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Objectives: The Learner will…,

10.3 Completing the Square. Objectives: The Learner will…,. Form a perfect-square trinomial from a given quadratic binomial. Write a given quadratic function in vertex form. NCSCOS. 1.01, 4.02. (. ). b. 2. +. 2. 10.3 Completing the Square.

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Objectives: The Learner will…,

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  1. 10.3 Completing the Square Objectives: The Learner will…, • Form a perfect-square trinomial from a given quadratic binomial. • Write a given quadratic function in vertex form. NCSCOS 1.01, 4.02

  2. ( ) b 2 + 2 10.3 Completing the Square Converting to Vertex Form by first Completing the Square. 1) Make a perfect-square trinomial from a binomial. 2 8 + x2 + 8x x2 + bx 2 b2 x2 + bx + x2 + 8x + 16 4 2) Then make a perfect-square binomial. b = (x + )2 (x + 4)2 2

  3. ( ) b 2 x2 + bx + 2 10.3 Completing the Square From a quadratic function, write a PS Binomial. Completing the Square b2 b 2 = x2 + bx + = (x + )2 4 y = x2 + 12x + 36 y = (x + 6)2 64 y = (x + 8)2 y = x2+ 16x + y = (x + 7)2 49 y = x2 + 14x + 9 y = x2+ 6x + y = (x + 3)2

  4. ( ) b 2 x2 + bx + 2 10.3 Completing the Square b2 b 2 = x2 + bx + = (x + )2 4 From a quadratic function, write a PS Binomial. y = x2– 4x + 4 y = (x – 2)2 y = (x – 3)2 y = x2– 6x + 9 y = (x – 4)2 y = x2– 8x + 16 y = (x – 5)2 y = x2– 10x + 25 y = (x – 7)2 y = x2– 14x + 49

  5. 10.3 Completing the Square Rewrite each function in the form of y = a(x – h)2 + k y = (x – 2)2 – 4 y = x2– 4x + 4 – 4 y = (x + 3)2 – 9 y = x2+ 6x + 9 – 9 y = (x – 4)2– 16 y = x2– 8x + 16 – 16 y = x2+ 10x + 25 – 25 y = (x + 5)2 – 25 y = x2– 14x + 49 – 49 y = (x – 7)2 – 49

  6. 10.3 Completing the Square Rewrite each function in the form of y = a(x – h)2 + k y = (x + 2)2 – 4 y = x2+ 4x + 4 – 4 y = (x – 3)2 – 9 y = x2– 6x + 9 – 9 y = (x + 4)2– 16 y = x2+ 8x + 16 – 16 y = x2– 10x + 25 – 25 y = (x – 5)2 – 25 y = x2+ 14x + 49 – 49 y = (x + 7)2 – 49

  7. 10.3 Completing the Square Rewrite each function in the form of y = a(x – h)2 + k y = (x – 6)2 – 36 y = x2– 12x + 36 – 36 y = (x + 6)2 – 36 y = x2+ 12x + 36 – 36 y = (x + 8)2 – 64 y = x2 + 16x + 64 – 64 y = (x – 12)2 – 144 y = x2– 24x + 144 – 144 y = (x – 10)2 –100 y = x2– 20x + 100 – 100

  8. 3. Find . ( ) b 2 2 10.3 Completing the Square Rules and Properties Converting to Vertex Form 1. quadratic function y = x2 + 6x + 5 2. Group x2 and x terms. y = (x2 + 6x) + 5 y =(x2+6x+9)+5–9 4. Add that number inside parenthesis and subtract outside parenthesis. y =(x+3)2+5–9 y = (x + 3)2 – 4 Vertex: (-3, -4) 5. Simplify. (vertex form) -3 AOS =

  9. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 4x + 3 Zeros: y = (x2– 4x) + 3 (x– 2)2– 1 = 0 y = (x2– 4x + 4) + 3 – 4 (x– 2)2 = 1 x – 2 =  1 y = (x– 2)2 + 3 – 4 x – 2 = 1 x – 2 = –1 y = (x– 2)2– 1 Vertex: (2, -1) x = 3 & –1 AOS = 2

  10. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 4x + 7 Zeros: y = (x2– 4x) + 7 (x– 2)2+ 3 = 0 y = (x2– 4x + 4) + 7 – 4 (x– 2)2 = –3 y = (x– 2)2 + 7 – 4 Can’t take the square root of negative number y = (x– 2)2 + 3 reason.., there are NO x-intercepts Vertex: (2, 3) AOS = 2

  11. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2 + 6x + 4 Zeros: y = (x2+ 6x) + 4 (x + 3)2– 5= 0 y = (x2+ 6x + 9) + 4 – 9 (x+ 3)2 = 5 y = (x+3)2+ 4 – 9 x+ 3 =  2.24 y = (x+ 3)2– 5 x + 3 = 2.24 x + 3 = –2.24 (-3, -5) Vertex: x = –0.76 & –5.24 -3 AOS =

