quadratic functionsmenu introductiongraphing quadraticssolve by factoringsolve by square rootcomplex...

Quadratic Functions Menu

Introduction

Graphing Quadratics

Solve by Factoring

Solve by Square Root

Complex Numbers

Completing the Square

Quadratic Formula (graphtheory)

Quadratic Inequalities

Modeling

Appendix

Quadratic Functions IntroductionMENUAPPENDIX

A QUADRATIC EQUATION is any equation of degree 2

That just means that there is an in the equation and no bigger exponents2x

cbxaxy 2))(( qxpxay khxay 2)(

QUADRATIC EQUATIONS:

STANDARD FORM FACTORED FORM

INTERCEPT FORM

VERTEX FORM

or

Quadratic Functions IntroductionMENUAPPENDIX

This equation is in standard form. Rewrite it in intercept form.

2 9 18y x x

( 3)( 6)y x x

This equation is in intercept form. Rewrite it in standard form.

( 8)( 6)y x x

2 2 48y x x

This equation is in vertex form. Rewrite it in standard form.

21( 4) 64y x 2 8 16 64y x x 2 8 48y x x ( 4)( 12)y x x

Then it is easy to change to FACTORED FORM

Quadratic Functions Graphing QuadraticsMENUAPPENDIX

The graph of a quadratic equation looks like this:

•This U-shaped graph is called a PARABOLA

•The VERTEX is the high or low point on the graph

•It can cross the X-Axis 0, 1, or 2 times

•We call The X-Intercepts: Zeros, Roots, Solutions

•The Axis of Symmetry is the “mirror” that passes through the vertex

•It can open UP or DOWN


If “a” is positive it opens upIf “a” is negative it opens down


Basic graphing method EXAMPLE #1 Graph the Quadratic Function using a 5-point table

22 4 6y x x

1. Find the x-coordinate of the vertex.

2. Plug the x-coordinate into the equation to find the y-coordinate of the vertex

3. Count up and down by 1’s to fill in the left side of the table

4. Plug the numbers in to find the right side of the table.

2ba

1 801

23

06

60


Graph the Quadratic Function. Label the VERTEX and AXIS of SYMMETRY:

Sketching a graph EXAMPLE #2

22 12 19y x x

2ba

122 2

3

(3, 1)

:opens UP

(3, 1)

x =

3


Graph the Quadratic Function. Label the VERTEX and AXIS of SYMMETRY:Sketching a graph EXAMPLE #3

23( 4) 5y x

:Vertex ( 4, 5)

:opens UP( 4, 5)

x =

-4


Graph the Quadratic Function. Label the VERTEX, AXIS of SYMMETRY, and X-INTERCEPTS

Sketching a graph EXAMPLE #4

4( 1)( 1)y x x

:X I ntercepts 1 1and

:opens UP( 1, 0)

x =

0

(1, 0)

:VERTEX average the I NTERCEPTS

(0, 4)

(0, 4)


The height of a ball thrown straight up in the air, by an 8 foot tall person, at 30 meters per second is given by the following equation:

EXAMPLE #5

29 1 152 3 2

y x

What is the maximum height the ball reaches?

Quadratic Functions FactoringMENUAPPENDIX

MONOMIAL: 1 Term, a combination of numbers and variables being multiplied

BINOMIAL: 2 Terms, two binomials being added together that are NOT “like terms”

TRINOMIAL: 3 Terms, three binomials being added together that are NOT “like terms”

Any expression with more than 3 “unlike terms” is called a POLYNOMIAL.

4

23 x

x8zyx 2410 2a

235 2 xx

67274069 2345 xxxxxy


POLYNOMIAL STANDARD FORM (for an expression)

• Like terms are combined• Highest exponent first, then next highest, etc• If there is more than 1 variable, write them alphabetically.

Bad:753 xx

329 xx 22 2566 bbaaab

Fixed:

72 x

xx 92 3 22 26 baba


POLYNOMIAL STANDARD FORM (for an equation)

• Like terms are combined• Highest exponent first, then next highest, etc• If there is more than 1 variable, write them alphabetically.

All the same rules, except, the equation must be equal to zero.

