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Do Do NowNow!!
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Do Do NowNow!!
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10 – 24 - 201310 – 24 - 2013 Do Do NowNow!!122 xx
Factor the trinomial.
a)
( ) ( )
b) 1252 2 xx
Factor the trinomial.
FCTPOLY
PRGM
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
b) Find Vertex _________
c) Identify
Axis of Symmetry _________
d) Find “Solutions”
x-intercepts __________
e) Opens UP or DOWN
f) Compare to y = x
Vertex shifts ______
Width _______
10 – 24 - 201210 – 24 - 2012
2
Now
10 – 25 - 201210 – 25 - 2012 Do Do NowNow!!
Graph and compare to )1)(3(2 xxy
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 = x2
Vertex shifts ______
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
Factor the trinomial.
1 a)
( ) ( )
a) Find Vertex _________
b) Identify
Axis of Symmetry _________
d) Find “Solutions”
x-intercepts __________
e) Opens UP or DOWN
f) Compare to y = x
Vertex shifts ______
Width _______
442 xxy
2
)6)(3( xxF.O.I.L.
1 b)
3)1(2)( 2 xxg
Rewrite in Quadratic Standard form
1 c)
11 – 29 - 201211 – 29 - 2012 Do Do Now!Now!
40412 2 xx
Factor the trinomial.
1) 0583 2 xx
Solve the equation
2)
FCTPOLY QUAD83PRGM PRGM
4 ( x – 2 ) ( 3x + 5 )1.6666666666666
1
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
QUAD83
5714. 3
236.44
4
236.6
AND
ANDAND
10 – 31 - 201210 – 31 - 2012
The Area of the rectangle is 30. What are the lengths of the sides?
x
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
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?
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
where is the object’s initial height (in feet).0h
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
TIME
HEIG
HTWindow re-set….
Xmin =
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
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
where is the object’s initial height (in feet).0h
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
TIME
HEIG
HTWindow re-set….
Xmin =
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
TIME
HEIG
HT
30
5
1 1.50.5
10
20
40
Modeling DroppedDropped 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
where is the object’s initial height (in feet).0h
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
0216 hth
where is the object’s initial height (in feet).
0h
Let’s assume that the cliff is 40 feet high.
TIME
HEIG
HT
Modeling Dropped Objects
0216 hth
Let’s assume that the cliff is 40 feet high. Write an equation?
40160 2 t
Make the substitution.
TIME
HEIG
HT
Graph it.
0216 hth
Let’s assume that the cliff is 40 feet high. Write an equation?
40160 2 t
Make the substitution.
TIME seconds
HEIG
HT
Graph it.
40
5
1 1.5.5
30
20
0216 hth
Let’s assume that the cliff is 40 feet high. Write an equation?
40160 2 t
+1.58 secondsHint: we want SOLUTIONS.
Think about… what are we trying to find?
Make the substitution.
TIME
HEIG
HT
Graph it.
40
5
1 1.5.5
30
20
-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
a
acbbx
2
42
Solve using the Quadratic Formula
01222 xx
Example 1)
Identify:A:
B:
C:
1
2
12
Plug them in to the formula
)1(2
)12)(1(4)2()2( 2 x
2
442
How to use a Discriminant to determine the number of solutions of a quadratic equation.
a
acbbx
2
42
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 )
PAGE 296 # 3 – 6, 31 – 33, 40 - 42,
MONDAY , NOV. 7thMONDAY , NOV. 7th
a
acbbx
2
42
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
0h
Collaborative Activity SheetChapter 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
0h
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?
x
x
270
150
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
Collaborative Activity Sheet 2Chapter 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
0h
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
0h
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?
x
x
270
150
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?
