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Warm Up Day 10 (8-21-09)

Warm Up Day 10 (8-21-09). Informal Algebra II Day 10 (8-21-09). Objective: 1. Identify Domain and Range 2. Know and use the Cartesian Plane 3. Graph equations using a chart 4. Determine if a Relation is a Function 5. Use the Vertical Line Test for Functions. Relations.

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Warm Up Day 10 (8-21-09)

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  1. Warm UpDay 10 (8-21-09)

  2. Informal Algebra IIDay 10 (8-21-09) Objective: 1. Identify Domain and Range 2. Know and use the Cartesian Plane 3. Graph equations using a chart 4. Determine if a Relation is a Function 5. Use the Vertical Line Test for Functions

  3. Relations • A relation is a mapping, or pairing, of input values with output values. • The set of input values is called the domain. • The set of output values is called the range.

  4. Domain is the set of all x values. Range is the set of all y values. Domain & Range {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Example 1: Domain- Range- D: {1, 2} R: {1, 2, 3}

  5. Example 2: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e} Range: {1, 2, 3} Page 107

  6. 3.2 Graphs

  7. Cartesian Coordinate System • Cartesian coordinate plane • x-axis • y-axis • origin • quadrants Page 110

  8. A Relation can be represented by a set of orderedpairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X<0, y>0 Origin (0,0) Quadrant IV X>0, y<0 Quadrant III X<0, y<0

  9. (4,3) (3,-4) (-3,5) (-4,-2) Plot:

  10. Every equation has solution points(points which satisfy the equation). 3x + y = 5 (0, 5), (1, 2), (2, -1), (3, -4) Some solution points: Most equations haveinfinitely manysolution points. Page 111

  11. Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4)and(7, 5);y = 3x -1 The collection of all solution points is the graph of the equation.

  12. Ex4 . Graph y = 3x – 1. x 3x-1 y Page 112

  13. Ex 5. Graph y = x² - 5 x x² - 5 y -3 -2 -1 0 1 2 3

  14. What are your questions?

  15. 3.3 Functions • A relation as a function provided there is exactly one output for each input. • It is NOT a function if at least one input has more than one output Page 116

  16. In order for a relationship to be a function… Functions EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT (DOMAIN) FUNCTIONMACHINE (RANGE) OUTPUT

  17. Example 6 Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

  18. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

  19. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

  20. Look at example 1 on page 116 • Do “Try This” a at the bottom of page 116

  21. Is this a function? 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)}

  22. The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

  23. Use the vertical line test to visually check if the relation is a function. (4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.

  24. Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line

  25. Examples • I’m going to show you a series of graphs. • Determine whether or not these graphs are functions. • You do not need to draw the graphs in your notes.

  26. YES! Function? #1

  27. YES! Function? #2

  28. Function? #3 NO!

  29. YES! Function? #4

  30. Function? #5 NO!

  31. YES! Function? #6

  32. Function? #7 NO!

  33. Function? #8 NO!

  34. YES! #9 Function?

  35. Function? YES! #10

  36. Function? #11 NO!

  37. YES! Function? #12

  38. Function Notation “f of x” Input = x Output = f(x) = y

  39. x y x f(x) Before… Now… y = 6 – 3x f(x) = 6 – 3x -2 -1 0 1 2 12 -2 -1 0 1 2 12 (x, f(x)) (x, y) 9 9 6 6 3 3 0 0 (input, output)

  40. Example 7 Find g(2) and g(5). g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)} 2 g(2) = g(5) = 3

  41. Example 8 Consider the functionh= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)} Find h(9), h(6), and h(0).

  42. Example 9. f(x) = 2x2 – 3 Find f(0), f(-3), f(5a).

  43. Example 10. F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a2 + 1

  44. Domain The set of all real numbers that you can plug into the function. D: {-3, -1, 0, 2, 4}

  45. What is the domain? g(x) = -3x2 + 4x + 5 Ex. D: all real numbers x + 3  0 Ex. x  -3 D: All real numbers except -3

  46. 1 = h ( x ) - x 5 1 = f ( x ) + x 2 What is the domain? x - 5  0 Ex. D: All real numbers except 5 Ex. x + 2 0 D: All Real Numbers except -2

  47. What are your questions?

  48. Homework • Page 108 14-20 even • Page 114 4-32 (multiples of 4)(omit #8) • Page 119 1-12 (yes or no) 14-28 even

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