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Chapter 3: Exponential and Logarithmic Functions

Chapter 3: Exponential and Logarithmic Functions. Bellwork. Name the following types of functions: 1.Y=(4x+2)/(2-x²) 2.Y=x²-4x+10 3.Y=5cos (3x) + 7 4.Y=9ˣ-8 5.Y=5x – 2 ***Quadratic, trigonometric, linear, exponential, or rational***.

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Chapter 3: Exponential and Logarithmic Functions

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  1. Chapter 3: Exponential and Logarithmic Functions

  2. Bellwork • Name the following types of functions: 1.Y=(4x+2)/(2-x²) 2.Y=x²-4x+10 3.Y=5cos (3x) + 7 4.Y=9ˣ-8 5.Y=5x – 2 ***Quadratic, trigonometric, linear, exponential, or rational***

  3. Exploration: Exponential FunctionsCLE 3126.3.4 Identify or analyze the distinguishing properties of exponential functions from tables, graphs, and equations. F(x)= aˣ • Let a = 1 The graph is _____________ • Let a<0 What happens when x is even? Odd? • Let a > 1 What does the graph look like? What do you notice about the table? • Let 0<a<1 What does the graph look like? What do you notice about the table?

  4. Exponential Growth and Decay

  5. Explore Further • F(x)=3ˣ • Describe the graph. Include any special point(s) on the graph. • Compare G(x) = 15ˣ and H(x) = 2ˣ to the above graph. • Discuss domain and range of each.

  6. Let’s Move • Given: f(x)=2ˣ • Change the equation so that the new graph will reflect f(x) about the y-axis. • How does this relate to our knowledge of the graph when 0<a<1? • Change the equation so that the graph shifts up 3. Compare the graphs and tables. • Change the equation so that the graph shifts left 5. Compare the graphs and tables.

  7. CLE 3126.3.4 Identify or analyze the distinguishing properties of exponential functions from tables, graphs, and equations. • F(x) = C(aˣ) • F(x)=3ˣ What is the value of C? • Where is the value of c graphically? • Change C to explore its purpose. Compare graphically and numerically(table) • Positive values • Negative values

  8. Journal • Given: F(x)=C(aˣˉ ͪ)+k Explain how a, C, h, and k work together to create different exponential graphs.

  9. Practice • Sketch f(x)=3(2ˣˉ²)+1 • Check with partner and calculator. • CLE 3126.3.4 Identify or analyze the distinguishing properties of exponential functions from tables, graphs, and equations.

  10. 3126.3.16 Solve real world problems that can be modeled using quadratic, exponential, or logarithmic functions (by hand and with appropriate technology). • Formula for compound interest: • bi-annually – interest every two years (n = .5) – quite rare • annually – interest once a year (n = 1) – somewhat rare • semi-annually - interest twice a year (n = 2) – somewhat rare • quarterly – interest every 3 months (n = 4) – somewhat common • monthly - interest every month (n = 12) – some banks do it • weekly – interest every week (n = 52) – quite rare • daily – interest every day (n = 365) – most banks do it

  11. Compound Interest • Jack invested $5000 in a long term Certificate of Deposit 50 years ago. The CD earns 4.5% interest compounded monthly. How much is in the CD today? • How much more money would Jack have earned if the CD had compounded daily?

  12. Compound Continuously Using the (1+r/n)^n part of the compound interest formula, let r=1(100%). Put the expression into y1. Look at the table. For larger and larger values of n to what value does the expression tend?

  13. The natural number: e • Compound continuously: A=Pe ͬ ͭ • Twenty-two years ago, I bought a CD that has been earning 4.2% interest compounding continuously. If it is now worth $7, 350, how much was it worth when I bought it?

  14. Match the Exponential Graph • http://www.teachmaths-inthinking.co.uk/exponential-graphs/exponential-games.htm

  15. Practice-Bellwork • Work p. 189-190 #33-47 odd (Individually)

  16. Vocabulary • Exponential Growth-development at an increasingly rapid rate in proportion to the growing total number or size; a constant rate of growth applied to a continuously growing base over a period of time • Exponential Decay-particular form of a very rapid decrease in some quantity

  17. Solve Exponential Problems • 3126.3.16 Solve real world problems that can be modeled using exponential or logarithmic functions (by hand and with appropriate technology).

  18. Half Life • I lost 4 grams of Strontium-90 in my garage. Strontium-90 has a half-life of 28.8 years. Write a formula for the function f(x) that expresses how much Strontium-90 is left after t years have gone by. • How much Strontium-90 will be left when my house is bulldozed to make way for some gaudy monstrosity in 70 years?

  19. Doubling Time • The approximate number of fruit flies in an experimental population after t hours is given by Q(t)=20e^(.03t) • What is the initial population of fruit flies? • What is the doubling time for the fruit fly population? • How large is the population of fruit flies after 72 hours?

  20. Journal • Explain half life and doubling time in your own words. • Edmodo group-AH Precalculus wh6a40

  21. Bellwork: Get out homework • You bought a car for $25,000. The average depreciation rate is 15% per year. Why is the equation for the value of the car after t years ………C(t)=25000(.85)^t • A certain population (20 units) grows at rate of 25% per minute. What is the equation for the population after t minutes?

  22. 3.2 Logarithmic Functions and Their Graphs CLE 3126.3.4 Identify or analyze the distinguishing properties of logarithmic functions from tables, graphs, and equations. Exponential Form: y=a^x Logarithmic Form: Restrictions on a?

  23. Compare Numerically and Graphically • Y= 2 ͯ • Y=log₂x • Sketch F(x)=log₆x without the calculator Compare the two logarithms on the calculator. What changes should be made to the equation to shift F(x) to the right?left? What changes should be made to F(x) to shift the equation up? Down?

  24. Sketch and discuss domain, VA, and x intercept • Y = 1+ log₂(x-3)

  25. Properties of Logarithms • logₐ1= • logₐa= • logₐa ͯ= • a^logₐx= • If logₐx=logₐy, then x=

  26. CLE 3126.3.4 Identify or analyze the distinguishing properties of logarithmic functions from tables, graphs, and equations. • What is the inverse of y=e ͯ? Properties of Natural Logarithms • ln 1= • ln e= • ln e ͯ= • e^ln x= • If ln x=ln y, then x=

  27. Practice 1.Write as a logarithm: 9^(3/2)=27 2. Write as an exponential: log₆36=2 3. Solve for x. log₂x=log₂15

  28. Sketch: Specify the domain, range, x-intercept, and any asymptotes • Y= -2ln(x+1) • Y= 2- log₂(x-5)

  29. Quick way to find Domain of a Logarithmic Function • Y=log(x+7) • Y=log(-x+3) • Y=ln(2x-4)

  30. 3.3 Log Properties • 3126.3.11 Prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems. • Change of base Formula logₐb =log (b)/log (a) or logₐb=ln (b)/ln (a) PROVE IT!

  31. Properties of Logarithms How do these properties relate to the exponential properties?

  32. Practice the Properties3126.2.18 Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.

  33. Journal : Describe the error • 3126.2.18 Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.

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