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Warm-Up

6 minutes. Warm-Up. Factor. 1) x 2 + 14x + 49. 2) x 2 – 22x + 121. 3) x 2 – 12x - 64. Solve each equation. 4) d 2 – 100 = 0. 5) z 2 – 2z + 1 = 0. 6) t 2 + 16 = -8t. Completing the Square. Objectives: Use completing the square to solve a quadratic equation. find. find.

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Warm-Up

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  1. 6 minutes Warm-Up Factor. 1) x2 + 14x + 49 2) x2 – 22x + 121 3) x2 – 12x - 64 Solve each equation. 4) d2 – 100 = 0 5) z2 – 2z + 1 = 0 6) t2 + 16 = -8t

  2. Completing the Square Objectives: Use completing the square to solve a quadratic equation

  3. find find Example 1 Complete the square for each quadratic expression to form a perfect-square trinomial. a) x2 – 10x x2 – 10x + 25 (x -5)2 b) x2 + 27x

  4. Practice Complete the square for each quadratic expression to form a perfect-square trinomial. Then write the new expression as a binomial squared. 1) x2 – 7x 2) x2 + 16x

  5. find Example 2 Solve x2 + 18x – 40 = 0 by completing the square. x2 + 18x = 40 x2 + 18x + 81 = 40 + 81 (x +9)2 = 121 x = 2 or x = -20

  6. find Example 3 Solve 3x2 - 6x = 5 by completing the square. 3(x2 - 2x) = 5 3(x2 - 2x + 1)= 5 + 3 3(x - 1)2 = 8

  7. Practice Solve by completing the square. 1) x2 + 10x – 24 = 0 2) 2x2 + 10x = 6

  8. Warm-Up Solve each equation by completing the square. 1) x2 + 10x + 16 = 0 2) x2 + 2x = 13

  9. Completing the Square Objectives: Use the vertex form of a quadratic function to locate the axis of symmetry of its graph

  10. Transformations y = af(x) gives a vertical stretch or compression of f y = f(ax) gives a horizontal stretch or compression of f y = f(x) + k gives a vertical translation of f y = f(x - k) gives a horizontal translation of f

  11. If the coordinates of the vertex of the graph of y = ax2 + bx + c, where are (h,k), then you can represent the parabola as y = a(x – h)2 + k, which is the vertex form of a quadratic function. Vertex Form

  12. Example 1 Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. vertex form: y = a(x – h)2 + k y = -6x2 + 72x - 207 y = -6(x2 - 12x) - 207 y = -6(x2 - 12x + 36) – 207 + 216 y = -6(x - 6)2 + 9 vertex: (6,9) axis of symmetry: x = 6

  13. Example 2 Given g(x) = 2x2 + 16x + 23, write the function in vertex form, and give the coordinates of the vertex and the equation of the axis of symmetry. Then describe the transformations from f(x) = x2 to g. g(x) = 2x2 + 16x + 23 vertex form: y = a(x – h)2 + k = 2(x2 + 8x) + 23 = 2(x2 + 8x + 16) + 23 – 32 = 2(x + 4)2 - 9 = 2(x – (-4))2 + (-9) vertex: (-4,-9) axis of symmetry: x = -4

  14. Practice Given g(x) = 3x2 – 9x - 2, write the function in vertex form, and give the coordinates of the vertex and the equation of the axis of symmetry. Then describe the transformations from f(x) = x2 to g.

  15. Homework

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