x
xx
x
Collaborative Activity Sheet 3Chapter 4 Solving – Graphing Quadratic functions
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 = x2
962 xxy
Graph and compare to 2xy
a) Graph
b) Find Vertex
c) Axis of Symmetry
d) x-intercepts
e) Opens UP or DOWN
f) Compare to y = x2
32
1 2 xxy
Graph and compare to 2xy
a) Graph
b) Find Vertex
c) Axis of Symmetry
d) x-intercepts
e) Opens UP or DOWN
f) Compare to y = x2
24
3 2 xxy
Graph and compare to 2xy
a) Graph
b) Find Vertex
c) Axis of Symmetry
d) x-intercepts
e) Opens UP or DOWN
f) Compare to y = x2
24
3 2 xxy
Graph and compare to 2xy
a) Graph
b) Find Vertex
c) Axis of Symmetry
d) x-intercepts
e) Opens UP or DOWN
f) Compare to y = x2
32
1 2 xxy
Graph and compare to
22122 2 xxy
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 = x2
Graph and compare to
22122 2 xxy
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 = x2
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 = x2
962 xxy
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 = x2
442 xxy
Graph and compare to
142 2 xxy
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 = x2
vertex(1, 3)
Axis of Symmetry
x = 1
x-intercepts
Graph and compare to
322 xxy
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 = x2
vertex(1, -4)
Axis of Symmetry
x = 1
x-intercepts
Graph and compare to
232 2 xxy
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 _______
Graph and compare to
142 xxy
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 _______
Quiz 4.1
Graph and compare to
442 xxy
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 _______
Quiz 4.1 RETAKE
Graph and compare to
442 xxy
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 _______
Quiz 4.1 RETAKE
Graph and compare to
4)2(4 2 xy
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 _______
Quiz 4.2
#1)
Graph and compare to
)4)(2(3 xxy
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 _______
#2)
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
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
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)
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
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
1a
1a
abx 2
abx 2
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
Example 1A Graphing a Quadratic Function
Graph and compare to
342 xxy
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
2xy
Vertex X Y 2 -1
Solutions 3 1
Graphing Calculator
PRGM
down to QUAD83
A= ?
B= ?
C=?
11
-4-4
33Axis of SymmetryThe line x = 2
Example 1B Graphing a Quadratic Function
Graph and compare to
122 xxy
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
2xy
Vertex X Y -1 0
Solutions -1 -1
Graphing Calculator
PRGM
down to QUAD83
A= ?
B= ?
C=?
11
22
11Axis of SymmetryThe line x = -1
Example 1C Graphing a Quadratic Function
Graph and compare to
582 2 xxy
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
2xy
Vertex X Y -2 3
Solutions -3.225 -.775
Graphing Calculator
PRGM
down to QUAD83
A= ?
B= ?
C=?
-2-2
-8-8
-5-5
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
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
Page 240
#
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 = -
(-4)
2(1) = 2
Graph and compare to
342 xxy
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
2xy
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
= -1
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
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.
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
Additional Example 1
a) y = x2 + 2x +1
4.1 Graphing Quadratic Functions in Standard Form4.1 Graphing Quadratic Functions in Standard Form
Additional Example 2
b) y = -2x2 - 8x - 5
4.24.2 Graphing Quadratic Functions in Vertex or Intercept Form
What you should learn:GoalGoal 11
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)(
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
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)(
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Vertex ( h, k )
So, Vertex ( 3, -4 )
Vertex FormVertex Form
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)(
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
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.
Vertex FormVertex Form
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
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Example 2Example 2
Graph y = (x+2)2 - 3
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Vertex
Opens UP, vertex is MIN
Since, 1a
the parabola is the same width as 2xy
Vertex FormVertex Form
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.
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept 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
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
VertexVertex
OpensOpens
Since, 1a
the parabola is the same width as 2xy
Vertex FormVertex Form
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
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Vertex ( 1, -16)
Opens UP, vertex is MIN
Since, 1a
the parabola is the same width as 2xy
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
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Vertex ( 1, 9)
Opens DOWN, vertex is MAX
Since, 11 a
the parabola is the same width as 2xy
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
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
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.
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
InterceptIntercept form
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
y = - (x – 1)(x + 3)
Additional Example Additional Example 22
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
y = (x + 1)(x - 3)
Additional Example Additional Example 33
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
Additional Example Additional Example 11
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this
equation?
assignmentassignment
4.2 Graphing Quadratic Functions in Vertex or Intercept form4.2 Graphing Quadratic Functions in Vertex or Intercept form
4.34.3 Solving Quadratic Equations by Factoring
What you should learn:GoalGoal 11
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
1 X
? 6
? 8
CONST
FCTPOLY
DEGREE: 2
COEF. OF X^2?
