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

Do Now! sheet Name date date date date date date

Do Now! sheet Name date date date date

Do Now! 10 – 24 - 2013 ( ) ( ) Factor the trinomial.
b) ( ) ( ) PRGM FCTPOLY

Do Now! 10 – 24 - 2012 ( ) ( ) F.O.I.L.( distribute )
Factor the trinomial. a) c) ( ) ( ) b) d)

10 – 23 - 2012 Do Now! Graph Vertex shifts ______ Width _______
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 _______

10 – 24 - 2012 2 Vertex shifts ______ Width _______
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 _______

Thursday Do Now! 10 – 25 - 2012 2 Vertex shifts ______ Width _______
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 _______

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 object’s initial height (in feet). 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?

Do Now! 10 – 29 - 2011 ( ) ( ) Factor the trinomial. 1 a) 2 F.O.I.L.
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)

Do Now! 11 – Factor the trinomial. Solve the equation 1) 2) PRGM FCTPOLY PRGM QUAD83 1 4 ( x – 2 ) ( 3x + 5 )

Do Now! 11 – 30 - 2012 1. Solve the equation.
2. Find the x-intercepts. QUAD83 QUAD83 AND 3. Find the Zero’s 4. What are the Solutions of the equation? QUAD83 QUAD83 AND AND

Wednesday Do Now! x 3x + 1 x 3x + 1 ( ) 10 – 31 - 2012 = 30
The Area of the rectangle is 30. What are the lengths of the sides? x 3x + 1 x ( ) 3x + 1 = 30

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 object’s initial height (in feet). 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
Student 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 object’s initial height (in feet). 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? Window re-set…. Xmin = Xmax = Xscl = Ymin = Ymax = Yscl = HEIGHT TIME

40 30 20 HEIGHT 10 5 0.5 1 1.5 TIME

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 object’s initial height (in feet). 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?

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 object’s initial height (in feet). Let’s assume that the cliff is 40 feet high. TIME

Let’s assume that the cliff is 40 feet high. Write an equation?
Make the substitution. HEIGHT Graph it. TIME

30 Make the substitution. 20 Graph it. HEIGHT 5 .5 1 1.5 TIME seconds

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? .5 1 1.5 Hint: we want SOLUTIONS. +1.58 seconds -1.58 seconds TIME

Changing the world takes more than everything any one person knows.
But not more than we know together. So let's work together.

Do Now! 11 – 8 - 2012 Quadratic Formula
Example 1) Solve using the Quadratic Formula Identify: A: B: C: 1 2 12 Plug them in to the formula

How to use a Discriminant to determine the number of solutions of a quadratic equation.
*if , (positive) then 2 real solutions. *if , (zero) then 1 real solutions. *if , (negative) then 2 imaginary solutions.

ASSIGNMENT PAGE 279 # 12 -27 ALL PAGE 296 # 3 – 6, 31 – 33, 40 - 42,
MONDAY , NOV. 7th ASSIGNMENT PAGE # ALL Complex Numbers ( i ) PAGE # 3 – 6, 31 – 33, , Quadratic Formula QUAD83 Solve the equation Discriminant How many solutions

Collaborative Activity Sheet 1 Collaborative Activity Sheet
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 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? 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? 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 = x( 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 = x( 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? 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?

Collaborative Activity Sheet 2
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 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? 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 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? x 270 150 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 = x( 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 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?

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 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? 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? 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. 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? x 270 150 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

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

Graph and compare to Graph Find Vertex Axis of Symmetry
d) x-intercepts e) Opens UP or DOWN f) Compare to y = x 2

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

(1, 3) vertex x = 1 Graph and compare to Graph Find Vertex Identify
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

(1, -4) vertex x = 1 Graph and compare to Graph Find Vertex Identify
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

Graph and compare to Vertex shifts ______ Width _______ 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 _______

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

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

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

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

Graphing Quadratic Functions
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

Vocabulary A parabola is the U-shaped graph of a quadratic function.
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

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

Characteristics of this graph are: The graph opens up if a > 0
PROPERTIES of the GRAPH of Characteristics of this graph are: The graph opens up if a > 0 The graph open down if a < 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

Graph and compare to The line x = 2 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

Graph and compare to The line x = -1 Example 1B
Graphing a Quadratic Function Graph and compare to Graphing Calculator PRGM down to QUAD83 A= ? B= ? C=? Vertex X Y -1 1 Solutions -1 2 Axis of Symmetry The line x = -1 1 4.1 Graphing Quadratic Functions in Standard Form

