms .

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Functions

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  1. Functions Exponentials and Logarithms

  2. Overview In this module you will be learning about two classes of functions called exponentials andlogarithms. Exponential functions have the independent variable (or input) as the exponent. For example, y = 2x. Note that this is type of function is different than power functions where the input is the base, such as y = x2. Many real-world applications use exponential functions ranging from finance to biology. Logarithmic functionsare inverse functions of the exponential, that is, they help us to find values of an unknown exponent, such as by solving 2x = 10. Using a logarithm, we can determine this mysterious input. Further, logarithms have a core set of algebraic properties that can help in simplifying an expression.

  3. Exponential Topics • Exponents and Laws of Exponents • Fractional Exponents • Solving Problems Involving Exponents • Parent Graphs of Exponential Functions • Transformations of Exponential Functions • Exponential Growth and Decay • Applications of Exponential Functions • The Special Number e • Solving Exponential Equations Algebraically • Solving Exponential Equations Graphically

  4. Logarithm Topics • Inverse Functions (review) • Defining the Inverse of an Exponential Function • Finding the Inverse of an Exponential Function • Two Special Logs • History of the Logarithm • Evaluating Logarithmic Expressions • Graphs of Logarithmic Functions • Properties of Logarithms • Curve Fitting • Solving Logarithmic Equations Algebraically • Solving Logarithmic Equations Graphically

  5. Exponentials Exponents and Laws of Exponents • Big Ideas: Before studying exponential functions it is important to have grasp of exponents and the laws that govern them. Exponents are nothing more than a way to express repeated multiplication. • xmxn=x(m+n) • xm/xn=x(m-n) • (xm)n=xmn • x0=1 • x-n=1/xn • (xy)n=xnyn and (x/y)n=xn/yn

  6. Exponentials Exponents and Laws of Exponents • Virtual Nerd - What are exponents? • http://www.virtualnerd.com/algebra-1/algebra-foundations/exponents-definition.php • Video from Virtual Nerd describing the meaning of exponents.

  7. Exponentials Exponents and Laws of Exponents • Khan Academy - Understanding Exponents • Watch these videos in order. • Khan Academy Video: Understanding Exponents • Khan Academy Video: Understanding Exponents 2 • Khan Academy Video: Level 1 Exponents • These videos show the meaning between repeated addition, repeated multiplication and when you should use multiplication rather than an exponent.

  8. Exponentials Exponents and Laws of Exponents • Virtual Nerd - Converting numbers to expanded form. • http://www.virtualnerd.com/pre-algebra/factors-fractions-exponents/exponential-to-expanded-form-conversion.php • Did you know that exponents are just a quick way to show repeated multiplication? In this tutorial, see how to expand out a value in exponential form to see what it really represents!

  9. Exponentials Exponents and Laws of Exponents • Virtual Nerd - Converting numbers from expanded form. • http://www.virtualnerd.com/pre-algebra/factors-fractions-exponents/powers/exponent-examples/expanded-to-exponential-form-conversion • Exponential form is a quick way to show that a number should be multiplied by itself a certain number of times. In this tutorial, see how to write a repeated multiplication in exponential form!

  10. Exponentials Exponents and Laws of Exponents • Khan Academy - Exponent Rules • https://www.khanacademy.org/math/arithmetic/exponents-radicals/exponent-properties/e/exponent_rules • Practice problems from Khan Academy on solving exponent problems by adding, subtracting, or multiplying exponents.

  11. Exponentials Exponents and Laws of Exponents • Illustrative Mathematics - Extending the Definitions of Exponents, Variation 1 • http://www.illustrativemathematics.org/illustration_pdfs/395.pdf • This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. While it may be unfamiliar to some students, it is good for them to learn the convention that negative time is simply any time before t = 0.

  12. Exponentials Exponents and Laws of Exponents • BrainGenie - Simplifying Expressions Using the Power to a Power Property • http://braingenie.ck12.org/skills/105540 • Problems look like (25) x (26) =___. These exercises are simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

  13. Exponentials Exponents and Laws of Exponents • BrainGenie - Simplifying Expressions Using the Power to a Power Property • http://braingenie.ck12.org/skills/105541 • Problems look like (b7)8 =___. These exercises are simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

  14. Exponentials Exponents and Laws of Exponents • BrightStorm - Multiplication and Division Properties of Exponents • http://youtu.be/kETKV0KJlvw • A video on YouTube from Brightstorm that shows how to simplify products and quotients involving exponents. • Starting at 1:30 into the video the teacher shows some easy examples of the properties/laws of exponents using simple numbers that make it easier to understand the rules.

