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Linear and Quadratic Functions

Ms. Anneke Timan. Linear and Quadratic Functions. Wayward Wednesday. How were the sound effects in Star Wars created? . It’s all about wave speeds in metal. Functions: Linear and Quadratic. 1. 2. Quadratic Functions. Linear Functions. 1. Linear Functions. y = mx + b. What is m?

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Linear and Quadratic Functions

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  1. Ms. Anneke Timan Linear and Quadratic Functions

  2. Wayward Wednesday How were the sound effects in Star Wars created? It’s all about wave speeds in metal.

  3. Functions:Linear and Quadratic 1 2 Quadratic Functions Linear Functions

  4. 1 Linear Functions y = mx + b

  5. What is m? -1/3 -3 1/3 3 f(x) = mx + b

  6. What is m? -1/3 -3 1/3 3 f(x) = mx + b run rise

  7. What is b? -2 -6 2 6 f(x) = mx + b

  8. What is b? -2 -6 2 6 f(x) = mx + b

  9. The equation y = mx + b describes a straight line with a slope m and and a y-intercept b. Linear Functions

  10. 2 Quadratic Functions y = ax2 + bx + c

  11. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What function describes Ms. Breimer’s trajectory?

  12. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What function describes Ms. Breimer’s trajectory? y x

  13. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What function describes Ms. Breimer’s trajectory? y y = –x2 + x + 2 x

  14. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is Ms. Breimer’s height when she is 0.5 m from the wall? y y = –x2 + x + 2 x

  15. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is Ms. Breimer’s height when she is 0.5 m from the wall? y y = –x2 + x + 2 y = –(0.5)2 + (0,5) + 2 x

  16. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is Ms. Breimer’s height when she is 0.5 m from the wall? y y = –x2 + x + 2 y = –(0.5)2 + (0,5) + 2 y = 2.25 She is 2.25m above the ground. x

  17. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is the height of our stair when Ms. Breimer is 0 m from the wall? y y = –x2 + x + 2 x

  18. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is the height of our stair when Ms. Breimer is 0 m from the wall? y y = –x2 + x + 2 y = –(0)2 + (0) + 2 x

  19. Ms. Breimer leaps from her stair to the ground during her epic dash to teach you! What is the height of our stair when Ms. Breimer is 0 m from the wall? y y = –x2 + x + 2 y = –(0)2 + (0) + 2 y = 2 The y-intercept is at (0, 2) x

  20. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 x

  21. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 x

  22. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) x

  23. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) To make this whole side of the equation zero, I can either make this factor zero or… x

  24. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) To make this whole side of the equation zero, I can either make this factor zero or… I can make the other factor zero x

  25. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) 0 = (x – 2) x

  26. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) 0 = (x – 2) 2 = x 2 x

  27. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) 0 = (x + 1) 2 x

  28. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) 0 = (x + 1) -1 = x -1 2 x

  29. Where does Ms. Breimer hit the ground? y y = –x2 + x + 2 0 = –x2 + x + 2 0 = –1(x2 – x – 2) 0 = –1(x – 2)(x + 1) The x-intercepts are at (2, 0) and (-1, 0) -1 2 x

  30. Practice in Teams Bryce hits a birdie over the net to Jarrod, and it follows the trajectory: y = –2x2 – 5x + 3 Where y is the height above the ground in meters and x is the distance from net in m (negative on Bryce’s side of the net and positive on Jarrod’s side). • At what height does the birdie go over the net (aka. what is the y-intercept?) • Where would the birdie land on Jarrod’s side unless Jarrod saves it? (aka. what are the x-intercepts?)

  31. Y depends on X Y = dependent variable X = independent variable Example: If you were graphing height and age, which is y and which is x? Y = age, X = height Y = height, X = age I have no idea What makes a function?

  32. Y depends on X Y = dependent variable X = independent variable Example: If you were graphing height and age, which is y and which is x? Y = age, X = height Y = height, X = age I have no idea What makes a function?

  33. Y depends on X Y = dependent variable X = independent variable 2. There is only one y-value for every x-value (though there can be multiple x-values for every y-value) What makes a function?

  34. 2. There is only one y-value for every x-value Example: Is it a function? What makes a function?

  35. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  36. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  37. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  38. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  39. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  40. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  41. 2. There is only one y-value for every x-value Example: Is it a function? Vertical line test: If a vertical line drawn anywhere only passes through once, it is a function. What makes a function?

  42. Domain and Range Domain: the x-values encompassed by the function

  43. Domain and Range Domain: the x-values encompassed by the function –3 < x < 1

  44. Domain and Range Range: the y-values encompassed by the function

  45. Domain and Range Range: the y-values encompassed by the function –7 < y <–3

  46. Just one khan academy activity today Page 196 #3 Page 197 #1, 3, 4 Homework:

  47. y = ax2 + bx + c What is the y-intercept of the curved line? 0.3 - 1.8 - 2 - 1

  48. y = ax2 + bx + c What is the y-intercept? 0.3 - 1.8 - 2 - 1

  49. What are the x-intercepts of f(x) shown on the left? • (2, 0) and (3, 0) • (-2, 0) and (-3, 0) • (5, 0) and (6, 0) • (-5, 0) and (-6, 0) f(x) = x2 + 5x + 6

  50. f(x) = x2 + 5x + 6 f(x) = (x + 2)(x + 3) • What are the x-intercepts? • (2, 0) and (3, 0) • (-2, 0) and (-3, 0) • (5, 0) and (6, 0) • (-5, 0) and (-6, 0)

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