  12. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = –8 – 6x + x2 Zeros: y = (x2– 6x)– 8 (x – 3)2– 17= 0 y = (x2– 6x + 9) – 8 – 9 (x – 3)2 = 17 y = (x–3)2– 8 – 9 (x – 3) = ± 4.12 y = (x– 3)2– 17 x – 3 = 4.12 x – 3 = – 4.12 Vertex: (3, -17) AOS = 3 x = 7.12 & –1.12

  13. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 2x + 5 Zeros: y = (x2– 2x) + 5 (x– 1)2 + 4= 0 y = (x2– 2x + 1) + 5 – 1 (x– 1)2 = – 4 y = (x–1)2 + 5 – 1 Can’t take the square root of negative number y = (x– 1)2 + 4 Vertex: (1, 4) reason.., there are NO x-intercepts 1 AOS =

  14. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 12x + 35 Zeros: y = (x2– 12x) + 35 (x – 6)2– 1= 0 y = (x2– 12x + 36) + 35 – 36 (x – 6)2 = 1 y = (x–6)2 + 35 – 36 x – 6 =  1 y = (x– 6)2– 1 x – 6 = 1 x – 6 = –1 Vertex: (6, -1) x = 7 & 5 6 AOS =

  15. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2 + 6x + 11 Zeros: y = (x2 + 6x) + 11 (x+ 3)2 + 2= 0 y = (x2 + 6x + 9) + 11 – 9 (x+ 3)2 = – 2 y = (x + 3)2 + 11 – 9 Can’t take the square root of negative number y = (x + 3)2 + 2 Vertex: (-3, 2) NO x-intercepts -3 AOS =

  16. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2 + 10x + 7 Zeros: y = (x2 + 10x) + 7 (x + 5)2– 18= 0 y = (x2 + 10x + 25) + 7 – 25 (x + 5)2 = 18 x + 5 =  4.24 y = (x + 5)2 + 7 – 25 x + 5 = 4.24 x + 5 = –4.24 y = (x + 5)2– 18 Vertex: (-5, -18) x = –0.76 & –9.24 AOS = -5

  17. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 8x + 7 Zeros: y = (x2– 8x) + 7 (x– 4)2 – 9 = 0 (x– 4)2 = 9 y = (x2– 8x + 16) + 7 – 16 x – 4 =  3 y = (x– 4)2 + 7 – 16 x – 4 = 3 x – 4 = –3 y = (x– 4)2 – 9 Vertex: (4, -9) x = 7 & 1 AOS = 4

  18. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 4x – 21 Zeros: y = (x2– 4x)– 21 (x– 2)2 – 25 = 0 (x– 2)2 = 25 y = (x2– 4x + 4) – 21 – 4 x – 2 =  5 y = (x– 2)2– 21 – 4 x – 2 = 5 x – 2 = –5 y = (x– 2)2 – 25 Vertex: (2, -25) x = 7 & –3 AOS = 2

  19. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2 + 4x – 21 Zeros: y = (x2 + 4x)– 21 (x+ 2)2 – 25 = 0 (x + 2)2 = 25 y = (x2 + 4x + 4) – 21 – 4 x + 2 =  5 y = (x + 2)2– 21 – 4 x + 2 = 5 x + 2 = –5 y = (x + 2)2 – 25 Vertex: (–2, –25) x = 3 & –7 AOS = –2

  20. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2 + 8x + 13 Zeros: y = (x2 + 8x) + 13 (x + 4)2– 3= 0 y = (x2 + 8x + 16) + 13 – 16 (x + 4)2 = 3 x + 4 =  1.73 y = (x + 4)2 + 13 – 16 x + 4 = 1.73 x + 4 = –1.73 y = (x + 4)2– 3 Vertex: (-4, -3) x = –2.27 & –5.73 AOS = -4

  21. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: y = x2– 5x + 2 Zeros: y = (x2– 5x) + 2 (x– 2.5)2 – 4.25 = 0 y = (x2– 5x + 6.25) + 2 – 6.25 (x– 2.5)2 = 4.25 x – 2.5 =  2.06 y = (x– 2.5)2 + 2 – 6.25 x – 2.5 = 2.06 x – 2.5 = –2.06 y = (x– 2.5)2 – 4.25 Vertex: (2.5, –4.25) x = 4.56 & 0.44 AOS = 2.5

  22. 10.3 Completing the Square Find the vertex, AOS, and zeros of a parabola by completing the square: Zeros: y = x2 + 7x + 4 (x+ 3.5)2 – 8.25 = 0 y = (x2+ 7x) + 4 (x+ 3.5)2 = 8.25 y = (x2+ 7x + 12.25) + 4 – 12.25 y = (x+ 3.5)2 + 4 – 12.25 x + 3.5 =  2.87 y = (x+ 3.5)2 – 8.25 x + 3.5 = 2.87 x + 3.5 = –2.87 (−3.5, –8.25) Vertex: −3.5 AOS = x = −6.37 & −0.63

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