  15. Exponentials Exponents and Laws of Exponents • BrainGenie - Simplifying Expressions Using the Multiple Exponent Properties Involving Products • http://braingenie.ck12.org/skills/105545 • Problems look like (73)6 x 78=___. These exercises are simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

  16. Exponentials Exponents and Laws of Exponents • BrainGenie - Simplifying Expressions Using the Quotient Property of Powers • http://braingenie.ck12.org/skills/105545 • Problems look like (x/y)7. These exercises are simplifying examples with numbers as bases (no variables). Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

  17. Exponentials Exponents and Laws of Exponents • BrainGenie - Simplifying Expressions Using Multiple Exponent Properties Involving the Quotient • http://braingenie.ck12.org/skills/105546 • Problems look like (412/43)2. Make sure you read the explanation if you get any problem wrong. Get 5 correct and then move on.

  18. Exponentials Exponents and Laws of Exponents • Brightstorm - Zero and Negative Exponents • http://youtu.be/pFW3z7EJZso • A video on YouTube from Brightstorm that shows how to simplify products involving zero and negative exponents. • This video is less than 2 minutes. Again, the teacher shows the rules with variables, but then does some simple number examples to explain the rules.

  19. Exponentials Exponents and Laws of Exponents • BrainGenie - Zero Exponents • http://braingenie.ck12.org/skills/105549 • Problems look like 60 =__. Get 5 in a row and move on. A 31 second video is included if you need additional help.

  20. Exponentials Exponents and Laws of Exponents • BrainGenie - Negative Exponents • http://braingenie.ck12.org/skills/105550 • Problems look like 5-2 =__. Get 5 in a row and move on. A 35 second video is included if you need additional help.

  21. Exponentials Exponents and Laws of Exponents • BrainGenie - Evaluating Exponential Expressions Involving the Zero Exponent Property and Others • http://braingenie.ck12.org/skills/105551 • Problems look like 20/72 =__. Get 5 in a row and move on. A 35 second video is included if you need additional help.

  22. Exponentials Exponents and Laws of Exponents • BrainGenie - Evaluating Exponential Expressions Involving the Negative Exponent Property and Others • http://braingenie.ck12.org/skills/105552 • Problems look like 53/5-4 =__. Get 5 in a row and move on. A 1 minute 27 second video is included if you need additional help.

  23. Exponentials Exponents and Laws of Exponents • Properties of Exponents • You’ll need a login to access this site (you can create a free login as a student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome) • http://www.sascurriculumpathways.com/portal/Launch?id=5061 • “You'll learn how to use the properties of exponents to simplify expressions and solve simple exponential equations. The lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.” • When you are doing the self-check problems use ^ when you need to enter an exponent. Pay special attention to the “Stop and Think” audio clips. PDFs at the end are especially good.

  24. Exponentials Exponents and Laws of Exponents • SAS Curriculum Pathways - Geometric Sequences • You’ll need a login to access this site (you can create a free login as a student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome) • http://www.sascurriculumpathways.com/portal/Launch?id=5062 • “You'll learn how to recognize geometric sequences, write a rule for a geometric sequence, and find the nth term in a geometric sequence. The lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.” When you are doing the self-check problems use ^ when you need to enter an exponent. Pay special attention to the “Stop and Think” audio clips.

  25. Exponentials Fractional Exponents • Big Ideas: Before studying exponential functions it is also important to understand that exponents don’t have to be integers (...-2,-1,0,1,2…). Exponents can be fractions or decimals. Fractional exponents are another way of representing square roots, cube roots, 4th roots… nth roots. Another name for roots are radicals. These fractional exponents also follow the laws of exponents mentioned: • xmxn=x(m+n) • xm/xn=x(m-n) • (xm)n=xmn • x0=1 • x-n=1/xn • (xy)n=xnyn and (x/y)n=xn/yn

  26. Exponentials Fractional Exponents • Khan Academy - Basic Fractional Exponents Video • http://youtu.be/lZfXc4nHooo • Pay special attention to the argument that is made around minute 2:00 and how he extends the idea at minute 3:00. If you don’t understand those then the example around 4:20 into the video should help you understand the concept of fractional exponents.

