Do Now!

1
Do Now!
sheet
Name
date
date
date
date
date
date
2
Do Now!
sheet
Name
date
date
date
date
3
Do Now!
10 24 - 2013
Factor the trinomial.
Factor the trinomial.
a)
b)
( ) ( )
PRGM
FCTPOLY
4
Do Now!
10 24 - 2012
F.O.I.L.( distribute )
Factor the trinomial.
c)
a)
( ) ( )
b)
d)
5
10 23 - 2012
Do Now!
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
6
10 24 - 2012
Now
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
7
Do Now!
Thursday
10 25 - 2012
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
8
Modeling Projectile Objects
When an object is projected, its height h (in
feet) above the ground after t seconds can be
modeled by the function
where is the objects initial height (in
feet).
Baseball Hit A baseball is hit by a batter.
1.) Write an equation giving the balls height h
(in feet) above the ground after t seconds. 2.)
Graph the equation. 3.) During what time
interval is the balls height above 3 feet?
9
Do Now!
10 29 - 2011
Factor the trinomial.
1 a)
2

• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

( ) ( )
F.O.I.L.
1 b)
Rewrite in Quadratic Standard form
Vertex shifts ______ Width _______
1 c)
10
Do Now!
11 29 - 2012
Factor the trinomial.
Solve the equation
1)
2)
FCTPOLY
PRGM
QUAD83
PRGM
1.6666666666666 1
4 ( x 2 ) ( 3x 5 )
11
11 30 - 2012
Do Now!
1. Solve the equation.
2. Find the x-intercepts.
QUAD83
QUAD83
AND
4. What are the Solutions of the equation?
3. Find the Zeros
QUAD83
QUAD83
AND
AND
12
Do Now!
Wednesday
10 31 - 2012
The Area of the rectangle is 30. What are the
lengths of the sides?
x
3x 1
x
3x 1
( )
30
13
Modeling Projectile Objects
When an object is projected, its height h (in
feet) above the ground after t seconds can be
modeled by the function
where is the objects initial height (in
feet).
Baseball Hit A baseball is hit by a batter.
1.) Write an equation giving the balls height h
(in feet) above the ground after t seconds. 2.)
Graph the equation. 3.) During what time
interval is the balls height above 3 feet?
14
Modeling Dropped Objects
Student
When an object is dropped, its height h (in feet)
above the ground after t seconds can be modeled
by the function
where is the objects initial height (in
feet).
CLIFF DIVING A cliff diver dives off a cliff
40 feet above water. 1.) Write an equation
giving the divers height h (in feet) above the
water after t seconds. 2.) Graph the equation.
(plot some points from the table) 3.) How long is
the diver in the air? (what are you looking
for?) 4.) The place that the diver starts is
called what? (mathematically) 5.) What are we
going to count by?
Window re-set. Xmin Xmax Xscl Ymin
Ymax Yscl
HEIGHT
TIME
15
Modeling Dropped Objects
Student
When an object is dropped, its height h (in feet)
above the ground after t seconds can be modeled
by the function
where is the objects initial height (in
feet).
CLIFF DIVING A cliff diver dives off a cliff
40 feet above water. 1.) Write an equation
giving the divers height h (in feet) above the
water after t seconds. 2.) Graph the equation.
(plot some points from the table) 3.) How long is
the diver in the air? (what are you looking
for?) 4.) The place that the diver starts is
called what? (mathematically) 5.) What are we
going to count by?
Window re-set. Xmin Xmax Xscl Ymin
Ymax Yscl
HEIGHT
TIME
16
40
30
20
HEIGHT
10
5
1.5
0.5
1
TIME
17
Modeling Dropped Objects
When an object is dropped, its height h (in feet)
above the ground after t seconds can be modeled
by the function
where is the objects initial height (in
feet).
CLIFF DIVING A cliff diver dives off a cliff
40 feet above water. 1.) Write an equation
giving the divers height h (in feet) above the
water after t seconds. 2.) Graph the equation.
3.) How long is the diver in the air?
18
Modeling Dropped Objects
When an object is dropped, its height h (in feet)
above the ground after t seconds can be modeled
by the function
HEIGHT
where is the objects initial height (in
feet).
Lets assume that the cliff is 40 feet high.
TIME
19
Lets assume that the cliff is 40 feet high.
Write an equation?
Make the substitution.
HEIGHT
Graph it.
TIME
20
40
Lets assume that the cliff is 40 feet high.
Write an equation?
30
Make the substitution.
20
Graph it.
HEIGHT
5
1
1.5
.5
TIME seconds
21
40
Lets assume that the cliff is 40 feet high.
Write an equation?
30
Make the substitution.
20
Graph it.
HEIGHT
How long will the diver be in the air?
5
Think about what are we trying to find?
Hint we want SOLUTIONS.
1
1.5
.5
1.58 seconds
-1.58 seconds
TIME
22
Changing the world takes more than everything any
one person knows. But not more than we know
together. So let's work together.
23
11 8 - 2012
Do Now!
Quadratic Formula
Example 1)
Solve using the Quadratic Formula
Identify A B C
1
2
12
Plug them in to the formula
24
How to use a Discriminant to determine the number
of solutions of a quadratic equation.
discriminant
if , (positive) then 2
real solutions.
if , (zero) then 1 real
solutions.
if , (negative) then 2
imaginary solutions.
25
MONDAY , NOV. 7th
ASSIGNMENT
PAGE 279 12 -27 ALL
Complex Numbers (
i ) PAGE 296 3 6, 31 33,
40 - 42,
Quadratic Formula
QUAD83 Solve the equation
Discriminant How many solutions
26
(No Transcript)
27
Collaborative Activity Sheet 1 Chapter 4 Solving
Graphing Quadratic functions
Collaborative Activity Sheet Chapter 4 Solving
Graphing Quadratic functions
A.) For a science competition, students must
design a container that prevents an egg from
breaking when dropped from a height of 50 feet.
1.) Write an equation giving the containers
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the container take to hit the ground?
A.) For a science competition, students must
design a container that prevents an egg from
breaking when dropped from a height of 50 feet.
1.) Write an equation giving the containers
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the container take to hit the ground?
B.) A bird flying at a height of 30 feet
carries a shellfish. The bird drops the
shellfish to break it and get the food inside.
1.) Write an equation giving the shellfish
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the shellfish take to hit the ground?
B.) A bird flying at a height of 30 feet