(all terms on 1 side)072 x

092 3 xx

026 22 baba

A2H NAME_____________________________ _____

WS Quadratic Factoring Period ___

ALLWAYS CHECK Ex. 366024 2 xx 1. 23 15 72x x 2. 230 5 10x x I f you can Factor out a GCF first.

3. 212 36x x 4. 4 3 232 16 72 28x x x x

How to factor a simple Ex. 1032 xx 1. 2 7 12x x 2.

2 8 12x x

quadratic of the form

cbxx 2 )1( a

A. Write a pair of blank binomials

B. List the factors of the “c” term

C. I dentif y which pair of f actors

could add to the middle term “b”

D. Fill in the blanks with those f actors

How to factor a tougher Ex. 212 16 3x x 1.

210 17 3x x 2. 26 15x x


cbxax 2 )1( a

A. Multiply “a” and “c”

B. List the factors of the this number,

and identif y which pair adds to the

“b” term

C. Replace the middle term with

these 2 terms

D. Use parenthesis to group the fi rst 3. 212 22 14x x 4.

240 11 2x x 5. 212 22 14x x

2 terms together and the last

2 together

E. “Udistribute” the largest f actor

f rom each pair of parenthesis

F. The binomials should be the same.

“undistributed” them


212(2 5 3)x x 23( 5 24)x x 25(6 2)x x

12 ( 3)x x 3 24 (8 4 18 7)x x x x

Always do this first if possible

A2H NAME__________________________________



3. 212 36x x 4. 4 3 232 16 72 28x x x x


2 8 12x x


cbxx 2 )1( a


B. List the f actors of the “c” term





210 17 3x x 2. 26 15x x


cbxax 2 )1( a


B. List the f actors of the this number,


“b” term


these 2 terms


240 11 2x x 5. 212 22 14x x


2 together






( 5)( 2)x x ( 3)( 4)x x ( 6)( 2)x x

A2H NAME_____________________________ _____



3. 212 36x x 4. 4 3 232 16 72 28x x x x


2 8 12x x


cbxx 2 )1( a







210 17 3x x 2. 26 15x x


cbxax 2 )1( a




“b” term


these 2 terms


240 11 2x x 5. 212 22 14x x


2 together






361,36

2,18

3,12

4,9

6,6

212 18 2 3x x x 2(12 18 ) (2 3)x x x

6 (2 3) 1(2 3)x x x

6 1 (2 3)x x

321510 2 xxx)32()1510( 2 xxx)32(1)32(5 xxx

)32)(15( xx

159106 2 xxx)53(3)53(2 xxx

)53)(32( xx

A2H NAME_____________________________ _____



3. 212 36x x 4. 4 3 232 16 72 28x x x x


2 8 12x x


cbxx 2 )1( a







210 17 3x x 2. 26 15x x


cbxax 2 )1( a




“b” term


these 2 terms


240 11 2x x 5. 212 22 14x x


2 together






)7116(2 2 xx 731462 2 xxx

)73()146(2 2 xxx

)73(1)73(22 xxx

)73)(12(2 xx

251640 2 xxx)25()1640( 2 xxx

)25(1)25(8 xxx)25)(18( xx

22(6 11 7)x x 22(6 14 3 7)x x x

22((6 14 ) (3 7))x x x

2(2 (3 7) 1(3 7))x x x (2)(3 7)(2 1)x x

What if I need to solve, but Use the quadratic f ormula!! a

acbbx

2

42

The equation won’t factor?

Ex. 22 5 6x x 1.

22 3 2 5x x x 2. 25 4 2x x

How to solve an equation by Ex. 3072 xx 1. 3926 2 xxx 2. xx 13402

factoring

02 cbxax

A. Get all terms to 1 side and arrange

them into standard f orm.

B. Make the lead coeffi cient positive

C. Factor the quadratic side

D. Set each binomial equal to = 0

E. Solve f or x. You will get 2 answers

B. Use these patterns:

Perfect square Ex. 4129 2 xx 1. 93025 2 xx 2. 11664 2 xx

222

222

2)(

2)(

bababa

bababa

Diff erence of Squares Ex. 49 2 x 1. 925 2 x 2. 164 2 x

22))(( bababa


2(3 )x 2(2)

(3 2)(3 2)x x 2(3 2)x

2(3 )x 2(2)

(3 2)(3 2)x x

2(5 3)x 2(8 1)x

(5 3)(5 3)x x (8 1)(8 1)x x

Quadratic Functions Square RootsMENUAPPENDIX

VOCABULARY:

ROOT

RADICAL

RADICAND

RATIONALIZE THE DENOMINATOR

Short for “square root”

Another name for square root

The number or expression you are trying to take the square root of. 4

5

2

Getting rid of a square root on the bottom of a fraction.