  27. Exponentials Fractional Exponents • Khan Academy - Negative Fractional Exponents • http://youtu.be/tn53EdOr6Rwhttp://youtu.be/bIFdW0NZ9W4 • The first of these videos takes the law for negative exponents and the law for fractional exponents and combines them into one rule. It is only 3 minutes long, watch it all. • The second video is just over 3 minutes. All he does in this video is makes the base of the negative fractional exponent a fraction instead of a whole number.

  28. Exponentials Fractional Exponents • Khan Academy - Fractional Exponents Practice Problem Sets • https://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/e/exponents_3 • Make sure you can do 5 in a row correctly before moving on.

  29. Exponentials Fractional Exponents • Khan Academy - Fractional Exponents with Numerator other than 1 • http://youtu.be/S34NM0Po0eA • This video extends fractional exponents beyond (½), (⅓), (¼) and (⅕) and now allows you to simplify expressions with exponents like (⅔) or (⅗).

  30. Exponentials Fractional Exponents • Khan Academy - Fractional Exponents with Numerator other than 1 Problem Set • https://www.khanacademy.org/math/arithmetic/exponents-radicals/negative-exponents-tutorial/e/exponents_4 • Make sure you can do 5 in a row correctly before moving on.

  31. Exponentials Fractional Exponents • Cool Math - Fractional Exponents • http://www.coolmath.com/prealgebra/14-intro-to-radicals/07-radicals-fractional-exponents-01-87.htm • This resource color codes the fractional exponent and its base and shows how the problems can be translated to radical expressions in order to simplify them. This resources gives you the basic concept and a few practice problems to try on your own.

  32. Exponentials Fractional Exponents • Math Is Fun - Fractional Exponents • http://www.mathsisfun.com/algebra/exponent-fractional.html • This resource is for the person who likes to read a thorough easy to understand explanation of what fractional exponents are in relation to radicals and see how the rules of exponents still apply. There is also a nice applet on this page that the user can manipulate to see how changing the fractional exponent affects the graph of the function.

  33. Exponentials Fractional Exponents • Sophia - Fractional Exponents Tutorial • http://www.sophia.org/fractional-exponents-pathway • Nine different teachers explain several topics related to Fractional Exponents and each topic has a 3 question multiple choice quiz to check for understanding. The topics include: • Numbers raised to a power that is a fraction • The relationship between roots and fractional exponents • Rewriting Fractional Exponents • Converting to Fractional Exponent Form • Multiplication Rule with Fractional Exponents • Division Rule with Fractional Exponents • Power Rule with Fractional Exponents • Just click “Learn this” to begin

  34. Exponentials Fractional Exponents • Wolfram Alpha Can simplify Fractional Expressions • http://www.wolframalpha.com • An online math tool that can give you multiple forms of the same answer. This is really useful if you need to check to see if your answer is the same as another form. You can type in (⅜)^(½) and it will give you a lot of different representations of the simplification of that number. In some instances it can even show you step-by-step directions on how they came up with those representations.

  35. Exponentials Solving Problems Involving Exponents • Big Ideas: All of the exponent rules you know now can be put into context of real-life situations. Quantities that are really small in comparison have negative exponents. Quantities that are large have positive exponents. They can be whole numbers or fractional. Sometimes a problem may require you to compare to of these such numbers together through multiplication or division.

  36. Exponentials Solving Problems Involving Exponents • BrainGenie - Simplifying Expressions Using Multiple Exponent Properties Involving Products • http://braingenie.ck12.org/skills/105553 • Solve at least 5 problems before moving on. If you get stuck watch the 1 minute 31 second video.

  37. Exponentials Solving Problems Involving Exponents • BrainGenie - Solving Word Problems Using Exponent Properties Involving Products • http://braingenie.ck12.org/skills/105544 • Solve at least 5 problems before moving on. If you get stuck watch the 1 minute 11 second video.

