Do Now! sheet date Name. Do Now! sheet date Name

Do Do NowNow!!

sheetdate date

Do Do NowNow!!

sheetdate date

date date

10 – 24 - 201310 – 24 - 2013 Do Do NowNow!!122 xx

Factor the trinomial.

( ) ( )

b) 1252 2 xx

FCTPOLY

10 – 24 - 201210 – 24 - 2012 Do Do NowNow!!342 xx

Factor the trinomial.)2)(4( xxa) c)

( ) ( )

F.O.I.L.( distribute )

20244 2 xx )8)(5(2 xxb) d)

Graph and compare to 2xy

a) Graph

b) Find Vertex _________

c) Identify

Axis of Symmetry _________

d) Find “Solutions”

x-intercepts __________

e) Opens UP or DOWN

f) Compare to y = x 2

Vertex shifts ______

Width _______

10 – 23 - 201210 – 23 - 2012 Do Do Now!Now!142 xxy

Graph and compare to

4)2(4 2 xy

2xy a) Graph

c) Identify

e) Opens UP or DOWN

f) Compare to y = x

Width _______

10 – 24 - 201210 – 24 - 2012

10 – 25 - 201210 – 25 - 2012 Do Do NowNow!!

Graph and compare to )1)(3(2 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

f) Compare to y = x2

Width _______

ThursdayThursday

Modeling ProjectileProjectile Objects

When an object is projected, its height h (in feet) above the ground after t seconds can be modeled by the function

02 14016 htth

where is the object’s initial height (in feet).0h

Baseball Hit A baseball is hit by a batter. 1.) Write an equation giving the ball’s height h (in feet) above the ground after t seconds. 2.) Graph the equation. 3.) During what time interval is the ball’s height above 3 feet?

10 – 29 - 201110 – 29 - 2011 Do Do NowNow!!

872 xx

( ) ( )

a) Find Vertex _________

b) Identify

e) Opens UP or DOWN

f) Compare to y = x

Width _______

442 xxy

)6)(3( xxF.O.I.L.

3)1(2)( 2 xxg

Rewrite in Quadratic Standard form

11 – 29 - 201211 – 29 - 2012 Do Do Now!Now!

40412 2 xx

1) 0583 2 xx

Solve the equation

FCTPOLY QUAD83PRGM PRGM

4 ( x – 2 ) ( 3x + 5 )1.6666666666666

Do Do NowNow!!

11 – 30 - 201211 – 30 - 2012

1649)( 2 xxf1. Solve the equation.

1272 xxy2. Find the x-intercepts.

24100 2 xx3. Find the Zero’s

140 2 xx

4. What are the Solutions of the equation?

QUAD83 QUAD83

QUAD83

5714. 3

236.44

ANDAND

10 – 31 - 201210 – 31 - 2012

The Area of the rectangle is 30. What are the lengths of the sides?

3x + 1

x 3x + 1( ) = 30

DDo o NNooww!!

WWeeddnneessddaayy

Modeling ProjectileProjectile Objects

When an object is projected, its height h (in feet) above the ground after t seconds can be modeled by the function

02 14016 htth

Baseball Hit A baseball is hit by a batter. 1.) Write an equation giving the ball’s height h (in feet) above the ground after t seconds. 2.) Graph the equation. 3.) During what time interval is the ball’s height above 3 feet?

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

0216 hth

CLIFF DIVING A cliff diver dives off a cliff 40 feet above water.

1.) Write an equation giving the diver’s 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?

Student

HTWindow re-set….

Xmin =

Xmax =

Xscl =

Ymin =

Ymax =

Yscl =

0216 hth

CLIFF DIVING A cliff diver dives off a cliff 40 feet above water.

1.) Write an equation giving the diver’s 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?

Student

HTWindow re-set….

Xmin =

Xmax =

Xscl =

Ymin =

Ymax =

Yscl =

1 1.50.5

Modeling DroppedDropped Objects

0216 hth

CLIFF DIVING A cliff diver dives off a cliff 40 feet above water. 1.) Write an equation giving the diver’s height h (in feet) above the water after t seconds. 2.) Graph the equation. 3.) How long is the diver in the air?

0216 hth

where is the object’s initial height (in feet).

Let’s assume that the cliff is 40 feet high.

0216 hth

Let’s assume that the cliff is 40 feet high. Write an equation?

40160 2 t

Make the substitution.

Graph it.