PROGRAM
1 X
? -1
? -12
CONST
)4)(3( xx
4.3 Solving Quadratic Equations by Factoring4.3 Solving Quadratic Equations by Factoring
Example 1) Example 2)
52 xxDirections: Factor the expression.
FCTPOLY
DEGREE: 2
COEF. OF X^2?
PROGRAM
1 X
? 1
? -5
CONST
)5( 2 xx
This means cannot be factored
4.3 Solving Quadratic Equations by Factoring4.3 Solving Quadratic Equations by Factoring
822 xx
(x - 2 )(x + 4)
FCTPOLY
DEGREE: 2
COEF. OF X^2?
PROGRAM
1 X
? 2
? -8
CONST
Example 4)Example 3)
03652 xx
(x - )(x + )49
Ex 1)
Directions: Solve the equation.
QUAD83PROGRAM
A ?
B ?
C ?
1
-5
-36
SOLUTIONS 9-4
FCTPOLY
DEGREE: 2
COEF. OF X^2?
PROGRAM
1 X
? -5
? -36
CONST
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.
Possible Factorizations
( 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.
162 x
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
What you should learn:GoalGoal 11
4.3 Solving Quadratic Equations by Factoring4.3 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)
23x
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.
Step 3: Try various combinations of these factors.
Possible Factorizations
( 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)
28x
( 4x )(2 x )
Step 3: Try various combinations of these factors.
Possible Factorizations
( 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
x = 5
+2 +2
x = 2
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
+1 +1
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
x = 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
1
y = x2 + 3x - 10
-5 ; 2
Example 6:
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
What must be true about a quadratic equation before you can solve it using the
zero product property?
assignmentassignment
Page 261# 47 – 88, 90
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4.54.5 Solving Quadratic Equations by Finding Square Roots
What you should learn:GoalGoal 11
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
16
3Example 3)433.
9
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
45Example 4) 4553
Solve the Quadratic Equation.
0162 xExample 1)
4x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
243.4x
Example 2)
464.3x
Example 3)
182 x
363 2 x-36-36
QUAD83
-18-18
0182 x
0363 2 x
QUAD83
QUAD83
Solve the Quadratic Equation.
40)1( 2 xExample 4)
-40 -40
03922 xx
325.7x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
40)1)(1( xx
40122 xx
325.5&
QUAD83
Solve the Quadratic Equation.
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
Solve the Quadratic Equation.
4)2( 2 xExample 6)
+4 +4
042 xx
0x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
4)2)(2( xx
4)44( 2 xx
4&
QUAD83
4442 xx
Properties of Square Roots (a > 0, b > 0)
baab
Product Property
b
a
b
a
Quotient Property
Example)
428
25
3
25
3
Example)
22
225
3
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
Simplify the expression.
81Example 1)
24Example 2) 64 64 62
16
3Example 3)
16
3
4
3
9
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
45Example 4) 95 95 53
Simplify the expression.
3
1Example 5)
5
2Example 6)
5
5
5
52
7
3Example 7)
7
3
7
7
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
3
3
3
3
7
21
Rationalizing the denominatorRationalizing the denominator – eliminate a radical as denominator by multiplying.
Which means… No radicals (square roots) in the denominator.
Simplify the expression.
33 Example 8)
102 Example 9) 20 54 52
634 Example 10)184 294
3
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
234
212
Solve the Quadratic Equation.
0162 xExample 1)
4x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
182 x
Example 2)
29 x
18x
23x
363 2 x
Example 3)
34 x
122 x
32x
0182 x162 x
0363 2 x
Solve the Quadratic Equation.
1712 2 xExample 4)
-1 -1
162 2 x2 2
82 x
8x
22x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
Pythagorean Theorem
222 cba
a
b
c
Solve the Quadratic Equation.
40)1(2 2 xExample 5)
+1 +1
201x
521x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
2
20)1( 2 x
201 x
2
7)5(
3
1 2xExample 6) 3
-5
21)5( 2 x
-5
215x
Solve the Quadratic Equation.
7)5(3
1 2 x
21)5( 2 x
21)5( x
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
Solve the Quadratic Equation.