  38. Exponentials Solving Problems Involving Exponents • BrainGenie - Solving Word Problems Using Exponent Properties Involving Quotients • http://braingenie.ck12.org/skills/105548 • Solve at least 5 problems before moving on. If you get stuck watch the 1 minute 21 second video.

  39. Exponentials Parent Graphs of Exponential Functions • Big Ideas: Before learning how complex exponential functions can be graphed and manipulated to model a real-world situation one must first conquer the key features of the parent function. The parent function of an exponential is in the form f(x)=abx.

  40. Exponentials Parent Graphs of Exponential Functions • SAS - Linear, Exponential and Quadratic Models • You’ll need a login to access this site (you can create a free login as a student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome) • http://www.sascurriculumpathways.com/portal/Launch?id=6016 • “You'll learn how to compare linear, exponential, and quadratic models; solve a system of linear and exponential equations; and solve a system of linear and quadratic equations. This lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.” You’ll use functions, tables and graphs to understand the difference between these three function families.

  41. Exponentials Parent Graphs of Exponential Functions • SAS Curriculum Pathways - Graphing Exponential Functions • You’ll need a login to access this site (you can create a free login as a student). Plug-ins Required: PDF Viewer, Java (don’t use Chrome) • http://www.sascurriculumpathways.com/portal/Launch?id=5063 • “You'll learn how to graph an exponential function and identify the key features of an exponential function. This lesson provides instructional content with interactive components and guided practice, and additional problems to check your learning.”

  42. Exponentials Parent Graphs of Exponential Functions • Cool Math - Graph exponential function of the form f(x)=abx • http://www.coolmath.com/algebra/Algebra2/06ExponentialsLogs/06_expgraph.htm • Explanation of how to graph an exponential function with different bases.

  43. Exponentials Parent Graphs of Exponential Functions • Virtual Nerd - Graph exponential function of the form f(x)=abx • http://www.virtualnerd.com/algebra-1/exponents-exponential-functions/growth-definition.php • “Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check out these kinds of exponential functions in this tutorial!”

  44. Exponentials Parent Graphs of Exponential Functions • SAS Curriculum Pathways - Graph exponential function of the form f(x)=abx • Exploring Graphs of Special Functions • http://www.sascurriculumpathways.com/portal/Launch?id=1436 • “Use the tool's built-in examples (or create your own) to explore families of special functions—absolute value, radical, rational, and exponential. You'll investigate the relationship between the parameters of an equation of a special function and its graph.”

  45. Exponentials Parent Graphs of Exponential Functions • SAS Curriculum Pathways - Graph exponential function of the form f(x)=abx • Describing the Rate of Change of an Exponential Functions • http://www.sascurriculumpathways.com/portal/Launch?id=95 • “Explore definitions and examples of exponential growth. After reviewing the rate of change of a linear function, investigate the rate of change of an exponential function. You'll summarize what you learn to answer this focus question: How can the rate of change of an exponential function be described?”

  46. Exponentials Parent Graphs of Exponential Functions • Purple Math - Graphing Exponential Functions • http://www.purplemath.com/modules/graphexp.htm • Pay special attention to how to shift a graph left, right, up or down, or how to reflect it over the x-axis or y-axis. The graphs on page 4 are especially useful.

  47. Exponentials Parent Graphs of Exponential Functions • BrainGenie - Graphing Exponential Functions by Making a Table • http://braingenie.ck12.org/skills/106063 • These are practice problems of how to graph an exponential function by using a table of values. The base can be a whole number or a fraction and the exponent is usually a multiple of x. Include a 1 minute 31 second video if needed.

  48. Exponentials Transformations of Exponential Functions • For Dummies - How to Graph and Transform Exponential Functions • http://www.dummies.com/how-to/content/how-to-graph-and-transform-an-exponential-function.html • This is only 1 page long and has a good explanation of how a graph looks with a whole number base or a fractional base. It also has a nice summary of which numbers represent vertical and horizontal shifts and which one is a vertical stretch. Read the entire page.

  49. Exponentials Transformations of Exponential Functions • BrainGenie - Transforming Exponential Functions • http://braingenie.ck12.org/skills/106064 • This is a set of practice problems that allows you to recognize a graph of a transformation of the original exponential function without having to make a table of values.

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