0216 hth

40160 2 t

TIME seconds

Graph it.

1 1.5.5

0216 hth

40160 2 t

+1.58 secondsHint: we want SOLUTIONS.

Think about… what are we trying to find?

Graph it.

1 1.5.5

-1.58 seconds

How long will the diver be in the air?

Changing the world takes more than everything any one person knows.

But not more than we know together.

So let's work together.

11 – 8 - 201211 – 8 - 2012 Do Do NowNow!!

Quadratic Formula

Solve using the Quadratic Formula

01222 xx

Example 1)

Identify:A:

Plug them in to the formula

)12)(1(4)2()2( 2 x

How to use a Discriminant to determine the number of solutions of a quadratic equation.

discriminant

*if , (positive) then 2 real solutions.042 acb

*if , (zero) then 1 real solutions.042 acb

*if , (negative) then 2 imaginary solutions.042 acb

ASSIGNMENTPAGE 279 # 12 -27 ALL Complex Numbers ( i )

# 3 – 6, 31 – 33, 40 - 42,

MONDAY , NOV. 7thMONDAY , NOV. 7th

DiscriminantHow many solutions

QUAD83Solve the equation

Quadratic Formula

Collaborative Activity Sheet 1Chapter 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 container’s 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?

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 ball’s 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?

0216 hvtth

Collaborative Activity SheetChapter 4 Solving – Graphing Quadratic functions

0216 hvtth

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?

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 x3.) Solve the equation. By what distance x should the length and width of the parking lot be expanded?

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?

3008016 2 tty

0216 hvtth

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 ball’s 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?

0216 hvtth

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 xc.) Solve the equation. By what distance x should the length and width of the parking lot be expanded?

3.) In a football game, a defensive player jumps up to block a pass by the opposing team’s 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 player’s teammates have to intercept the ball before it hits the ground?

0216 hvtth

4.) The aspect ratio of a widescreen TV is the ratio of the screen’s width to its height, or 16:9 . 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 screen’s diagonal.) Draw a picture.

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?

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

962 xxy

a) Graph

b) Find Vertex

c) Axis of Symmetry

d) x-intercepts

e) Opens UP or DOWN

1 2 xxy

a) Graph

b) Find Vertex

c) Axis of Symmetry

d) x-intercepts

e) Opens UP or DOWN

3 2 xxy

a) Graph

b) Find Vertex

c) Axis of Symmetry

d) x-intercepts

e) Opens UP or DOWN

3 2 xxy

a) Graph

b) Find Vertex

c) Axis of Symmetry

d) x-intercepts

e) Opens UP or DOWN

1 2 xxy

22122 2 xxy

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

22122 2 xxy

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

962 xxy

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

442 xxy

142 2 xxy

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

vertex(1, 3)

Axis of Symmetry

x-intercepts

322 xxy

a) Graph

b) Find Vertex

c) Identify

Axis of Symmetry

x-intercepts

e) Opens UP or DOWN

vertex(1, -4)

Axis of Symmetry

x-intercepts

232 2 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

142 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

Quiz 4.1

442 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

Quiz 4.1 RETAKE

442 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

Quiz 4.1 RETAKE

4)2(4 2 xy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

Quiz 4.2

)4)(2(3 xxy

a) Graph

c) Identify

e) Opens UP or DOWN

Width _______

4.14.1 Graphing Quadratic Functions

What you should learn:GoalGoal 11

GoalGoal 22

Graph quadratic functions.

Use quadratic functions to solve real-life problems.

4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form

Vocabulary

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.

Quadratic Functions in Standard FormStandard Form is written as

,Where a

cbxaxy 2

PARENT FUNCTION for Quadratic Functions

The parent function for the family of all quadratic functions is f(x) = . 2x

Axis of Symmetry

divides the parabola into mirror images and passes through the vertex.

Vertex is (0, 0)

PROPERTIES of the GRAPH of cbxaxy 2

Characteristics of this graph are:

1.The graph opens up if a > 0

2.The graph open down if a < 0

3. The graph is wider than if

4. The graph is narrower than if

5. The x-coordinate of the vertex is

6. The Axis of Symmetry is the vertical line

2xy 2xy

Example 1A Graphing a Quadratic Function

342 xxy

Vertex X Y 2 -1

Solutions 3 1

Graphing Calculator

down to QUAD83

33Axis of SymmetryThe line x = 2

Example 1B Graphing a Quadratic Function

122 xxy

Vertex X Y -1 0

Solutions -1 -1

Graphing Calculator

down to QUAD83

11Axis of SymmetryThe line x = -1

Example 1C Graphing a Quadratic Function

582 2 xxy

Vertex X Y -2 3

Solutions -3.225 -.775

Graphing Calculator

down to QUAD83

Axis of SymmetryThe line x = -2

Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section

How is the Vertex of a parabola related to its Axis of Symmetry?

assignmentassignment

Example 1 Graphing a Quadratic Function

The coefficients are a = 1, b = -4, c = 3

Since a > 0, the parabola opens up.

