Download 6 -6 Factoring by Grouping

SAT Problem of the Day
SAT Problem of the Day
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
Example 1
Complete the square for each quadratic expression
to form a perfect-square trinomial.
2
b
find  
2
a) x2 – 10x
x2 – 10x + 25
(x - 5)2
2
b)
x2
b
find  
2
+ 27x
2
 27 
x2  27x  

2


2
27 

x



2


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
Example 2
Solve x2 + 18x – 40 = 0 by completing the square.
x2 + 18x = 40
x2 + 18x + 81 = 40 + 81
(x + 9)2 = 121
x  9  11
x = 2 or x = -20
2
b
find  
2
Practice
Solve by completing the square.
1) x2 + 10x – 24 = 0
2) 2x2 + 10x = 6
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
2
b
find  
2
Practice
Solve by completing the square.
1) x2 - 7x = 14
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
8
(x  1) 
3
2
8
x 1  
3
8
x 1
3
2
b
find  
2
Vertex Form
If the coordinates of the vertex of the graph of y =
ax2 + bx + c, where a  0, 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 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 = -6x2 + 72x - 207
y = a(x – h)2 + k
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,
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.
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
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.
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