Warm-Up Factor. 6 minutes 1) x x ) x 2 – 22x ) x 2 – 12x - 64 Solve each equation. 4) d 2 – 100 = 0 5) z 2 – 2z + 1 = 0 6) t = -8t

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

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

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

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

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

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

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

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

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

Vertex Form If the coordinates of the vertex of the graph of y = ax 2 + 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.

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

Example 2 Given g(x) = 2x x + 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) = x 2 to g. g(x) = 2x x + 23 = 2(x 2 + 8x) + 23 = 2(x 2 + 8x = 2(x + 4) = 2(x – (- 4)) 2 + (-9) + 16)+ 23– 32 vertex: (-4,-9) axis of symmetry: x = -4 vertex form: y = a(x – h) 2 + k

Practice Given g(x) = 3x 2 – 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) = x 2 to g.

Homework