To find the x-coordinate of the vertex, substitute 1 for a and -4 for b in the formula:

x = -b

2a = -

2(1) = 2

342 xxy

To find the y-coordinate of the vertex, substitute 2 for x in the original equation, and solve for y.

y = x2 - 4x + 3

=(22) - 4(2) + 3

= 4 - 8 + 3

The vertexvertex 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.

Additional Example 1

a) y = x2 + 2x +1

b) y = -2x2 - 8x - 5

4.24.2 Graphing Quadratic Functions in Vertex or Intercept Form

GoalGoal 22

Graph quadratic functions in VERTEX form or INTERCEPT form.

Find the Minimum value or the Maximum value

4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form

F.O.I.L.Review theReview the)2)(2( xx1. )4)(4( xx2.

)5)(5( xx

2)5( x3.

Example Example 1A1A Graphing a Quadratic Function in

Vertex formVertex formkhxay 2)(

Vertex ( h, k )

So, Vertex ( 6, 1 )

Vertex FormVertex Form

1)6)(6( xx

136662 xxx

37122 xx

37122 xxy

Split and FOIL

Combine like terms

use QUAD83 to find the Solutions and confirm Vertex

Rewrite in Standard FormRewrite in Standard Form

1)6( 2 xyGraph

Example Example 1B1B Graphing a Quadratic Function in

Vertex formVertex form

Graph y = 2(x-3)2 - 4

khxay 2)(

Vertex ( h, k )

So, Vertex ( 3, -4 )

4)3)(3(2 xx

4)96(2 2 xx

418122 2 xx

14122 2 xxy

Split and FOIL

distribute

Combine like terms

use QUAD83 to find the Solutions and confirm Vertex

Rewrite in Standard FormRewrite in Standard Form

Example Example 1B1B Graphing a Quadratic Function in

Vertex formVertex form

Graph y = 2(x-3)2 - 4

Use the form y = a(x-h)2 + k, wherea = 2, h = 3, and k = -4. Since a>0,the parabola opens up.

khxay 2)(

Vertex ( h, k )

So, Vertex ( 3, -4 )

Since, 1a the parabola is narrower than 2xy

Now, Graph it on the calculator.

Vertex is a Minimum Pt.

Plot the vertexvertex (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

Example 2Example 2

Graph y = (x+2)2 - 3

Vertex

Opens UP, vertex is MIN

Since, 1a

the parabola is the same width as 2xy

Axis of Symmetry x = -2

“Solutions” x-intercepts(-3.73, 0) and (-.268, 0)

ZERO’s

(-2, -3)

4.24.2 Graphing Quadratic Functions in Vertex or Intercept Form

What you should learn:

GoalGoal 11 Graph quadratic functions in ….. INTERCEPTINTERCEPT form.

VERTEX form to STANDARD formReview Review rewriterewrite

3)2( 2 xy1. 5)4(2 2 xy2.

3)2)(2( xx

3)44( 2 xx

742 xxy

5)4)(4(2 xx

5)168(2 2 xx

27162 2 xxy

532162 2 xx

continued

DO THESE PROBLEMS

Additional Example Additional Example 33

VertexVertex

OpensOpens

Since, 1a

2)1( 2 xyGraph

Axis of SymAxis of Sym x = 1

SolutionsSolutions -.414 and 2.414

DOWN, vertex is MAX

WidthWidth

ShiftShift Rt 1 --- Up 2

( 1, 2)

Graph y = (x + 3)(x - 5)

Example 1Example 1

Vertex ( 1, -16)

Opens UP, vertex is MIN

Since, 1a

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.

Intercept Form

Example 2Example 2

Vertex ( 1, 9)

Opens DOWN, vertex is MAX

Since, 11 a

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.