carries a shellfish. The bird drops the
shellfish to break it and get the food inside.
1.) Write an equation giving the shellfish
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the shellfish take to hit the ground?
C.) Some harbor police departments have
firefighting boats with water cannons. The boats
are use to fight fires that occur within the
harbor. The function y - 0.0035x( x
143.9) models the path of water shot by a water
cannon where x is the horizontal distance ( in
feet ) and y is the corresponding height ( in
feet ). 1.) Write an equation (in standard
form) modeling the path of water. 2.) Graph the
equation. 3.) How far does the water cannon
shoot?
C.) Some harbor police departments have
firefighting boats with water cannons. The boats
are use to fight fires that occur within the
harbor. The function y - 0.0035x( x
143.9) models the path of water shot by a water
cannon where x is the horizontal distance ( in
feet ) and y is the corresponding height ( in
feet ). 1.) Write an equation (in standard
form) modeling the path of water. 2.) Graph the
equation. 3.) How far does the water cannon
shoot?
D.) A football is kicked upward by a player in
the game. The height h (in feet) of the ball
after t seconds is given by the function Where
v (the velocity for the ball when kicked) is 96
mph, the initial height of the ball is 3
feet. 1.) Write an equation giving the
balls height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does it take for the ball to hit the ground?
D.) A football is kicked upward by a player in
the game. The height h (in feet) of the ball
after t seconds is given by the function Where
v (the velocity for the ball when kicked) is 96
mph, the initial height of the ball is 3
feet. 1.) Write an equation giving the
balls height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does it take for the ball to hit the ground?
28
Collaborative Activity Sheet 2 Chapter 4 Solving
Graphing Quadratic functions
E.) A stunt man working on a movie set falls
from a window that is 70 feet above an air
cushion positioned on the ground. 1.) Write an
equation that models the height of the stunt man
as he falls. 2.) Graph the equation. 3.) How
long does it take him to hit the ground?
A.) For a science competition, students must
design a container that prevents an egg from
breaking when dropped from a height of 50 feet.
1.) Write an equation giving the containers
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the container take to hit the ground?
F.) A science center has a rectangular parking
lot. The Science center wants to add 18,400
square feet to the area of the parking lot by
expanding the existing parking lot as
shown 1.) Find the area of the existing
parking lot. 2.) Write an equation that you can
use to find the value of x 3.) Solve the
equation. By what distance x should the length
and width of the parking lot be expanded?
B.) A bird flying at a height of 30 feet
carries a shellfish. The bird drops the
shellfish to break it and get the food inside.
1.) Write an equation giving the shellfish
height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does the shellfish take to hit the ground?
C.) Some harbor police departments have
firefighting boats with water cannons. The boats
are use to fight fires that occur within the
harbor. The function y - 0.0035x( x
143.9) models the path of water shot by a water
cannon where x is the horizontal distance ( in
feet ) and y is the corresponding height ( in
feet ). 1.) Write an equation (in standard
form) modeling the path of water. 2.) Graph the
equation. 3.) How far does the water cannon
shoot?
G.) An object is propelled upward from the top
of a 300 foot building. The path that the object
takes as it falls to the ground can be modeled by
Where t is the time (in seconds) and y is the
corresponding height ( in feet) of the object.
1.) Graph the equation. 2.) How long is it in
the air?
D.) A football is kicked upward by a player in
the game. The height h (in feet) of the ball
after t seconds is given by the function Where
v (the velocity for the ball when kicked) is 96
mph, the initial height of the ball is 3
feet. 1.) Write an equation giving the
balls height (h) above the ground after (t)
seconds. 2.) Graph the equation. 3.) How long
does it take for the ball to hit the ground?
29
Collaborative Activity Sheet 3 Chapter 4 Solving
Graphing Quadratic functions
1.) A football is kicked upward by a player in
the game. The height h (in feet) of the ball
after t seconds is given by the function Where
v (the velocity for the ball when kicked) is 65
mph, the initial height of the ball is 3
feet. a.) Write an equation giving the
balls height (h) above the ground after (t)
seconds. b.) Graph the equation. c.) How long
does it take for the ball to hit the ground? d.)
Is the Vertex a Max or Min?
3.) In a football game, a defensive player jumps
up to block a pass by the opposing teams
quarterback. The player bats the ball downward
with his hand at an initial vertical velocity of
-50 feet per second when the ball is 7 feet above
the ground. How long do the defensive players
teammates have to intercept the ball before it
hits the ground?
4.) The aspect ratio of a widescreen TV is the
ratio of the screens width to its height, or
169 . What are the width and the height of a 32
inch widescreen TV? (hint Use the Pythagorean
theorem and the fact that TV sizes such as 32
inches refer to the length of the screens
diagonal.) Draw a picture.
2.) A science center has a rectangular parking
lot. The Science center wants to add 18,400
square feet to the area of the parking lot by
expanding the existing parking lot as
shown a.) Find the area of the existing
parking lot. b.) Write an equation that you can
use to find the value of x c.) Solve the
equation. By what distance x should the length
and width of the parking lot be expanded?
5.) You are using glass tiles to make a picture
frame for a square photograph with sides 10
inches long. You want to frame to form a uniform
border around the photograph. You have enough
tiles to cover 300 square inches. What is the
largest possible frame width x?
x
x
x
x
30
Graph and compare to