SQUARE ROOTS

Click for Algebra I Square Root presentation


96 16 6

73

3 74 15

4 6

7

3

3

3 21

3

2160

720

7 20

20

14 520

7 510


212 13x 22( 4) 6 92x 2 25x 2 25x

5x

22( 4) 98x 2( 4) 49x 4 7x

4 7x 3, 11x


2 18 34x 22( 3) 1 51x

22( 3) 50x 2( 3) 25x

3 5x

2 16x 4x

8, 2x


2 124

7 7x

2 12 28x

7

2 16x 4x

Quadratic Functions Complex NumbersMENUAPPENDIX

Getting ready for imaginary numbers

1. ( 1)( 1)( 1)

2. ( 1)( 1)( 1)( 1)( 1)( 1)( 1)( 1)

233. ( 1)

4084. ( 1)

1

1

1

11

2

3

4

5

( 1) 1

( 1) 1

( 1) 1

( 1) 1

( 1) 1


Taking the square root of a negative number

No real number times itself will give you a negative

So to take the square root of a negative, you have to factor -1 outRoot -1 shows up any time we try to take the square root of a negative

It’s not a real number, so we call it the IMAGINARY NUMBER: i

36 16 40

136 116 140 16 14 1102

i6 i4 i102


Vocab:

Complex Number A Real and complex number added/subtracted together.

i The square root of -1

Imaginary Number Any number that ends with i. (i and it’s coefficient) Also called a “pure imaginary”

number.

Standard form for a Complex Number a+bi


Addition and Subtraction with Complex numbers

The rules for adding, subtracting and multiplying and dividing complex numbers are the same for any variable

)26()43( ii )26()43( xx x29 i29


Simplifying an exponent on i

Simplify each of the following:

21

1i

11 12i

3i

4i

5i

31 111 11 i

41 1111 11 1

51 114

i1 i

i


1

1

4

3

2

i

ii

i

ii 7i

18i

49i

203i

34 ii i

24444 iiiii 11i i3i i



31

9

324

50

1.

2.

3.

4.

i

i

i

i


3i i1i i4i 1

2i 1


Multiplication with Complex Numbers

)26()43( ii 2824618 iii

281818 ii

Just FOIL it out

)1(81818 i

81818 ii1826

But i2 is -1


1. 2.

3. 4.

)52()43( ii )52)(43( ii

)52()43( ii )43)(43( ii

i15 2208156 iii 22076 ii

2076 ii726

i91 21612129 iii 2169 i

169 25


The Imaginary component disappeared!

This happens anytime you multiply complex numbers like this

( a + bi ) ( a – bi )

They make the imaginary number

disappear.

So anytime you multiply two

complex conjugates

the answer will be a REAL NUMBER

4. )43)(43( ii 21612129 iii

2169 i169

25

These are called COMPLEX CONJUGATES


It is common and useful to graph them in what is called the COMPLEX PLANE We won’t go into why here.

This is much easier than it sounds

REAL

IMAGINARY

Graph the real number along the horizontal (x)

and the imaginary number along the vertical (y)

i43

3

4


GRAPH:

i24 i24 i24 i24 i24

i24

R

I I I

RR


GRAPH:

i24 i24 i24 i24 i24

i24

R

I I I

RR

Every complex number has an ABSOLUTE VALUE.