Intercept Form)2)(4( xxyGraph

Example 3

Graphing a Quadratic Function in InterceptIntercept form

Graph y = -1

2(x - 1)(x + 3)

Use the intercept form y = a(x - p)(x - q), where

a = -1

2, p = 1, and q = -3

The x-intercepts are (1,0) and (-3,0)

The axis of symmetry is x = -1

InterceptIntercept form

Cont’ Example 3

The x-coordinate of the vertex is -1. The y-coordinate is:

y = -1

2(-1 - 1)(-1 + 3) = 2

Graph the parabola.

InterceptIntercept form

y = - (x – 1)(x + 3)

y = (x + 1)(x - 3)

ExampleExample 4 4Writing Quadratic Functions in StandardStandard Form

Write y = 2(x – 3)(x + 8) in standard form

y = 2(x - 3)(x + 8) Write original equation

= 2(x2 + 5x - 24) Multiply using FOIL

= 2x2 + 10x - 48 Use distributive property

Write the quadratic function in standard form

y = (x + 1)2 - 8

y = x2 + 2x - 7

y = -4(x + 2)(x - 2)

y = -4x2 + 16

Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this

equation?

4.34.3 Solving Quadratic Equations by Factoring

GoalGoal 22

Factor quadratic expressions and solve quadratic equations by factoring.

Find zeros of quadratic functions.

4.3 Solving Quadratic Equations by Factoring4.3 Solving Quadratic Equations by Factoring

862 xx 122 xx

(x + 2 )(x + 4)

Directions: Factor the expression.

FCTPOLY

DEGREE: 2

COEF. OF X^2?

PROGRAM

FCTPOLY

DEGREE: 2

COEF. OF X^2?

PROGRAM

)4)(3( xx

Example 1) Example 2)

52 xxDirections: Factor the expression.

FCTPOLY

DEGREE: 2

COEF. OF X^2?

PROGRAM

)5( 2 xx

This means cannot be factored

822 xx

(x - 2 )(x + 4)

FCTPOLY

DEGREE: 2

COEF. OF X^2?

PROGRAM

Example 4)Example 3)

03652 xx

(x - )(x + )49

Directions: Solve the equation.

QUAD83PROGRAM

SOLUTIONS 9-4

FCTPOLY

DEGREE: 2

COEF. OF X^2?

PROGRAM

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.

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: 862 xx

Step 1: Enter x as the first term of each factor.Step 1: Enter x as the first term of each factor.

( x )( x )

Step 2: List pairs of factors of the constant, 8.Step 2: List pairs of factors of the constant, 8.

Factors of 8 8, 1 4, 2 -8, -1 -4, -2

Step 3: Try various combinations of these factors.Step 3: Try various combinations of these factors.

Possible Factorizations

( x + 8)( x + 1)

( x + 4)( x + 2)

( x - 8)( x - 1)

( x - 4)( x - 2)

Sum of Outside and Inside Products (should equal 6x)

x + 8x = 9x

2x + 4x = 6x

-x - 8x = - 9x

-2x - 4x = - 6x

Example 2:782 xx

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.

( x + 7)( x + 1)

( x - 7)( x - 1)

Sum of Outside and Inside Products (should equal 8x)

x + 7x = 8x

-x - 7x = - 8x

862 xx 52 xx

652 xx 9922 xx

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.

Look at the 2nd sign:

(x + )(x + )

(x - )(x - )

(x - )(x + )

(x + )(x - )

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.

))((22 BABABA

Example 1) 42 x

2.) or you could look at this as the trinomial…

402 xx

1.) Difference of the Two Squares,22 2x

Factoring the Difference of Two Squares

Difference of the Two Squares,

1.) 162 x We must express each term as the square of some monomial. Then use the formula for factoring

22 BA 162 x 22 4x )4)(4( xx

You can check it by using FOIL on the binomial.

2.) or you could look at this as the trinomial…

1602 xx

(x )(x )+ - 44

Example 2:

672 xx 64122 xx

72222 xx 16152 xx

(x + )(x + )

(x - )(x - )

(x - )(x + )

(x + )(x - )

16 4 16

4 18 1 16

22 128 yxyx 22 283 yxyx

(x + )(x + ) (x - )(x + )2y6y 4y 7y

Factor.

Factoring Trinomials whose Leading Coefficient is NOT one.

