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# Storyboard math

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• Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
• Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
• Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
• This is a piecewise function
• D: all reals R: [0, 1] Another cool function: y = sin(abs(x)) Y = sin(x) * abs(x)
• Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)
• D: [-3, -1) U (-1, 3] R: {-1, 1}
• D: [-3, -1) U (-1, 3] R: {-1, 1}
• ### Storyboard math

1. 1. Warm Up Activity
2. 2. Relation and Function Objective: 1. Identify Domain and Range 2. Use the Cartesian Plane in plotting points 3. Graph equations using a chart 4. Determine if a Relation is a Function 5. Use the Vertical Line Test for Functions
3. 3. Relations <ul><li>A relation is a mapping, or pairing, of input values with output values. </li></ul><ul><li>The set of input values is called the domain . </li></ul><ul><li>The set of output values is called the range . </li></ul>
4. 4. Domain & Range <ul><li>Domain is the set of all x values. </li></ul><ul><li>Range is the set of all </li></ul><ul><li>y values. </li></ul>Example 1: Domain- D: {1, 2} Range- R: {1, 2, 3} {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
5. 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}
6. 6. Can you give example/s of relation you use or experience daily?
7. 7. Graphs
8. 8. Cartesian Coordinate System <ul><li>Cartesian coordinate plane </li></ul><ul><li>x-axis </li></ul><ul><li>y-axis </li></ul><ul><li>origin </li></ul><ul><li>quadrants </li></ul>
9. 9. A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X<0, y>0 Quadrant III X<0, y<0 Quadrant IV X>0, y<0 Origin (0,0)
10. 10. Plot: (-3,5) (-4,-2) (4,3) (3,-4)
11. 11. 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 have infinitely many solution points.
12. 12. 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.
13. 13. Ex4 . Graph y = 3x – 1. x 3x-1 y
14. 14. Ex 5. Graph y = x ² - 5 x x ² - 5 y -3 -2 -1 0 1 2 3
15. 15. What are your questions?
16. 16. Functions <ul><li>A relation as a function provided there is exactly one output for each input. </li></ul><ul><li>It is NOT a function if at least one input has more than one output </li></ul>
17. 17. Functions INPUT (DOMAIN) OUTPUT (RANGE) FUNCTION MACHINE In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT
18. 18. Example 6 <ul><li>No two ordered pairs can have the same first coordinate </li></ul><ul><li>(and different second coordinates). </li></ul>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)}
19. 19. Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output
20. 20. Input Output -3 3 1 -2 4 1 4 Identify the Domain and Range. Then tell if the relation is a function. Domain = {-3, 1,4} Range = {3,-2,1,4} Function? No: input 1 is mapped onto Both -2 & 1 Notice the set notation!!!
21. 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. 22. The Vertical Line Test <ul><li>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. </li></ul>
23. 23. (-3,3) (4,4) (1,1) (1,-2) Use the vertical line test to visually check if the relation is a function. Function? No, Two points are on The same vertical line.
24. 24. (-3,3) (4,-2) (1,1) (3,1) Use the vertical line test to visually check if the relation is a function. Function? Yes, no two points are on the same vertical line
25. 25. Examples <ul><li>I’m going to show you a series of graphs. </li></ul><ul><li>Determine whether or not these graphs are functions. </li></ul><ul><li>You do not need to draw the graphs in your notes. </li></ul>
26. 26. #1 Function? YES!
27. 27. Function? #2 YES!
28. 28. Function? #3 NO!
29. 29. Function? #4 YES!
30. 30. Function? #5 NO!
31. 31. #6 Function? YES!
32. 32. Function? #7 NO!
33. 33. Function? #8 NO!
34. 34. #9 Function? YES!
35. 35. Function? #10 YES!
36. 36. Function? #11 NO!
37. 37. Function? #12 YES!
38. 38. Function Notation “ f of x” Input = x Output = f(x) = y
39. 39. y = 6 – 3x -2 -1 0 1 2 12 9 6 0 3 f(x) = 6 – 3x -2 -1 0 1 2 12 9 6 0 3 Before… Now… (x, y) (input, output) (x, f(x)) x y x f(x)
40. 40. Find g (2) and g (5). g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)} g(2) = 3 g(5) = 2 Example 7
41. 41. Consider the function h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)} Example 8 Find h(9), h(6), and h(0).
42. 42. Example 9 f(x) = 2x 2 – 3 Find f(0), f(-3), f(5a).
43. 43. F(x) = 3x 2 +1 Example 10 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a 2 + 1
44. 44. Domain The set of all real numbers that you can plug into the function. D: {-3, -1, 0, 2, 4}
45. 45. What is the domain? g(x) = -3x 2 + 4x + 5 D: all real numbers Ex. Ex . x + 3  0 x  -3 D: All real numbers except -3
46. 46. What is the domain? x - 5  0 Ex. D: All real numbers except 5 D: All Real Numbers except -2 Ex. x + 2  0 h x x ( )   1 5 f x x ( )   1 2
47. 47. What are your questions?