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
31
Graph and compare to

• Graph
• Find Vertex
• Axis of Symmetry
• d) x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
32
Graph and compare to

• Graph
• Find Vertex
• Axis of Symmetry
• d) x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
33
Graph and compare to

• Graph
• Find Vertex
• Axis of Symmetry
• d) x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
34
Graph and compare to

• Graph
• Find Vertex
• Axis of Symmetry
• d) x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
35
Graph and compare to

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
36
Graph and compare to

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
37
Graph and compare to

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
38
Graph and compare to

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

2
39
Graph and compare to
Axis of Symmetry
(1, 3)

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

vertex
x-intercepts
2
x 1
40
Graph and compare to
Axis of Symmetry

• Graph
• Find Vertex
• Identify
• Axis of Symmetry
• d) Find Solutions
• x-intercepts
• e) Opens UP or DOWN
• f) Compare to y x

x-intercepts
(1, -4)
vertex
2
x 1
41
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
42
Quiz 4.1
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
43
Quiz 4.1 RETAKE
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
44
Quiz 4.1 RETAKE
Graph and compare to

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
45
Quiz 4.2
Graph and compare to
1)

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
46
Graph and compare to
2)

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

2
Vertex shifts ______ Width _______
47
(No Transcript)
48
(No Transcript)
49
4.1
Graphing Quadratic Functions
What you should learn
Goal
1
Graph quadratic functions.
Goal
2
Use quadratic functions to solve real-life
problems.
4.1 Graphing Quadratic Functions in Standard Form
50
Vocabulary
Quadratic Functions in Standard Form is written as
,Where a ¹ 0
A parabola is the U-shaped graph of a quadratic
function. The vertex of a parabola is the lowest
point of a parabola that opens up, and the
highest point of a parabola that opens down.
4.1 Graphing Quadratic Functions in Standard Form
51
PARENT FUNCTION for Quadratic Functions The
parent function for the family of all quadratic
functions is f(x) .
Axis of Symmetry divides the parabola into mirror
images and passes through the vertex.
Vertex is (0, 0)
4.1 Graphing Quadratic Functions in Standard Form
52
PROPERTIES of the GRAPH of

• Characteristics of this graph are
• The graph opens up if a gt 0
• The graph open down if a lt 0
• The graph is wider than if
  
• The graph is narrower than if
• The x-coordinate of the vertex is
• The Axis of Symmetry is the vertical line