Objectives

1. Factor trinomials by trial and error.1. Factor trinomials by trial and error.

4.44.4 Solving Quadratic Equations by Factoring

Factoring by the Trial-and-Error MethodFactoring by the Trial-and-Error Method

How would we factor: 28203 2 xx

Notice that the leading coefficient is 3, and we can’t divide it out

( 3x )( x )

example: 28203 2 xx

Step 1: find the two First terms whose product is .

( 3x )( x )

Step 2: Find two Last terms whose product is 28.

Factors of 28 -1(-28) - 2(-14) - 4(-7)

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.

( 3x - 1)( x - 28)

( 3x - 28)( x - 1)

( 3x - 2)( x - 14)

( 3x - 14)( x - 2)

Sum of Outside and Inside Products (should equal -20x)

-84x - x = - 85x

-3x - 28x = - 31x

-42x - 2x = - 44x

-6x - 14x = - 20x

( 3x - 4)( x - 7)

( 3x - 7)( x - 4)

-21x - 4x = - 25x

-12x - 7x = - 19x

example: 3108 2 xx

Step 1: find the two First terms whose product is .

( 8x )( x )

Step 2: Find two Last terms whose product is -3.

Factors of -3 1(-3) -1(3)

( 4x )(2 x )

( 8x + 1)( x - 3)

( 8x - 3)( x + 1)

( 8x - 1)( x + 3)

( 8x + 3)( x - 1)

Sum of Outside and Inside Products (should equal -10x)

-24x + x = - 23x

8x - 3x = 5x

24x - x = 23x

- 8x + 3x = - 5x

( 4x + 1)(2 x - 3)

( 4x - 3)( 2x + 1)

-12x + 2x = - 10x

4x - 6x = - 2x

( 4x - 1)( 2x + 3)

( 4x + 3)( 2x - 1)

12x - 2x = 10x

-4x + 6x = 2x

169 2 xx 7124 2 xx

276 2 xx 1572 2 xx

(3x + )(3x + )

(2x - )(3x - )

(2x + )(2x - )

(2x + )(x - )

11 7 1

1 2 3 5

Factoring Trinomials whose Leading Coefficient is NOT one.

Ex 1) Ex 2)

Ex 3)Ex 4)

The Zero-Product PrincipleIf 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, ( ???)(###) = 0Example)

Then either (???) is zero, or (###) is zero.

Example 1)

0)2)(5( xxAccording to the principle,

this product can be equal to zero, if either…

0)5( x 0)2( xor+5 +5

The resulting two statements indicate that the solutions are 5 and 2.

Solve the equation.

Example 2)

0472 2 xx

Factor the Trinomial using the methods we know.

0)12( x 0)4( xor

x = 1/2

- 4 - 4

x = - 4

The resulting two statements indicate that the solutions are 1/2 and - 4.

Solve the Equation (standard form) by Factoring

(2x )(x ) = 0- +1 4

2x = 1

Example 3) 962 xxMove all terms to one side with zero on the other. Then factor.

0)3( x+3 +3

The resulting two statements indicate that the solutions are 3.

(x )(x ) = 0- -3 3

0962 xx

The trinomial is a perfect square, so we only need to solve once.

Solve the Equation (standard form) by Factoring

Factoring out the greatest common factor.

But, before we do that…do you remember the Distributive Property?

)32(5 xx

xx 1510 2

When factoring out the GCF, what we are going to do is UN-Distribute.

What I mean is that when you use the Distributive Property, you are multiplying.

But when you are factoring, you use division.

example: 305 2 xFactor:

1st determine the GCF of all the terms.

52nd pull 5 out, and divide both terms by 5.

)6(5 2 x

Factor each polynomial using the GCF.Factor each polynomial using the GCF.

xx 54 ex) )5( 3 xx

xx 217 2 ex) )3(7 xx

xxx 10515 23 ex)

)23(5 2 xxx

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: xxx 42153 23

factor out 3x )145(3 2 xxx

3x(x )(x )2 7-+

54333 2 aa 234 96262 xxx

(a - )(a - ) (x + )(x - )2 9 3 1622x3

18112 aa 48132 xx3( ) ( ) 22x

Factor.Example 2: Example 3:

Factoring GCF First

Factor 2x2 - 12x + 18

2x2 - 12x + 18 = 2(x2 -6x +9)

= 2(x - 3)(x - 3)

= 2(x - 3)2

Step1) GCF

Example 1) xx 33 3

Factoring out the GCF and then factoring the Difference of two Squares.

What’s the GCF?

