1 / 79

Chapter 2 – Functions

Chapter 2 – Functions. 18 Days. Table of Contents. 2.1 Definition of a Function 2.2 Graphs of Functions 2.3 Quadratic Functions 2.4 Operations on Functions 2.5 Inverse Functions 2.6 Variation. 2.1 Definition of a Function. Two Days. Functions*. For the function Find f(a) Find f(a-1)

cecilia-tucker
Download Presentation

Chapter 2 – Functions

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. Chapter 2 – Functions 18 Days

  2. Table of Contents • 2.1 Definition of a Function • 2.2 Graphs of Functions • 2.3 Quadratic Functions • 2.4 Operations on Functions • 2.5 Inverse Functions • 2.6 Variation

  3. 2.1 Definition of a Function Two Days

  4. Functions* • For the function • Find f(a) • Find f(a-1) • Find • Find

  5. Homework p 148 (# 5,8,14-28 even, 45,47,49,52,57)

  6. Day 2 Def Function, Domain, Range, Increasing/Decreasing, Vert Line Test, Def Linear Function Evaluating (p148 #13)

  7. Definition of a Function • A function F from a set D to a set E is a correspondence that assigns each element x of D to exactly one element y in E.

  8. Domain and Range • Domain – The set D is the domain of the function. Domain is the set of all possible inputs. • Range – The set E is the range of the function. Range is the set of all possible outputs. • The element y in E is the value of f at x also called the image of x under f.

  9. Function Mapping • We say that f maps D into E. • Two functions f and g are equal if and only if f(x) = g(x) for all x in D.

  10. Graphs of Functions • The graph of a function f is the graph of the equation y = f(x) for all x in the domain of f. • The vertical line test can be used to determine if a graph represents a function. • What does the vertical line test represent in terms of a function mapping?

  11. Increasing and Decreasing Functions -f is increasing when f(a)<f(b) and a<b. -f is decreasing when f(b)>f(c) and b<c. -f is constant when f(x)=f(y) for all x and y.

  12. Evaluating Functions • Given • Determine the domain of g. • Evaluate g(-3) • Evaluate

  13. Sketching Functions • What is the difference between sketching and graphing a function? • Why would we sketch a function as opposed to graph a function?

  14. Sketching Functions • Sketch the following functions and determine the domain, range, and intervals of decreasing, increasing, and constant value:

  15. Finding Linear Functions • We can find linear functions in the same way that we find the equation of a line. • If f is a linear function such that f(-3)=6 and f(2)=-12, find f(x) where x is any real number.

  16. Applications Problems • Pg 150 #57, 59

  17. Homework p 148 (# 15,32,34,35,46,48,50,53,54,60,63,65)

  18. 2.2 Graphs of Functions Four Days

  19. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  20. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  21. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  22. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  23. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  24. Parent Functions -Name of Family -Parent Equation -General Equation -Locator Point -Domain -Range

  25. Graph Shifting and Reflections • Parent: • Shift up k units: • Shift down k units: • Shift right h units: • Shift left h units • Combined Shift: • (right h units, up k units)

  26. Graph Shifting and Reflections • Parent: • Reflection in x-axis: • Vertical Stretch a>1 • Vertical Shrink 0<a<1 • Horizontal Stretch 0<c<1 : • Horizontal Compression c>1: • Combined Transformation:

  27. Graph Shifting and Reflections • Graph the following using translations:

  28. Homework Shifts and Reflections WS

  29. Day 2 – Even and Odd functions. Vertical and Horizontal stretching and compressing of graphs.

  30. Even and Odd Functions • f is an even function if f(-x)=f(x) for all x in the domain. • Even functions have symmetry with respect to the y-axis. • Ex: • f is an odd function if f(-x)=-f(x) for all x in the domain. • Odd functions have symmetry with respect to the origin. • Ex:

  31. Family Functions and Shifts • A parent function is the simplest function in a family of certain characteristics. • A translation shifts the graph horizontally, vertically, or both. Resulting in a graph of the same shape in a different location. • A reflection over the x-axis changes y-values to their opposites.

  32. Family Functions and Shifts • A vertical stretch multiplies all y-values by the same factor greater than 1. • A vertical shrink reduces all y-values by the same factor between 0 and 1. • Each member of a family of functions is a transformation, or change, of the parent function. • A horizontal compression divides all x-values by the same factor greater than 1. • A horizontal stretch divides all x-values by the same factor between 0 and 1.

  33. Graph Shifting and Reflections • Parent: • Shift up k units: • Shift down k units: • Shift right h units: • Shift left h units • Combined Shift: • (right h units, up k units)

  34. Graph Shifting and Reflections • Parent: • Reflection in x-axis: • Vertical Stretch a>1 • Vertical Shrink 0<a<1 • Horizontal Stretch 0<c<1 : • Horizontal Compression c>1: • Combined Transformation:

  35. Homework pg 164 (# 2,3,5,7,8,13,15,17,20,31-36,39 a-f, 41,42,45)

  36. Day 3 – Piecewise functions and questions from the previous 2 days. Application of Piecewise functions (pg 168 #66)

  37. Piecewise Functions • Piecewise functions are defined by more than one expression over different intervals. • Absolute Value is actually a piecewise defined function.

  38. Piecewise Functions • Lets graph the following piecewise defined function.

  39. Piecewise Functions • Lets graph the following piecewise defined function.

  40. Applications of Piecewise Functions • An electric company charges its customers $0.0577 per kWh for the first 1000kWh, $0.0532 for the next 4000kWh, and $0.0511 for any over 5000kWh. Write a piecewise defined function C for a customer’s bill of x kWhs. • How much will a customer’s bill be if they used 4300kWh of electricity?

  41. Homework pg 167 (# 47-50,53,54,55,56,63-65)

  42. Day 4 – Graphing Piecewise functions WS. Working day for students.

  43. Homework Graphing Piecewise Functions WS

  44. 2.3 Quadratic Functions Two Days

  45. Day 1 – Standard form of a quadratic. Vertex form of a quadratic. Completing the square. Finding x and y intercepts.

  46. Quadratic Functions • Standard form of a Quadratic: • Vertex form of a Quadratic:

  47. Competing the Square

  48. Finding x and y intercepts • To find the x-intercept, set y=0. Solve for x. • To find the y-intercept, set x=0. Solve for y. • Find the x and y intercepts of the following:

  49. Homework Vertex and Intercepts WS

  50. Day 2 – Vertex formula and Theorem on Max/Min Values.

More Related