1 / 58

Intermediate Algebra Chapter 9

Intermediate Algebra Chapter 9. Exponential and Logarithmic Functions. Intermediate Algebra 9.1-9.2. Review of Functions. Def: Relation. A relation is a set of ordered pairs. Designated by: Listing Graphs Tables Algebraic equation Picture Sentence. Def: Function.

chapa
Download Presentation

Intermediate Algebra Chapter 9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Intermediate AlgebraChapter 9 • Exponential • and • Logarithmic Functions

  2. Intermediate Algebra 9.1-9.2 • Review of Functions

  3. Def: Relation • A relation is a set of ordered pairs. • Designated by: • Listing • Graphs • Tables • Algebraic equation • Picture • Sentence

  4. Def: Function • A function is a set of ordered pairs in which no two different ordered pairs have the same first component. • Vertical line test – used to determine whether a graph represents a function.

  5. Defs: domain and range • Domain: The set of first components of a relation. • Range: The set of second components of a relation

  6. Examples of Relations:

  7. Objectives • Determine the domain, range of relations. • Determine if relation is a function.

  8. Intermediate Algebra 9.2 • Inverse Functions

  9. Inverse of a function • The inverse of a function is determined by interchanging the domain and the range of the original function. • The inverse of a function is not necessarily a function. • Designated by • and read f inverse

  10. One-to-One function • Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.

  11. Horizontal Line Test • A function is a one-to-one function if and only if no horizontal line intersects the graph of the function at more than one point.

  12. Inverse of a function

  13. Inverse of function

  14. Objectives: • Determine the inverse of a function whose ordered pairs are listed. • Determine if a function is one to one.

  15. Intermediate Algebra 9.3 • Exponential Functions

  16. Michael Crichton – The Andromeda Strain (1971) • The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. It this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”

  17. Definition of Exponential Function • If b>0 and b not equal to 1 and x is any real number, an exponential function is written as

  18. Graphs-Determine domain, range, function, 1-1, x intercepts, y intercepts, asymptotes

  19. Graphs-Determine domain, range, function, 1-1, x intercepts, y intercepts, asymptotes

  20. Growth and Decay • Growth: if b > 1 • Decay: if 0 < b < 1

  21. Properties of graphs of exponential functions • Function and 1 to 1 • y intercept is (0,1) and no x intercept(s) • Domain is all real numbers • Range is {y|y>0} • Graph approaches but does not touch x axis – x axis is asymptote • Growth or decay determined by base

  22. Natural Base e

  23. Calculator Keys • Second function of divide • Second function of LN (left side)

  24. Property of equivalent exponents • For b>0 and b not equal to 1

  25. Compound Interest • A= amount P = Principal t = time • r = rate per year • n = number of times compounded

  26. Compound interest problem • Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years.

  27. Objectives: • Determine and graph exponential functions. • Use the natural base e • Use the compound interest formula.

  28. Dwight Eisenhower – American President • “Pessimism never won any battle.”

  29. Intermediate Algebra 9.4,9.5,9.6 • Logarithmic Functions

  30. Definition: Logarithmic Function • For x > 0, b > 0 and b not equal to 1 toe logarithm of x with base b is defined by the following:

  31. Properties of Logarithmic Function • Domain:{x|x>0} • Range: all real numbers • x intercept: (1,0) • No y intercept • Approaches y axis as vertical asymptote • Base determines shape.

  32. Shape of logarithmic graphs • For b > 1, the graph rises from left to right. • For 0 < b < 1, the graphs falls from left to right.

  33. Common Logarithmic Function The logarithmic function with base 10

  34. Natural logarithmic functionThe logarithmic function with a base of e

  35. Calculator Keys • [LOG] • [LN]

  36. Objective: • Determine the common log or natural log of any number in the domain of the logarithmic function.

  37. Change of Base Formula • For x > 0 for any positive bases a and b

  38. Problem: change of base

  39. Objective • Use the change of base formula to determine an approximation to the logarithm of a number when the base is not 10 or e.

  40. Intermediate Algebra 10.5 • Properties • of • Logarithms

  41. Basic Properties of logarithms

  42. For x>0, y>0, b>0 and b not 1Product rule of Logarithms

  43. For x>0, y>0, b>0 and b not 1Quotient rule for Logarithms

  44. For x>0, y>0, b>0 and b not 1Power rule for Logarithms

  45. Objectives: • Apply the product, quotient, and power properties of logarithms. • Combine and Expand logarithmic expressions

  46. Theorems summary Logarithms:

  47. Norman Vincent Peale • “Believe it is possible to solve your problem. Tremendous things happen to the believer. So believe the answer will come. It will.”

  48. Intermediate Algebra 9.7 • Exponential • and • Logarithmic • Equations

  49. Objective: • Solve equations that have variables as exponents.

  50. Exponential equation

More Related