The absolute value is the length of the hypotenuse

2 24 2 20 2 5


STANDARD FORM:

bia Complex numbers should always look like this:

AND, since i is a square root, you cannot leave it on the bottom of a fraction


Put these into standard form

2(3 4 ) 5 9i i 6 8 5 9i i

11 i

(3 5 )(7 2 )i i 221 6 35 10i i i

221 29 10i i 21 29 10i

31 29i

#1 #2



32i

ii

2

3

2

i

i 3

2i

85i

ii

2

8

5

i

i 8

5i

3 92

ii

ii

3 9

2i

9 32 2

i

#3

#4

#52

2

3 9

2

i i

i



52 3i

2 32 3

ii

2

10 15

4 6 6 9

i

i i i

2

10 15

4 9

i

i

10 15

4 9i

10 1513

i

10 1513 13

i

85 4i

5 45 4

ii

2

40 32

25 20 20 16

i

i i i

40 3225 16

i

40 32

41i

40 3241 41

i

#6

#7

Quadratic Functions Complete the SquareMENUAPPENDIX

Click for Complete the Square Presentation

Day 1 Day 2

Quadratic Functions Quadratic FormulaMENUAPPENDIX

a

acbbx

2

42

The Quadratic FormulaBecause the quadratic formula is simply a formula for completing the square…

We can use it to sole ANY quadratic equation.

723 2 xx 52

734)2(2 2

x

0723 2 xx

10

882

x

10

2222

x5

22

5

1


a

acbbx

2

42

The Quadratic FormulaEven if the solutions are COMPLEX

0222 xx12

214)2(2 2

x

2

42

x

2

22

i

x i1


The graph tells us the types of solutions:

1282 xxy#1 #2 #3

1682 xxy 2082 xxy)2)(6( xxy )4)(4( xxy

2,6: solutions 4,4: solutions isolutions 24: factortdoesn'


Give the number and type of solutions based on the graph shown:

#4#5 #6

#7 #8#9

1 REAL solution

1 REAL solution

2 COMPLEX solutions 2 REAL solutions

2 COMPLEX solutions 2 REAL solutions


Determine the number and type of solutions without graphing:

1252 xxy

12

1214)5(5 2

x

2

235 x

144 2 xxy

42

144)4(4 2

x

2

04 x

252 xxy

12

214)5(5 2

x

2

175x

We can stop here, because we can already tell the solutions are going to be… COMPLE

X

1 REAL

2 REAL


The DISCRIMINANT

a

acbbx

2

42

acb 42

The DISCRIMINANT

If the discriminant is…

POSITIVE…

ZERO…

NEGATIVE…

There are 2 REAL solutions

There is 1 REAL solution

There are 2 COMPLEX solutions


Describe the discriminant based on the graph:

D: negative

D: zero D: positive

D: positive

D: negative

D: zero

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

x

y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-18-16-14-12-10

-8-6-4-2

2468

1012141618

x

y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

y

(1, -27)

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

y

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

x

y 500

-500

A2H NAME__________________________________

WS Quadratic Graphing (graph theory) Period ___

I dentify the x- intercepts for the graph of the equation given

1. )7)(5( xxy 2. )2)(4)(7( xxy 3. )9)(2( xxy

4. )15)(23( xxy 5. )7)(1)(3)(5( xxxxy 6. 1892 xxy

I dentify the equation shown in the graph:

7. 8. 9.

10. I n your calculator set the y- min and y- max to - 50 and 50. Then graph the equation listed and sketch

each one in the blank graph provided.

)6)(2( xxy

)6)(2(2 xxy

)6)(2(3 xxy

)6)(2(1 xxy

11. Use what you know about graphs to

identify the equation whose graph is shown here:

Quadratic Functions Quadratic InequalitiesMENUAPPENDIX

Quadratic Functions ModelingMENUAPPENDIX

Quadratic Functions Appendix MENU

OPENERSASSIGNMENTSEXTRA PROBLEMS

p. 253 Graphing Quadratic Equations

#21Graph, label the vertex and axis of symmetry.

22 12 19y x x

Vertex:12

2 2

3

22(3) 12(3) 19y

1y

: (3,1)vertex

(3,1)

X=

3


#27Graph, label the vertex and axis of symmetry.

2( 2) 1y x

Vertex:( , )h k

: (2, 1)vertex (2, 1)

X=

2


#35 Graph, label the vertex and axis of symmetry AND X-INTERCEPTS

1 ( 4)( 1)3

y x x

Vertex:

: ( 2.5, 0.75)vertex ( 2.5, 0.75) X

=-2

.5

4 12

2.5( 4,0) ( 1,0)


#39 Write the quadratic function in standard form.

( 3)( 4)y x x F O i L

2( 3 4 12)y x x x

2( 12)y x x

2 12y x x

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