SAT Problem of the Day

SAT Problem of the Day

SAT Problem of the Day

SAT Problem of the Day

5.4 Completing the Square5.4 Completing the Square5.4 Completing the Square5.4 Completing the SquareObjectives: •Use completing the square to solve a quadratic equation•Use the vertex form of a quadratic function to locate the axis of symmetry of its graph

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

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

0

2) 2x2 + 10x = 6

Example 3Solve x2 + 9x – 22 = 0 by completing the square.

x2 + 9x = 22

2b2

find

x2 + 9x + (81/4) = 22 + (81/4)

(x + 9/2)2 = 169/4

x = 2 or x = -11

x + 9/2 = +13/2 or -13/2

PracticeSolve by completing the square.1) x2 - 7x = 14

Example 4Solve 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

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 5Write 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 6Given 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

Application

A softball is thrown upward with an initial velocity of 32 feet per second from 5 feet above ground. The ball’s height in feet above the ground is modeled by

h(t) = -16t2 + 32t + 5, where t is the time in seconds after the ball is released. Complete the square and rewrite h in vertex form. Then find the maximum height of the ball.

Objectives: •Use the vertex form of a quadratic function to locate the vertex, the axis of symmetry, and describe the graph.

Collins Type II

• As an exit ticket, explain what exactly h and k represent (vertex form) for the application problem.– Use specific terms from the problem

Objectives: •Use the vertex form of a quadratic function to locate the vertex, the axis of symmetry, and describe the graph.

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.

Objectives: •Use the vertex form of a quadratic function to locate the vertex, the axis of symmetry, and describe the graph.

Homework

Lesson 5.4 exercises 39-45 ODD