SAT Problem of the Day

SAT Problem of the Day

5.4 Completing the Square Objectives: 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

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

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

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

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

find Example 3 Solve x2 + 9x – 22 = 0 by completing the square. x2 + 9x = 22 x2 + 9x + (81/4) = 22 + (81/4) (x +9/2)2 = 169/4 x +9/2 = +13/2 or -13/2 x = 2 or x = -11

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

find Example 4 Solve 3x2 - 6x = 5 by completing the square. 3(x2 - 2x) = 5 3(x2 - 2x + 1)= 5 + 3 3(x - 1)2 = 8

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

Example 5 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

Example 6 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

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.

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. • 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

SAT Problem of the Day