4.1 Graphing Quadratic Functions in Standard Form
53
Example 1A
Graphing a Quadratic Function
Graph and compare to
Graphing Calculator PRGM down to QUAD83 A
? B ? C?
Vertex X Y 2
-1
1
Solutions 3
1
-4
Axis of Symmetry
3
The line x 2
4.1 Graphing Quadratic Functions in Standard Form
54
Example 1B
Graphing a Quadratic Function
Graph and compare to
Graphing Calculator PRGM down to QUAD83 A
? B ? C?
Vertex X Y -1
0
1
Solutions -1
-1
2
Axis of Symmetry
The line x -1
1
4.1 Graphing Quadratic Functions in Standard Form
55
Example 1C
Graphing a Quadratic Function
Graph and compare to
Graphing Calculator PRGM down to QUAD83 A
? B ? C?
Vertex X Y -2
3
-2
Solutions -3.225
-.775
Axis of Symmetry
-8
The line x -2
-5
4.1 Graphing Quadratic Functions in Standard Form
56
Reflection on the Section
very important question
How is the Vertex of a parabola related to its
Axis of Symmetry?
assignment
Page 240
4.1 Graphing Quadratic Functions in Standard Form
57
Example 1
Graphing a Quadratic Function
Graph and compare to
The coefficients are a 1, b -4, c 3 Since a
gt 0, the parabola opens up. To find the
x-coordinate of the vertex, substitute 1 for a
and -4 for b in the formula
4.1 Graphing Quadratic Functions in Standard Form
58
To find the y-coordinate of the vertex,
substitute 2 for x in the original equation, and
solve for y.
4.1 Graphing Quadratic Functions in Standard Form
59
The vertex is (2, -1). Plot two points, such as
(1,0) and (0,3). Then use symmetry to plot two
more points (3,0) and (4,3). Draw the parabola.
4.1 Graphing Quadratic Functions in Standard Form
60
Additional Example 1
4.1 Graphing Quadratic Functions in Standard Form
61
Additional Example 2
62
4.2
Graphing Quadratic Functions in Vertex or
Intercept Form
What you should learn
Graph quadratic functions in VERTEX form or
INTERCEPT form.
Goal
1
Find the Minimum value or the Maximum value
Goal
2
F.O.I.L.
Review the
3.
1.
2.
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
63
Example 1A
Vertex Form
Graphing a Quadratic Function in Vertex form
Vertex ( h, k )
So, Vertex ( 6, 1 )
Graph
Rewrite in Standard Form
Split and FOIL
Combine like terms
use QUAD83 to find the Solutions and confirm
Vertex
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
64
Example 1B
Vertex Form
Graphing a Quadratic Function in Vertex form
Vertex ( h, k )
So, Vertex ( 3, -4 )
Rewrite in Standard Form
Split and FOIL
distribute
Combine like terms
use QUAD83 to find the Solutions and confirm
Vertex
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
65
Example 1B
Vertex Form
Graphing a Quadratic Function in Vertex form
Vertex ( h, k )
So, Vertex ( 3, -4 )
Vertex is a Minimum Pt.
Since,
the parabola is narrower than
Now, Graph it on the calculator.
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
66
Plot the vertex (h,k) ? (3,-4) Plot
x-intercepts 4.41 and 1.59 Plot two more
points, such as (2,-2) and (4, -2). Draw the
parabola. Compare to Parent
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
67
Example 2
Vertex Form
Vertex
(-2, -3)
x -2
Axis of Symmetry
Opens UP, vertex is MIN
ZEROs
Solutions x-intercepts
(-3.73, 0) and
(-.268, 0)
Since,
the parabola is the same width as
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
68
Graphing Quadratic Functions in Vertex or
Intercept Form
continued
4.2
What you should learn
Graph quadratic functions in .. INTERCEPT form.
DO THESE PROBLEMS
Goal
1
VERTEX form to STANDARD form
Review rewrite
1.
2.
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
69
Additional Example 3
Vertex Form
Graph
( 1, 2)
Vertex
Axis of Sym
x 1
Opens
DOWN, vertex is MAX
Solutions
-.414 and 2.414
Shift
Rt 1 --- Up 2
Since,
Width
the parabola is the same width as
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
70
Example 1
Intercept Form
x-intercepts (-3, 0) (5, 0)
To find the Vertex -3 5 divided by 2
Then, substitute in for x to find the y
coordinate.
Vertex ( 1, -16)
Opens UP, vertex is MIN
Since,
the parabola is the same width as
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
71
Example 2
Intercept Form
Graph
x-intercepts (4, 0) (-2, 0)
To find the Vertex 4 (-2) divided by 2
Then, substitute in for x to find the y
coordinate.
Vertex ( 1, 9)
Opens DOWN, vertex is MAX
Since,
the parabola is the same width as
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
72
Example 3
Intercept form
Graphing a Quadratic Function in Intercept form
The x-intercepts are (1,0) and (-3,0) The axis of
symmetry is x -1
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
73
Cont Example 3
Intercept form
The x-coordinate of the vertex is -1. The
y-coordinate is
Graph the parabola.
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
74
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
75
Additional Example 2
y - (x 1)(x 3)
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
76
Additional Example 3
y (x 1)(x - 3)
77
Example 4