7236
5 22
xxExample 7)
12x
2
4.5 Solving Quadratic Equations by Finding Square Roots4.5 Solving Quadratic Equations by Finding Square Roots
726
3 2
x
1442 x
726
5 22
6
2xx
2
Reflection on the SectionReflection on the SectionReflection on the SectionReflection 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 Roots4.5 Solving Quadratic Equations by Finding 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
b) Find Vertex _________
c) Identify
Axis of Symmetry _________
d) Find “Solutions”
x-intercepts __________
e) Opens UP or DOWN
f) Compare to y = x
4.64.6 Complex Numbers
What you should learn:GoalGoal 11
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
12 i
The imaginary number i can be used to write the square root of any negative number.
4.6 Complex Numbers4.6 Complex Numbers
Simplify the expression.
4Example 1)
12Example 2) i464.3
16
3
Example 3)i433.
i2
45Example 4) i708.6
4.6 Complex Numbers4.6 Complex Numbers
Error Go to MODE then down to
Now, try again.4
Notice Notice
bia
Adding and Subtracting Complex Numbers )23()4( ii Example 1)
)81()57( ii Example 2) i36
i7
4.6 Complex Numbers4.6 Complex Numbers
)7)(5( iiExample 3)
Multiplying Complex Numbers
Dividing Complex Numbers
i
i
21
35
Example 4)
i6.22.
i236
Solve the Quadratic Equation.
1512 2 xExample 1)
+15 +15
0162 2 x83QUAD
ix 828.2
4.6 Complex Numbers4.6 Complex Numbers
NO REAL SOLUTIONSNO REAL SOLUTIONS
PRGM
down to QUAD
A= ?
B= ?
C=?
Solve the Quadratic Equation.
1512 2 xExample 1)
+15 +15
0162 2 x83QUAD
8x
22ix
4.6 Complex Numbers4.6 Complex Numbers
NO REAL SOLUTIONSNO REAL SOLUTIONS
Graphing Calculator
PRGM
down to QUAD83
A= ?
B= ?
C=?
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Describe the procedure for each of the four basic operations on complex numbers.
assignmentassignment
5.4 Complex Numbers5.4 Complex Numbers
Write the expression as a Complex Number in standard form.
i
i
21
35
Example 1)
)21)(21( ii
5
131 i
i
i
21
21
4.6 Complex Numbers4.6 Complex Numbers
Simplify the expression.
4Example 1)
12Example 2) 341 32i
16
3
Example 3)
4
3i
4
3i
i2
45Example 4) 951 53i
5.4 Complex Numbers5.4 Complex Numbers
41
16
31
Additional Example 1
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
122 xxy
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens: UP or DOWN
Additional Example 2
b) y = -2x2 - 8x - 5
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens: UP or DOWN
Graph y = (x + 3)(x - 5)
Additional Example 3Additional Example 3
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens: UP or DOWN
y = - 2(x – 1)(x + 3)
Additional Example 4Additional Example 4
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens: UP or DOWN
Additional Example 5 Additional Example 5
Vertex Vertex formform
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
Graph the Quadratic Equation
Vertex
Axis of Symmetry
Opens: UP or DOWN
4)3(2 2 xy
Additional Example Additional Example 66
5.1 Graphing Quadratic Functions5.1 Graphing Quadratic Functions
Vertex Vertex formform
Graph the Quadratic Equation
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
Ex 4)
4.74.7 Completing the Square
What you should learn:GoalGoal 11
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
2
2
2
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
2
2
bc
cxx 162
2
2
16
28 64
Perfect square trinomial
28x square of a binomial
4.7 Completing the Square4.7 Completing the Square
Solving a Quadratic Equation
Solve by Completing the Square
0222 xx
xx 22 = 2
Ex)
4.7 Completing the Square4.7 Completing the Square
Solving a Quadratic Equation
Solve by Completing the Square
0164 2 xx
041
232 xx
4 4 4 4Ex)
4.7 Completing the Square4.7 Completing the Square
Solving a Quadratic Equation
Write the equation in Vertex Form
742 xxyEx)
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Why was completing the square used to find the maximum value of a function?
assignmentassignment
4.7 Completing the Square4.7 Completing the Square
2
242
x
2
242
x2
242
x
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
What you should learn:GoalGoal 11
GoalGoal 22
Solve quadratic equations using the quadratic formula.