Graph and compare to Example 1C Graphing a Quadratic Function
Graphing Calculator PRGM down to QUAD83 A= ? B= ? C=? Vertex X Y -2 3 -2 Solutions -3.225 -.775 -8 Axis of Symmetry The line x = -2 -5 4.1 Graphing Quadratic Functions in Standard Form

very important question
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

The coefficients are a = 1, b = -4, c = 3
Example 1 Graphing a Quadratic Function Graph and compare to 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: 4.1 Graphing Quadratic Functions in Standard Form

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

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

Additional Example 1 4.1 Graphing Quadratic Functions in Standard Form

Additional Example 2

Graphing Quadratic Functions in Vertex or Intercept Form
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 Review the F.O.I.L. 3. 1. 2. 4.2 Graphing Quadratic Functions in Vertex or Intercept form

Graphing a Quadratic Function in Vertex form
Example 1A Vertex Form Graphing a Quadratic Function in Vertex form Vertex ( h, k ) Graph So, Vertex ( 6, 1 ) 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

Graphing a Quadratic Function in Vertex form
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

the parabola is narrower than
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

Plot the vertex (h,k)  (3,-4)
Plot x-intercepts and 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

the parabola is the same
Example 2 Vertex Form Vertex (-2, -3) x = -2 Axis of Symmetry Opens UP, vertex is MIN ZERO’s “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

DO THESE PROBLEMS Graph quadratic functions in ….. INTERCEPT form.
continued Graphing Quadratic Functions in Vertex or Intercept Form 4.2 What you should learn: DO THESE PROBLEMS Graph quadratic functions in ….. INTERCEPT form. Goal 1 Review rewrite VERTEX form to STANDARD form 1. 2. 4.2 Graphing Quadratic Functions in Vertex or Intercept form

Additional Example 3 Vertex Form Graph Vertex ( 1, 2) Axis of Sym x = 1 Opens DOWN, vertex is MAX Solutions -.414 and 2.414 Shift Rt Up 2 Width Since, the parabola is the same width as 4.2 Graphing Quadratic Functions in Vertex or Intercept form

Example 1 Intercept Form x-intercepts: (-3, 0) (5, 0) To find the Vertex [ ] 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

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

The x-intercepts are (1,0) and (-3,0) The axis of symmetry is x = -1
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

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

4.2 Graphing Quadratic Functions in Vertex or Intercept form

y = - (x – 1)(x + 3) Additional Example 2
4.2 Graphing Quadratic Functions in Vertex or Intercept form

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

Write y = 2(x – 3)(x + 8) in standard form
Example 4 Writing Quadratic Functions in Standard Form Write y = 2(x – 3)(x + 8) in standard form

Additional Example 1 Write the quadratic function in standard form
4.2 Graphing Quadratic Functions in Vertex or Intercept form

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

Solving Quadratic Equations by Factoring
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

Directions: Factor the expression.
Example 1) Example 2) PROGRAM PROGRAM FCTPOLY FCTPOLY 2 DEGREE: 2 DEGREE: COEF. OF X^2 ? X CONST COEF. OF X^2 ? X CONST 1 1 ? 6 ? -1 ? 8 ? -12 (x )(x + 4) 4.3 Solving Quadratic Equations by Factoring

Directions: Factor the expression.
Example 3) Example 4) PROGRAM FCTPOLY PROGRAM FCTPOLY 2 DEGREE: 2 DEGREE: COEF. OF X^2 ? X CONST COEF. OF X^2 ? X CONST 1 1 ? 2 ? 1 ? -8 ? -5 (x )(x + 4) This means cannot be factored 4.3 Solving Quadratic Equations by Factoring

(x - )(x + ) 2 1 ? -5 ? -36 9 4 Directions: Solve the equation.
Ex 1) PROGRAM FCTPOLY PROGRAM QUAD83 2 DEGREE: A ? 1 COEF. OF X^2 ? X CONST B ? -5 1 C ? -36 ? -5 SOLUTIONS ? -36 9 -4 (x )(x ) 9 4

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
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

( x + 8)( x + 1) x + 8x = 9x ( x + 4)( x + 2) 2x + 4x = 6x
Step 3: Try various combinations of these factors. Sum of Outside and Inside Products (should equal 6x) Possible Factorizations ( x + 8)( x + 1) x + 8x = x ( x + 4)( x + 2) 2x + 4x = x -x - 8x = x ( x - 8)( x - 1) ( x - 4)( x - 2) -2x - 4x = x

( x )( x ) ( x + 7)( x + 1) x + 7x = 8x ( x - 7)( x - 1)
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 + 7)( x + 1) x + 7x = 8x ( x - 7)( x - 1) -x - 7x = x

(x + )(x + ) (x - )(x + ) (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 )

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.