• Writing Quadratic Functions in Standard Form

Write y 2(x 3)(x 8) in standard form
78
Additional Example 1
Write the quadratic function in standard form
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
79
Reflection on the Section
Give an example of a quadratic equation in vertex
form. What is the vertex of the graph of this
equation?
assignment
4.2 Graphing Quadratic Functions in Vertex or
Intercept form
80
4.3
Solving Quadratic Equations by Factoring
What you should learn
Goal
1
Factor quadratic expressions and solve quadratic
equations by factoring.
Goal
2
Find zeros of quadratic functions.
4.3 Solving Quadratic Equations by Factoring
81
Directions Factor the expression.
Example 1)
Example 2)
PROGRAM
PROGRAM
FCTPOLY
FCTPOLY
2
2
DEGREE
DEGREE
X
CONST
X
CONST
COEF. OF X2 ?
COEF. OF X2 ?
1
1
? -1
? 6
? -12
? 8
(x 2 )(x 4)
4.3 Solving Quadratic Equations by Factoring
82
Directions Factor the expression.
Example 4)
Example 3)
FCTPOLY
PROGRAM
FCTPOLY
PROGRAM
2
DEGREE
2
DEGREE
X
CONST
COEF. OF X2 ?
X
CONST
COEF. OF X2 ?
1
1
? 2
? 1
? -8
? -5
(x - 2 )(x 4)
This means cannot be factored
4.3 Solving Quadratic Equations by Factoring
83
Directions Solve the equation.
Ex 1)
FCTPOLY
PROGRAM
QUAD83
PROGRAM
2
DEGREE
A ?
1
X
CONST
COEF. OF X2 ?
B ?
-5
1
C ?
-36
? -5
SOLUTIONS
? -36
9 -4
(x - )(x )
4
9
84

• A monomial is a polynomial with only one term.
• A binomial is a polynomial with two terms.
• A trinomial is a polynomial with three terms.
• Factoring can be used to write a trinomial as a
  product of binomials.

85
We are doing the reverse of the F.O.I.L. of two
binomials. So, when we factor the trinomial, it
should be two binomials.
Example 1
Step 1 Enter x as the first term of each factor.
( x )( x )
Step 2 List pairs of factors of the constant, 8.
Factors of 8
8, 1
4, 2
-8, -1
-4, -2
86
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
6x)
Possible Factorizations
x 8x 9x
( x 8)( x 1)
( x 4)( x 2)
2x 4x 6x
-x - 8x - 9x
( x - 8)( x - 1)
( x - 4)( x - 2)
-2x - 4x - 6x
87
Example 2
Step 1 Enter x as the first term of each factor.
( x )( x )
Step 2 List pairs of factors of the constant, 7.
Factors of 7
-7, -1
7, 1
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
8x)
Possible Factorizations
x 7x 8x
( x 7)( x 1)
( x - 7)( x - 1)
-x - 7x - 8x
88
(x )(x )
(x - )(x )
Look at the 2nd sign

• If it is positive, both signs in binomials will
  be the same. (same as the 1st sign.)
• If it is negative, the signs in binomials will be
  different.

(x - )(x - )
(x )(x - )
89
The Difference of Two Squares If A and B are real
numbers, variables, or algebraic expressions,
then In words The difference of the squares
of two terms is factored as the product of the
sum and the difference of those terms.
90
Factoring the Difference of Two Squares
Example 1)
1.) Difference of the Two Squares,
2.) or you could look at this as the trinomial
91
Difference of the Two Squares,
Example 2
We must express each term as the square of some
monomial. Then use the formula for factoring
1.)
You can check it by using FOIL on the binomial.
2.) or you could look at this as the trinomial
(x )(x )
-
4
4