)1(3 2 xx

22 )1()(3 xx

)1)(1(3 xxx

))((22 BABABA

Example 2) xx 312 3

Factoring out the GCF and then factoring the Difference of two Squares.

What’s the GCF?

)14(3 2 xx

22 )1()2(3 xx

)12)(12(3 xxx

))((22 BABABA

Factor the quadratic expression.

2x2 - 50

2(x + 5)(x - 5)5x2 + 10x + 5

5(x + 1)2

4y2 + 4y

4y(y + 1)

Additional Examples

Example 3:

2)3( x

Factoring Perfect Square Trinomials

962 xx

(x )(x )+ + 3 3

Since both binomials are the same you can say

Example 4:

2)5( x

Factoring Perfect Square Trinomials

25102 xx

(x )(x )- - 5 5

Since both binomials are the same you can say

Example 5:

y = x2 + x - 20

-5 ; 4

y = x2 - 1

y = x2 + 3x - 10

-5 ; 2

Example 6:

What must be true about a quadratic equation before you can solve it using the

zero product property?

# 47 – 88, 90

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4.54.5 Solving Quadratic Equations by Finding Square Roots

GoalGoal 22

Solve quadratic equations by finding square roots.

Use quadratic equations to solve real-life problems.

4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots

Simplify the expression.

81Example 1)

24Example 2) 899.4

3Example 3)433.

45Example 4) 4553

Solve the Quadratic Equation.

0162 xExample 1)

243.4x

Example 2)

464.3x

Example 3)

363 2 x-36-36

QUAD83

-18-18

0182 x

0363 2 x

QUAD83

40)1( 2 xExample 4)

-40 -40

03922 xx

325.7x

40)1)(1( xx

40122 xx

325.5&

QUAD83

10)3(2 2 xExample 5)

-10 -10

08122 2 xx

764.x4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots

10)3)(3(2 xx

10)96(2 2 xx

236.5&

QUAD83

1018122 2 xx

4)2( 2 xExample 6)

042 xx

4)2)(2( xx

4)44( 2 xx

QUAD83

4442 xx

Properties of Square Roots (a > 0, b > 0)

Product Property

Quotient Property

Example)

81Example 1)

24Example 2) 64 64 62

3Example 3)

45Example 4) 95 95 53

1Example 5)

2Example 6)

3Example 7)

Rationalizing the denominatorRationalizing the denominator – eliminate a radical as denominator by multiplying.

Which means… No radicals (square roots) in the denominator.

33 Example 8)

102 Example 9) 20 54 52

634 Example 10)184 294

0162 xExample 1)

Example 2)

363 2 x

Example 3)

0182 x162 x

0363 2 x

1712 2 xExample 4)

162 2 x2 2

Pythagorean Theorem

222 cba

40)1(2 2 xExample 5)

20)1( 2 x

1 2xExample 6) 3

21)5( 2 x

21)5( x

xxExample 7)

1442 x

For what purpose would you use the product or quotient properties of square roots when solving quadratic equations

using square roots?

WARM-UP WARM-UP Vertex Vertex formform

5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions

Graph the Quadratic Equation

4)3(2 2 xy

a) Graph

c) Identify

e) Opens UP or DOWN

f) Compare to y = x

4.64.6 Complex Numbers

GoalGoal 22

Solve quadratic equations with complex solutions and…

…Perform operations with complex numbers.

4.6 Complex Numbers4.6 Complex Numbers

Imaginary numbers

i , defined as 1i

Note that

The imaginary number i can be used to write the square root of any negative number.

4Example 1)

12Example 2) i464.3

Example 3)i433.

45Example 4) i708.6

Error Go to MODE then down to

Now, try again.4

Notice Notice

Adding and Subtracting Complex Numbers )23()4( ii Example 1)

)81()57( ii Example 2) i36

)7)(5( iiExample 3)

Multiplying Complex Numbers

Dividing Complex Numbers

Example 4)

i6.22.

1512 2 xExample 1)

+15 +15

0162 2 x83QUAD

ix 828.2

NO REAL SOLUTIONSNO REAL SOLUTIONS

down to QUAD

1512 2 xExample 1)

+15 +15

0162 2 x83QUAD

Graphing Calculator

down to QUAD83

Describe the procedure for each of the four basic operations on complex numbers.

Write the expression as a Complex Number in standard form.