Factoring the Difference of Two Squares
Example 1) 1.) Difference of the Two Squares, 2.) or you could look at this as the trinomial…

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

(x )(x ) 6 1 (x )(x + ) 4 16 (x )(x ) 4 18 (x )(x ) 1 16

Factor. (x )(x ) 6y 2y (x )(x + ) 4y 7y

Factoring Trinomials whose Leading Coefficient is NOT one.
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

Factoring by the Trial-and-Error Method
How would we factor: Notice that the leading coefficient is 3, and we can’t divide it out ( 3x )( x )

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) - 2(-14) - 4(-7)

( 3x - 1)( x - 28) -84x - x = - 85x ( 3x - 28)( x - 1)
Step 3: Try various combinations of these factors. Sum of Outside and Inside Products (should equal -20x) Possible Factorizations ( 3x - 1)( x - 28) -84x - x = x ( 3x - 28)( x - 1) -3x - 28x = x ( 3x - 2)( x ) -42x - 2x = x ( 3x )( x - 2) -6x - 14x = x -21x - 4x = x ( 3x - 4)( x - 7) -12x - 7x = x ( 3x - 7)( x - 4)

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)

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 = x ( 8x - 3)( x + 1) 8x - 3x = x ( 8x - 1)( x + 3) 24x - x = 23x ( 8x + 3)( x - 1) - 8x + 3x = x -12x + 2x = x ( 4x + 1)(2 x - 3) ( 4x - 3)( 2x + 1) 4x - 6x = x ( 4x - 1)( 2x + 3) 12x - 2x = x -4x + 6x = 2x ( 4x + 3)( 2x - 1)

Factoring Trinomials whose Leading Coefficient is NOT one.
Ex 1) Ex 2) (3x )(3x ) 1 1 (2x )(2x - ) 7 1 Ex 3) Ex 4) (2x )(3x ) 1 2 (2x )(x ) 3 5

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.

or +5 +5 +2 +2 x = 5 x = 2 According to the principle,
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.

(2x )(x ) = 0 + Solve the Equation (standard form) by Factoring - 1 4
Example 2) Factor the Trinomial using the methods we know. (2x )(x ) = 0 - 1 + 4 or +1 +1 - 4 - 4 2x = 1 x = 1/2 x = - 4 The resulting two statements indicate that the solutions are 1/2 and - 4.

(x )(x ) = 0 - Solve the Equation (standard form) by Factoring - 3 3
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.

Factoring out the greatest common factor.
But, before we do that…do you remember the Distributive Property? When factoring out the GCF, what we are going to do is UN-Distribute.

1st determine the GCF of all the terms.
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.

Factor each polynomial using the GCF.
ex) ex) ex)

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 ) 2 - 7 +

Factor. 3( ) ( ) (a - )(a - ) (x + )(x - ) 2 9 3 16 Example 2:
3( ) ( ) (a )(a ) 2 9 (x )(x ) 3 16

Example 4 Factoring GCF First Step 1) GCF

Factoring out the GCF and then factoring the Difference of two Squares.
Example 1) What’s the GCF?

Factoring out the GCF and then factoring the Difference of two Squares.
Example 2) What’s the GCF?

Additional Examples Example 3:

Factoring Perfect Square Trinomials
Example 4: (x )(x ) + 3 + 3 Since both binomials are the same you can say

Factoring Perfect Square Trinomials
Example 5: (x )(x ) - 5 - 5 Since both binomials are the same you can say

Example 6:

What must be true about a quadratic equation before you can solve it using the zero product property? assignment Page 261 # 47 – 88, 90

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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

Simplify the expression.
Example 1) Example 2) Example 3) Example 4) 4.5 Solving Quadratic Equations by Finding Square Roots

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

-40 -40 Solve the Quadratic Equation. QUAD83 & Example 4)
4.5 Solving Quadratic Equations by Finding Square Roots

-10 -10 Solve the Quadratic Equation. QUAD83 & Example 5)

+4 +4 Solve the Quadratic Equation. QUAD83 & Example 6)

Properties of Square Roots (a > 0, b > 0)
Product Property Quotient Property Example) Example) 4.5 Solving Quadratic Equations by Finding Square Roots

Example 1) Example 2) Example 3) Example 4) 4.5 Solving Quadratic Equations by Finding Square Roots

No radicals (square roots) in the denominator.
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

Example 8) Example 9) Example 10) 4.5 Solving Quadratic Equations by Finding Square Roots