92
(x )(x )
1
6
(x - )(x )
4
16
(x - )(x - )
4
18
(x )(x - )
1
16
93
Factor.
(x )(x )
2y
6y
(x - )(x )
4y
7y
94
4.4
Solving Quadratic Equations by Factoring
What you should learn
Goal
1
Factoring Trinomials whose Leading Coefficient
is NOT one.
Objectives
1. Factor trinomials by trial and error.
4.3 Solving Quadratic Equations by Factoring
95
Factoring by the Trial-and-Error Method
How would we factor
Notice that the leading coefficient is 3, and we
cant divide it out
( 3x )( x )
96
example
Step 1 find the two First terms whose product is
.
( 3x )( x )
Step 2 Find two Last terms whose product is 28.
The number 28 has pairs of factors that are
either both positive or both negative. Because
the middle term, -20x, is negative, both factors
must be negative.
Factors of 28
-1(-28)
- 4(-7)
- 2(-14)
97
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
-20x)
Possible Factorizations
-84x - x - 85x
( 3x - 1)( x - 28)
( 3x - 28)( x - 1)
-3x - 28x - 31x
( 3x - 2)( x - 14)
-42x - 2x - 44x
( 3x - 14)( x - 2)
-6x - 14x - 20x
-21x - 4x - 25x
( 3x - 4)( x - 7)
-12x - 7x - 19x
( 3x - 7)( x - 4)
98
example
Step 1 find the two First terms whose product is
.
( 8x )( x )
( 4x )(2 x )
Step 2 Find two Last terms whose product is -3.
Factors of -3
1(-3)
-1(3)
99
Step 3 Try various combinations of these factors.
Sum of Outside and Inside Products (should equal
-10x)
Possible Factorizations
( 8x 1)( x - 3)
-24x x - 23x
( 8x - 3)( x 1)
8x - 3x 5x
( 8x - 1)( x 3)
24x - x 23x
( 8x 3)( x - 1)
- 8x 3x - 5x
-12x 2x - 10x
( 4x 1)(2 x - 3)
( 4x - 3)( 2x 1)
4x - 6x - 2x
( 4x - 1)( 2x 3)
12x - 2x 10x
-4x 6x 2x
( 4x 3)( 2x - 1)
100
Factoring Trinomials whose Leading Coefficient
is NOT one.
Ex 1)
Ex 2)
(3x )(3x )
1
1
(2x )(2x - )
1
7
Ex 3)
Ex 4)
(2x - )(3x - )
1
2
(2x )(x - )
3
5
101
The Zero-Product Principle
If the product of two algebraic expressions is
zero, then at least one of the factors is equal
to zero.
If AB 0, then A 0 or B 0.
If, ( ???)() 0
Example)
Then either (???) is zero, or () is zero.
102
Solve the equation.
Example 1)
According to the principle,
this product can be equal to zero, if either
or
5
5
2
2
x 5
x 2
The resulting two statements indicate that the
solutions are 5 and 2.
103
Solve the Equation (standard form) by Factoring
Example 2)
Factor the Trinomial using the methods we know.
(2x )(x ) 0
-

1
4
or
1
1
- 4
- 4
2x 1
x - 4
x 1/2
The resulting two statements indicate that the
solutions are 1/2 and - 4.
104
Solve the Equation (standard form) by Factoring
Example 3)
Move all terms to one side with zero on the
other. Then factor.
(x )(x ) 0
-
-
3
3
The trinomial is a perfect square, so we only
need to solve once.
3
3
x 3
The resulting two statements indicate that the
solutions are 3.
105
Factoring out the greatest common factor.
But, before we do thatdo you remember the
Distributive Property?
When factoring out the GCF, what we are going to
do is UN-Distribute.
106
What I mean is that when you use the Distributive
Property, you are multiplying. But when you are
factoring, you use division.
Factor
example
1st determine the GCF of all the terms.
5
2nd pull 5 out, and divide both terms by 5.
107
Factor each polynomial using the GCF.
ex)
ex)
ex)
108
Sometimes polynomials can be factored using more
than one technique. When the Leading Coefficient
is not one. Always begin by trying to factor
out the GCF.
Example 1
factor out 3x
3x(x )(x )
7
-
2

109
Factor.
Example 2
Example 3
3( )
( )
(a - )(a - )
2
9
(x )(x - )
3
16
110
Example 4
Factoring GCF First
Step 1) GCF
111
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 1)
Whats the GCF?
112
Factoring out the GCF and then factoring the
Difference of two Squares.
Example 2)
Whats the GCF?
113
Additional Examples
Example 3
114
Factoring Perfect Square Trinomials
Example 4
(x )(x )

3
3

Since both binomials are the same you can say
115
Factoring Perfect Square Trinomials
Example 5
(x )(x )
-
5
5
-
Since both binomials are the same you can say
116
Example 6
117
Reflection on the Section
What must be true about a quadratic equation
before you can solve it using the zero product
property?
assignment
Page 261 47 88, 90
118
www.animationfactory.com
Backdrops - These are full sized backdrops, just
scale them up! - Can be Copy-Pasted out of
Templates for use anywhere!
Title Backdrop
Slide Backdrop
Print Backdrop
Transitional Backdrop

• Additional Graphics
• Scale them up or down!
• .GIF clipart is animated.
• .JPG clipart can be scaled up and take up little
  file space.
• .PNG clipart can be scaled unusually large
  without distortion.