Use quadratic formula to solve real-life situations.
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
Quadratic Formula
When solving a quadratic equation like
02 cbxax
use
a
acbbx
2
42
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
Solve the Quadratic Equation.
01222 xxExample 1)
83QUAD
4.6 Complex Numbers4.6 Complex Numbers
NO REAL SOLUTIONSNO REAL SOLUTIONS What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.
a
acbbx
2
42
Identify:A:
B:
C:
1
2
12 Plug them in to the formula
Solve the Quadratic Equation.
Example 1 continued)
4.6 Complex Numbers4.6 Complex Numbers
a
acbbx
2
42
)1(2
)12)(1(4)2()2( 2 x
ix 317.31
2
442
ori317.31and
Solve the Quadratic Equation.
022122 2 xxExample 2)
83QUAD
4.6 Complex Numbers4.6 Complex Numbers
NO REAL SOLUTIONSNO REAL SOLUTIONS What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.
a
acbbx
2
42
Identify:A:
B:
C:
-2
-12
-22 Plug them in to the formula
Solve the Quadratic Equation.
Example 2 continued)
4.6 Complex Numbers4.6 Complex Numbers
a
acbbx
2
42
)2(2
)22)(2(4)12()12( 2
x
ix 414.13
4
3212
ori414.13and
How to use a Discriminant to determine the number of solutions of a quadratic equation.
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
a
acbbx
2
42
discriminant
*if , then 2 real solutions.042 acb
*if , then 1 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
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
*if , then 2 real solutions.042 acb
*if , then 1 real solutions.042 acb
*if , then 2 imaginary solutions.042 acb
Example 2) 0422 xxacb 42 substitute
)4)(1(4)2( 2
20 So, 2 Real Solutions
Solve the Quadratic Equation.
1512 2 xExample 3)
+15 +15
0162 2 x83QUAD
4.6 Complex Numbers4.6 Complex Numbers
NO REAL SOLUTIONSNO REAL SOLUTIONS
What we are going to do now is to use the QUADRATIC FORMULA to find the Imaginary solutions.
a
acbbx
2
42
Identify:A:
B:
C:
2
0
16 Plug them in to the formula
Solve the Quadratic Equation.
Example 3 continued)
4.6 Complex Numbers4.6 Complex Numbers
a
acbbx
2
42
)2(2
)16)(2(4)0()0( 2 x
4
128x
828.2x
2
42
x
2
42
x2
42
x
Pre-Stuff…Simplify for x.
2
22
x2
22
x
2
0
x
2
4
x 20
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
2
244
x
)21(2 x )21(2 x
Pre-Stuff… Solve for x.
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
2
214
)21(2 x
322 Ex1)
Factor out GCF
312
21015Ex2)
2235
Ex3)
Solve the Quadratic Equation.
052 2 xxExample 1)
4
411x
a
acbbx
2
42
)2(2
)5)(2(4)1(1 2 x
You can put these into calculator for Decimal answers.
4
411x
4
411x
Split this….Split this….
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
Solve the Quadratic Equation.
0169 2 xxExample 2)
a
acbbx
2
42
)9(2
)1)(9(4)6(6 2 x
3
21
3
21x
Split this….
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
18
726 x
18
266
18
216
3
21x
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Describe how to use a discriminant to determine the number of solutions of a
quadratic equation.
assignmentassignment
4.8 The Quadratic Formula and the Discriminant4.8 The Quadratic Formula and the Discriminant
a
acbbx
2
42
discriminant
*if , then 2 real solutions.042 acb
*if , then 1 real solutions.042 acb
*if , then 2 imaginary solutions.042 acb
4.94.9 Graphing and Solving Quadratic Inequalities
What you should learn:GoalGoal 11
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
What is the procedure used to solve quadratic inequality in two variables?
assignmentassignment
4.9 Graphing and Solving Quadratic Inequalities4.9 Graphing and Solving Quadratic Inequalities
Modeling with Quadratic Functions
What you should learn:GoalGoal 11
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Give four ways to find a quadratic model for a set of data points.
assignmentassignment
Modeling with Quadratic FunctionsModeling with Quadratic Functions