Example 1) Example 2) Example 3) 4.5 Solving Quadratic Equations by Finding Square Roots

-1 -1 2 2 Solve the Quadratic Equation. Example 4)

Pythagorean Theorem c a b

+1 +1 Solve the Quadratic Equation. Example 5)

3 -5 -5 Solve the Quadratic Equation. Example 6)

Example 7) 4.5 Solving Quadratic Equations by Finding Square Roots

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

Vertex form WARM-UP 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

Complex Numbers 4.6 What you should learn:
Solve quadratic equations with complex solutions and… Goal 1 Goal 2 …Perform operations with complex numbers. 4.6 Complex Numbers

i , defined as Imaginary numbers Note that
The imaginary number i can be used to write the square root of any negative number. 4.6 Complex Numbers

Error Notice Simplify the expression. Go to MODE then down to
Example 1) Go to MODE then down to Now, try again. Notice Example 2) Example 3) Example 4) 4.6 Complex Numbers

Adding and Subtracting Complex Numbers
Example 1) Example 2) Multiplying Complex Numbers Example 3) Dividing Complex Numbers Example 4) 4.6 Complex Numbers

+15 +15 PRGM down to QUAD A= ? B= ? C=? Solve the Quadratic Equation.
Example 1) +15 +15 NO REAL SOLUTIONS PRGM down to QUAD A= ? B= ? C=? 4.6 Complex Numbers

+15 +15 Graphing Calculator PRGM down to QUAD83 A= ? B= ? C=?
Solve the Quadratic Equation. Example 1) +15 +15 Graphing Calculator PRGM down to QUAD83 A= ? B= ? C=? NO REAL SOLUTIONS 4.6 Complex Numbers

Describe the procedure for each of the four basic operations on complex numbers. assignment 5.4 Complex Numbers

Write the expression as a Complex Number in standard form.
Example 1) 4.6 Complex Numbers

Example 1) Example 2) Example 3) Example 4) 5.4 Complex Numbers

Graph the Quadratic Equation
Additional Example 1 Graph the Quadratic Equation Vertex Axis of Symmetry Opens: UP or DOWN 5.1 Graphing Quadratic Functions

Additional Example 2 Graph the Quadratic Equation Vertex
Axis of Symmetry Opens: UP or DOWN 5.1 Graphing Quadratic Functions

Graph the Quadratic Equation
Additional Example 3 Graph the Quadratic Equation Vertex Axis of Symmetry Opens: UP or DOWN 5.1 Graphing Quadratic Functions

y = - 2(x – 1)(x + 3) 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

Vertex form Additional Example 5 Graph the Quadratic Equation Vertex

Vertex form Additional Example 6 Graph the Quadratic Equation Vertex

Solve the Quadratic Equation
Ex 1) Ex 3) Ex 2) Ex 4)

Completing the Square 4.7 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

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

= 2 Solving a Quadratic Equation Solve by Completing the Square Ex)

4 4 4 4 Solving a Quadratic Equation Solve by Completing the Square
Ex) 4 4 4 4 4.7 Completing the Square

Solving a Quadratic Equation
Write the equation in Vertex Form Ex)

Why was completing the square used to find the maximum value of a function? assignment 4.7 Completing the Square

Pre-Stuff… Simplify for x.
4.8 The Quadratic Formula and the Discriminant

The Quadratic Formula and the Discriminate
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

Quadratic Formula use When solving a quadratic equation like

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

or Solve the Quadratic Equation. and Example 1 continued)
4.6 Complex Numbers

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

or Solve the Quadratic Equation. and Example 2 continued)
4.6 Complex Numbers

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

*if , then 2 real solutions.
*if , then 2 imaginary solutions. Example 2) substitute 20 So, 2 Real Solutions 4.8 The Quadratic Formula and the Discriminant

+15 +15 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 16 Plug them in to the formula 4.6 Complex Numbers

Example 3 continued) 4.6 Complex Numbers

Pre-Stuff… Simplify for x.

Pre-Stuff… Solve for x. Factor out GCF Ex3) Ex1) Ex2)

Example 1) Split this…. You can put these into calculator for Decimal answers. 4.8 The Quadratic Formula and the Discriminant

Example 2) Split this…. 4.8 The Quadratic Formula and the Discriminant

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

Graphing and Solving Quadratic Inequalities
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

What is the procedure used to solve quadratic inequality in two variables? assignment 4.9 Graphing and Solving Quadratic Inequalities

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

assignment Reflection on the Section
Give four ways to find a quadratic model for a set of data points. assignment Modeling with Quadratic Functions

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