119
Solving Quadratic Equations by Finding Square
Roots
4.5
What you should learn
Goal
1
Solve quadratic equations by finding square roots.
Goal
2
Use quadratic equations to solve real-life
problems.
4.5 Solving Quadratic Equations by Finding Square
Roots
120
Simplify the expression.
Example 1)
Example 2)
Example 3)
Example 4)
4.5 Solving Quadratic Equations by Finding Square
Roots
121
Solve the Quadratic Equation.
Example 1)
QUAD83
Example 2)
-18
-18
Example 3)
QUAD83
-36
-36
QUAD83
4.5 Solving Quadratic Equations by Finding Square
Roots
122
Solve the Quadratic Equation.
Example 4)
-40
-40
QUAD83

4.5 Solving Quadratic Equations by Finding Square
Roots
123
Solve the Quadratic Equation.
Example 5)
-10
-10
QUAD83

4.5 Solving Quadratic Equations by Finding Square
Roots
124
Solve the Quadratic Equation.
Example 6)
4
4
QUAD83

4.5 Solving Quadratic Equations by Finding Square
Roots
125
Properties of Square Roots (a gt 0, b gt 0)
Product Property
Quotient Property
Example)
Example)
4.5 Solving Quadratic Equations by Finding Square
Roots
126
Simplify the expression.
Example 1)
Example 2)
Example 3)
Example 4)
4.5 Solving Quadratic Equations by Finding Square
Roots
127
Rationalizing the denominator eliminate a
radical as denominator by multiplying.
Simplify the expression.
Which means No radicals (square roots) in
the denominator.
Example 5)
Example 6)
Example 7)
4.5 Solving Quadratic Equations by Finding Square
Roots
128
Simplify the expression.
Example 8)
Example 9)
Example 10)
4.5 Solving Quadratic Equations by Finding Square
Roots
129
Solve the Quadratic Equation.
Example 1)
Example 2)
Example 3)
4.5 Solving Quadratic Equations by Finding Square
Roots
130
Solve the Quadratic Equation.
Example 4)
-1
-1
2
2
4.5 Solving Quadratic Equations by Finding Square
Roots
131
Pythagorean Theorem
c
a
b
132
Solve the Quadratic Equation.
Example 5)
1
1
4.5 Solving Quadratic Equations by Finding Square
Roots
133
Solve the Quadratic Equation.
3
Example 6)
-5
-5
4.5 Solving Quadratic Equations by Finding Square
Roots
134
Solve the Quadratic Equation.
Example 7)
4.5 Solving Quadratic Equations by Finding Square
Roots
135
Reflection on the Section
For what purpose would you use the product or
quotient properties of square roots when solving
quadratic equations using square roots?
4.5 Solving Quadratic Equations by Finding Square
Roots
136
WARM-UP
Vertex form
Graph the Quadratic Equation

• Graph
• Find Vertex _________
• Identify
• Axis of Symmetry _________
• d) Find Solutions
• x-intercepts __________
• e) Opens UP or DOWN
• f) Compare to y x

