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0207 ch 2 day 7
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Transcript

  • 1. 2.5 Quadratic Functions; Maxima & MinimaPsalm 55:22  Cast your burden on the LORD, and hewill sustain you; he will never permit the righteousto be moved.
  • 2. Standard Form of a quadratic equation
  • 3. Standard Form of a quadratic equation 2 f (x) = ax + bx + c
  • 4. Standard Form of a quadratic equation 2 f (x) = ax + bx + c Vertex Form of a quadratic equation
  • 5. Standard Form of a quadratic equation 2 f (x) = ax + bx + c Vertex Form of a quadratic equation 2 f (x) = a ( x − h ) + k vertex at ( h, k )
  • 6. Standard Form of a quadratic equation 2 f (x) = ax + bx + c Vertex Form of a quadratic equation 2 f (x) = a ( x − h ) + k vertex at ( h, k )Note: your book is wrong in that it names Vertex Form as Standard Form.Ignore your book on this point.
  • 7. (-b,c) and (i,j) are both Local Minimums (f,g) is a Local Maximum
  • 8. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2
  • 9. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Algebraic Approach:
  • 10. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Algebraic Approach: f (x) = 3( x 2 − 2x ) + 10
  • 11. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Algebraic Approach: f (x) = 3( x 2 − 2x ) + 10 f (x) = 3( x − 2x + 1) + 10 − 3 2
  • 12. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Algebraic Approach: f (x) = 3( x 2 − 2x ) + 10 f (x) = 3( x − 2x + 1) + 10 − 3 2 2 f (x) = 3( x − 1) + 7 vertex : (1, 7 )
  • 13. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Algebraic Approach: f (x) = 3( x 2 − 2x ) + 10 f (x) = 3( x − 2x + 1) + 10 − 3 2 2 f (x) = 3( x − 1) + 7 vertex : (1, 7 ) f (0) = 10 y − intercept : ( 0,10 )
  • 14. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Graphic Approach:
  • 15. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Graphic Approach: 2 y1 = 3x − 6x + 10
  • 16. Find the coordinates of the vertex and they-intercept for f (x) = 3x − 6x + 10 2Graphic Approach: 2 y1 = 3x − 6x + 10 Now ... how do we find the x-intercepts? (discuss and do algebraic vs. graphic)
  • 17. Put y = ax + bx + c into vertex form 2
  • 18. Put y = ax + bx + c into vertex form 2 ⎛ 2 b ⎞y = a ⎜ x + x ⎟ + c ⎝ a ⎠
  • 19. Put y = ax + bx + c into vertex form 2 ⎛ 2 b ⎞y = a ⎜ x + x ⎟ + c ⎝ a ⎠ 2 2 ⎛ 2 b b ⎞ aby = a ⎜ x + x + 2 ⎟ + c − 2 ⎝ a 4a ⎠ 4a
  • 20. Put y = ax + bx + c into vertex form 2 ⎛ 2 b ⎞y = a ⎜ x + x ⎟ + c ⎝ a ⎠ 2 2 ⎛ 2 b b ⎞ aby = a ⎜ x + x + 2 ⎟ + c − 2 ⎝ a 4a ⎠ 4a 2 2 ⎛ b ⎞ ⎛ b ⎞y = a ⎜ x + ⎟ + ⎜ c − ⎟ ⎝ 2a ⎠ ⎝ 4a ⎠
  • 21. Put y = ax + bx + c into vertex form 2 ⎛ 2 b ⎞y = a ⎜ x + x ⎟ + c ⎝ a ⎠ 2 2 ⎛ 2 b b ⎞ aby = a ⎜ x + x + 2 ⎟ + c − 2 ⎝ a 4a ⎠ 4a 2 2 ⎛ b ⎞ ⎛ b ⎞y = a ⎜ x + ⎟ + ⎜ c − ⎟ ⎝ 2a ⎠ ⎝ 4a ⎠ 2y = a(x - h) + k
  • 22. Put y = ax + bx + c into vertex form 2 ⎛ 2 b ⎞y = a ⎜ x + x ⎟ + c ⎝ a ⎠ 2 2 ⎛ 2 b b ⎞ aby = a ⎜ x + x + 2 ⎟ + c − 2 ⎝ a 4a ⎠ 4a 2 2 ⎛ b ⎞ ⎛ b ⎞y = a ⎜ x + ⎟ + ⎜ c − ⎟ ⎝ 2a ⎠ ⎝ 4a ⎠ 2 2 ⎛ −b b ⎞y = a(x - h) + k vertex: ⎜ 2a , c − 4a ⎟ ⎝ ⎠
  • 23. ⎛ −b b 2 ⎞If the vertex is then the extrema ⎜ 2a , c − 4a ⎟ ⎝ ⎠point (max or min) occurs at:
  • 24. ⎛ −b b 2 ⎞If the vertex is then the extrema ⎜ 2a , c − 4a ⎟ ⎝ ⎠point (max or min) occurs at: ⎛ −b ⎛ −b ⎞ ⎞ ⎜ 2a , f ⎜ 2a ⎟ ⎟ ⎝ ⎝ ⎠ ⎠
  • 25. ⎛ −b b 2 ⎞If the vertex is then the extrema ⎜ 2a , c − 4a ⎟ ⎝ ⎠point (max or min) occurs at: ⎛ −b ⎛ −b ⎞ ⎞ ⎜ 2a , f ⎜ 2a ⎟ ⎟ ⎝ ⎝ ⎠ ⎠This is called the Vertex Formula forquadratics (not to be confused with theVertex Form of a quadratic).
  • 26. Pages 201, 202: Do # 60, 62, 64 algebraically & verify graphically (do as many as a class as time permits) HW #6“Nothing pains some people more than havingto think.” Martin Luther King Jr.