Warm-UpFactor.

6 minutes

1) x2 + 14x + 49

2) x2 – 22x + 121

3) x2 – 12x - 64

Solve each equation.4) d2 – 100 =

05) z2 – 2z + 1 = 06) t2 + 16 = -8t

Completing the SquareCompleting the Square Completing the SquareCompleting the SquareObjectives: •Use completing the square to solve a quadratic equation

Example 1Complete the square for each quadratic expression to form a perfect-square trinomial.

a) x2 – 10x

2b2

find

x2 – 10x + 25(x - 5)2

b) x2 + 27x

2b2

find 2

2 272

x 27x

2

x272

Practice

1) x2 – 7x 2) x2 + 16x

Complete the square for each quadratic expression to form a perfect-square trinomial. Then write the new expression as a binomial squared.

Example 2Solve x2 + 18x – 40 = 0 by completing the square.

x2 + 18x = 40

2b2

find

x2 + 18x + 81 = 40 + 81

(x + 9)2 = 121x 9 11

x = 2 or x = -20

Example 3Solve 3x2 - 6x = 5 by completing the square.

3(x2 - 2x) = 5

2b2

find

3(x2 - 2x + 1) = 5 + 3

3(x - 1)2 = 8

8x 1

3

8x 1

3

2 8(x 1)

3

PracticeSolve by completing the square.1) x2 + 10x – 24 =

0

2) 2x2 + 10x = 6

Warm-UpSolve each equation by completing the square.1) x2 + 10x + 16 = 0

2) x2 + 2x = 13

Completing the SquareCompleting the SquareCompleting the SquareCompleting the SquareObjectives: •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 fy = f(ax) gives a horizontal stretch or compression of fy = f(x) + k gives a vertical translation of f

y = f(x - k) gives a horizontal translation of f

Vertex FormIf 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.

a 0,

Example 1Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry.

y = -6x2 + 72x - 207y = -6(x2 - 12x) - 207y = -6(x2 - 12xy = -6(x - 6)2 + 9vertex: (6,9)

axis of symmetry: x = 6

+ 36)

– 207 +216

vertex form: y = a(x – h)2 + k

Example 2Given 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

= 2(x2 + 8x) + 23

= 2(x2 + 8x= 2(x + 4)2 - 9= 2(x – (- 4))2 + (-9)

+ 16)

+ 23

– 32

vertex: (-4,-9)

axis of symmetry: x = -4

vertex form: y = a(x – h)2 + k

PracticeGiven 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.

Homework