5.1 Graphing Quadratic Functions
137
4.6
Complex Numbers
What you should learn
Solve quadratic equations with complex solutions
and
Goal
1
Goal
2
Perform operations with complex numbers.
4.6 Complex Numbers
138
Imaginary numbers i , defined as
Note that
The imaginary number i can be used to write the
square root of any negative number.
4.6 Complex Numbers
139
Simplify the expression.
Error
Example 1)
Go to MODE
then down to
Now, try again.
Notice
Example 2)
Example 3)
Example 4)
4.6 Complex Numbers
140
Adding and Subtracting Complex Numbers
Example 1)
Example 2)
Multiplying Complex Numbers
Example 3)
Dividing Complex Numbers
Example 4)
4.6 Complex Numbers
141
Solve the Quadratic Equation.
Example 1)
15
15
NO REAL SOLUTIONS
PRGM down to QUAD A ? B ? C?
4.6 Complex Numbers
142
Solve the Quadratic Equation.
Example 1)
15
15
Graphing Calculator PRGM down to QUAD83 A
? B ? C?
NO REAL SOLUTIONS
4.6 Complex Numbers
143
Reflection on the Section
Describe the procedure for each of the four basic
operations on complex numbers.
assignment
5.4 Complex Numbers
144
Write the expression as a Complex Number in
standard form.
Example 1)
4.6 Complex Numbers
145
Simplify the expression.
Example 1)
Example 2)
Example 3)
Example 4)
5.4 Complex Numbers
146
Additional Example 1
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
147
Additional Example 2
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
148
Additional Example 3
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
149
Additional Example 4
Graph the Quadratic Equation
y - 2(x 1)(x 3)
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
150
Additional Example 5
Vertex form
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
151
Additional Example 6
Vertex form
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens UP or DOWN
5.1 Graphing Quadratic Functions
152
Solve the Quadratic Equation
Ex 1)
Ex 3)
Ex 2)
Ex 4)
153
4.7
Completing the Square
What you should learn
Goal
1
Solve quadratic equations by completing the
square.
Goal
2
Use completing the square to write quadratic
functions in vertex form.
4.7 Completing the Square
154
Completing the Square
Find the value of c that makes
a perfect square trinomial. Then
write the expression as a square of a binomial.
Perfect square trinomial
square of a binomial
4.7 Completing the Square
155
Solving a Quadratic Equation
Solve by Completing the Square
Ex)
2
4.7 Completing the Square
156
Solving a Quadratic Equation
Solve by Completing the Square
Ex)
4
4
4
4
4.7 Completing the Square
157
Solving a Quadratic Equation
Write the equation in Vertex Form
Ex)
158
Reflection on the Section
Why was completing the square used to find the
maximum value of a function?
assignment
4.7 Completing the Square
159
Pre-Stuff Simplify for x.
4.8 The Quadratic Formula and the Discriminant
160
4.8
The Quadratic Formula and the Discriminate
What you should learn
Goal
1
Solve quadratic equations using the quadratic
formula.
Goal
2
Use quadratic formula to solve real-life
situations.
4.8 The Quadratic Formula and the Discriminant
161
Quadratic Formula When solving a quadratic
equation like
use
4.8 The Quadratic Formula and the Discriminant
162
Solve the Quadratic Equation.
Example 1)
NO REAL SOLUTIONS
What we are going to do now is to use the
QUADRATIC FORMULA to find the Imaginary
solutions.
Identify A B C
1
2
12
Plug them in to the formula
4.6 Complex Numbers
163
Solve the Quadratic Equation.
Example 1 continued)
or
and
4.6 Complex Numbers
164
Solve the Quadratic Equation.
Example 2)
NO REAL SOLUTIONS
What we are going to do now is to use the
QUADRATIC FORMULA to find the Imaginary
solutions.
Identify A B C
-2
-12
-22
Plug them in to the formula
4.6 Complex Numbers
165
Solve the Quadratic Equation.
Example 2 continued)
or
and
4.6 Complex Numbers
166
How to use a Discriminant to determine the number
of solutions of a quadratic equation.
discriminant
if , then 2 real solutions.
if , then 1 real solutions.
if , then 2 imaginary
solutions.
Example 1)
substitute
156
So, 2 Real Solutions
4.8 The Quadratic Formula and the Discriminant
167
if , then 2 real solutions.
if , then 1 real solutions.
if , then 2 imaginary
solutions.
Example 2)
substitute
20
So, 2 Real Solutions
4.8 The Quadratic Formula and the Discriminant
168
Solve the Quadratic Equation.
Example 3)
15
15
What we are going to do now is to use the
QUADRATIC FORMULA to find the Imaginary
solutions.
NO REAL SOLUTIONS
Identify A B C
2
0
16
Plug them in to the formula
4.6 Complex Numbers
169
Solve the Quadratic Equation.
Example 3 continued)
4.6 Complex Numbers
170
Pre-Stuff Simplify for x.
4.8 The Quadratic Formula and the Discriminant
171
Pre-Stuff Solve for x.
Ex3)
Factor out GCF
Ex1)
Ex2)
4.8 The Quadratic Formula and the Discriminant
172
Solve the Quadratic Equation.
Example 1)
Split this.
You can put these into calculator for Decimal
answers.
4.8 The Quadratic Formula and the Discriminant
173
Solve the Quadratic Equation.
Example 2)
Split this.
4.8 The Quadratic Formula and the Discriminant
174
Reflection on the Section
Describe how to use a discriminant to determine
the number of solutions of a quadratic equation.
discriminant
if , then 2 real solutions.
if , then 1 real solutions.
if , then 2 imaginary
solutions.
assignment
4.8 The Quadratic Formula and the Discriminant
175
4.9
Graphing and Solving Quadratic Inequalities
What you should learn
Goal
1
Graph quadratic inequalities in two variables.
Goal
2
Solve quadratic inequalities in one variable.
4.9 Graphing and Solving Quadratic Inequalities
176
Reflection on the Section
What is the procedure used to solve quadratic
inequality in two variables?
assignment
4.9 Graphing and Solving Quadratic Inequalities
177
Modeling with Quadratic Functions
What you should learn
Goal
1
Write quadratic functions given characteristics
of their graphs.
Goal
2
Use technology to find quadratic models for data.
Modeling with Quadratic Functions
178
Reflection on the Section
Give four ways to find a quadratic model for a
set of data points.
assignment
Modeling with Quadratic Functions