Example 1)

)21)(21( ii

4Example 1)

12Example 2) 341 32i

Example 3)

45Example 4) 951 53i

122 xxy

Vertex

Axis of Symmetry

Opens: UP or DOWN

b) y = -2x2 - 8x - 5

Vertex

Axis of Symmetry

Opens: UP or DOWN

Graph y = (x + 3)(x - 5)

Additional Example 3Additional Example 3

Vertex

Axis of Symmetry

Opens: UP or DOWN

y = - 2(x – 1)(x + 3)

Additional Example 4Additional Example 4

Vertex

Axis of Symmetry

Opens: UP or DOWN

Additional Example 5 Additional Example 5

Vertex Vertex formform

Vertex

Axis of Symmetry

Opens: UP or DOWN

4)3(2 2 xy

Vertex Vertex formform

Vertex

Axis of Symmetry

Opens: UP or DOWN

2)1( 2 xy

Solve the Quadratic Equation

16)4( 2 xEx 1)

2)6(3 2 xEx 2) 437)1( 2 x

Ex 3) 160)6(4 2 x

4.74.7 Completing the Square

GoalGoal 22

Solve quadratic equations by completing the square.

Use completing the square to write quadratic functions in vertex form.

4.7 Completing the Square4.7 Completing the Square

64162 xx

Completing the Square

22 bb xbxx

Find the value of c that makes a perfect square trinomial. Then write the expression as a square of a binomial.

cxx 6.12

cxx 162

Perfect square trinomial

28x square of a binomial

Solving a Quadratic Equation

Solve by Completing the Square

0222 xx

xx 22 = 2

Solve by Completing the Square

0164 2 xx

232 xx

4 4 4 4Ex)

Write the equation in Vertex Form

742 xxyEx)

Why was completing the square used to find the maximum value of a function?

Pre-Stuff…Simplify for x.

ix 449.21 ix 449.21

4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant

4.84.8 The Quadratic Formula and the Discriminate

GoalGoal 22

Solve quadratic equations using the quadratic formula.

Use quadratic formula to solve real-life situations.

Quadratic Formula

When solving a quadratic equation like

02 cbxax

01222 xxExample 1)

83QUAD

NO REAL SOLUTIONSNO REAL SOLUTIONS What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.

Identify:A:

12 Plug them in to the formula

Example 1 continued)

)12)(1(4)2()2( 2 x

ix 317.31

ori317.31and

022122 2 xxExample 2)

83QUAD

NO REAL SOLUTIONSNO REAL SOLUTIONS What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.

Identify:A:

-22 Plug them in to the formula

)22)(2(4)12()12( 2

ix 414.13

ori414.13and

How to use a Discriminant to determine the number of solutions of a quadratic equation.

discriminant

*if , then 2 real solutions.042 acb

*if , then 2 imaginary solutions.042 acb

Example 1) 03122 xxacb 42 substitute

)3)(1(4)12( 2

156 So, 2 Real Solutions

Example 2) 0422 xxacb 42 substitute

)4)(1(4)2( 2

20 So, 2 Real Solutions

1512 2 xExample 3)

+15 +15

0162 2 x83QUAD

What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.

Identify:A:

16 Plug them in to the formula

)16)(2(4)0()0( 2 x

828.2x

Pre-Stuff…Simplify for x.

)21(2 x )21(2 x

Pre-Stuff… Solve for x.

)21(2 x

322 Ex1)

Factor out GCF

21015Ex2)

052 2 xxExample 1)

)5)(2(4)1(1 2 x

You can put these into calculator for Decimal answers.

Split this….Split this….

0169 2 xxExample 2)

)1)(9(4)6(6 2 x

Split this….

Describe how to use a discriminant to determine the number of solutions of a

quadratic equation.

discriminant

4.94.9 Graphing and Solving Quadratic Inequalities

GoalGoal 22

Graph quadratic inequalities in two variables.

Solve quadratic inequalities in one variable.

4.9 Graphing and Solving Quadratic Inequalities4.9 Graphing and Solving Quadratic Inequalities

What is the procedure used to solve quadratic inequality in two variables?

4.9 Graphing and Solving Quadratic Inequalities4.9 Graphing and Solving Quadratic Inequalities

Modeling with Quadratic Functions

GoalGoal 22

Write quadratic functions given characteristics of their graphs.

Use technology to find quadratic models for data.

Modeling with Quadratic FunctionsModeling with Quadratic Functions

Give four ways to find a quadratic model for a set of data points.

Modeling with Quadratic FunctionsModeling with Quadratic Functions

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