MUHS Grade Improvement Learning Module

Teacher Lauren Thagard

Subject Mathematics - Algebra 1

Contact Info lthagard@muhs.com

Office Hours Weekdays 9am - 4pm

1roduction

I This is module 1 of 2 available to students to complete for a 10% grade increase. Please know, each portion

1 of this module MUST be completed in its ENTIRETY in order to receive credit. There are four assignments and one assessment to be completed. I will be grading this module based on students showing CLEAR and

, ORGANIZED work and the final answer needs to be circled or boxed. Please be advised how your grade will j be adjusted based on the following guidelines set by the school:

I 90% or better - No modules need to .be completed, your 3rd quarter grade will be your final grade © i G0-89% - No need to complete any modules unless you are looking to improve your grade. Completion of

one module of your choice to bring you up 10% MAX: ' 50-59% - Complete one of your choice to bring your grade up to a 60% .. I 40-49% - Complete BOTH modules to bring your grade up to a 60%. I 39% or lower- Sign-up for Summer School required; contact Gretchen Wesbr~ck at gwesbrock@muh~.com

•

ta ndard

I Alg l.M.F.IF.C.07b - The Highly Proficient·s'tudent c~n graph and compare llne-ar, quadratic, piecewise, exponential, step, and absolute value functions in various forms. Algl.M.A .REI.B .04ab - The Highly Proficient student can determine and use the most efficient method for solving quadratic equations and justify their choice. The Highly Proficient student can recognize when a quadratic equation has no real solutions. T -

1 Alg l.M .A.APR.8.03 - The Highly Proficient student can identify the zeros of a quadratic l function and use the zeros to construct the graph of the function.

· 2arning Objectives

1. SWBAT to find the axis of symmetry, the vertex, and the graph the graph of a particular quadratic equation. ' ,

-2. SWBAT determine the number of solutions a particular graph has as well as the discriminate.

f-I 3. SWBAT factor a quadratic where the value of a = 1 I

'

i 4. SWBAT use the quadratic formul;i to solve quadratic equations. I

I I

:cture Notes/Handouts

Description Location •

1

- Steps to Graph a Quadratic Equation M2Ll -Steps to Graph a Quadratic Equation NOTES I t- - ------ -------'-- ----------4-------------1 : Quadratic Roots and the Discriminate M2L2 - Quadratic Roots \ and the Discriminate NOTES \ Factoring Quadratics when a = 1 · ~ M2L3 - Factoring Quadratics when a = 1 NOTES \ The Quadratic Formula M2L4 - The Quadratic Formula NOTES

: -;5ential Question~

1. What is a quadrat ic equation? How are they graphed?What is the vertex? How are the coordinates ~fa vertex on a quadr~tic,equatio!l calculated? What is an axis of symmetry? How do you calculate the axis of symmetry? • ' 2. What does it mean to factor a quadratic equation? How~do you use the factors of a I polynomial to find the solutions? What is the discriminant? How is knowing the discriminant I useful in determining the number of solutions in a quadratic? 1 3. What is a quadratic equation? How is it solved? What is the quadratic formula? How is it used I

I to solve a quadratic equation? I I

\ 4. What does solutions to a quadratic represent? How can factoring be used to find solutions? I How do I know if the graph opens up or down? I I

d ptional/Supplemental Resources

Description

I Khan Academy Video - Features of a Quadratic · I O Skip to 7:26 on the video to see equations in standard form I - , I Khan Academy Video - The Discriminate 1

! Khan Academy Video - Solve Quadratics when a = 1

: Khan Academy Video - The Quadratic Formula I . I •

I

Location

khanacademy.org

khanacademy.org

khanacademy.org

khanacademy.org

L- ------~ ---'----'--'-"'-----'---'-...,;,_-'--=--------L.--'----'--------'----'

,, . t ~.ssrgnmen s

Description ··Location • , I

I 1. Graphing Quadratics homework , ·- - ~ M2L1 - Steps to Graph a . ~ .

I ' .. Quadratic Equation HW I .. I 2. Quadratic Roots and the Discriminate homework ') '·

M2L2 - Quadratic Roots I

and the Discriminate HW I . I .. I 3. Factoring Quadratics when a=l homework

~-M2L3 - Factoring I Quadratics when a = 1 HW i

I 4. Quadratic Formula homework M2l4 - The Quadratic

l Formula HW

: 5.

i .._

6. I i

r-;_

8.

I I

:_ssessments/Evaluations

Description Location ,- -..----------------=---.----------------, j 1. Module 2 Assessment Ml Assessment

'---1--------------_.:..:;.::_;_ __ --+-____:..-'---,---------------j I ! 2.

I I i

3.

I 4.

L _ _ ,.___ ______________ ..:.....;;: _ __._ _____________ ___,

~ubmission :..,. ''i.;.)·-~i: ·,.::•_ j ' ,.~,~ ..... ~ ) .,, •:, • •✓ -~

- ., .y ~- .. ~ ~- . ' -~,..,... . i • •

M odules are due by May 4th at 7:00am . .:r_his willc1llow~tet;1chers time to·grade and communicate with students ' ... , •\ ;If .. ,..« f;; :.- •• •

regarding any changes that need to be made. ·Electronic subm_ission will'be dcfne through email, Google ;,I· • ,, • ,,, " I :V ✓ , ,,. • • ; • •

Classroom, or using the Google Drive. f eachers wilrcontrol how tHey want to receive· electronic.submissions. ,. .. » ~ " .• "' '. . • Modules will be assessed usjqg a Pass/Fail model. Students who choose to do the module arid receive a "Fail" - , ~ ~,.... • , • • . r

will receive the original Qua'riter13 .grade fo~ t_he Semester. ' ·.{ ' .. , ., · I ~ i ' Y

I I ·· · . Modules that are completed as a packet wil!_be _returned-to JVlingus µnioJ'l. Tbere·wm·be additional • f, • • ,.,. ~ ' -

communication regarding a drop-off schedule. _. · , ; _., ' ,~ /·<·~

1 I

J t

t '

tv)o~lttlG 1- lts~OYJ 1- ·. 0rccph QtA.a.Jr@ cs \ Name: I l.__o_at_e·_. _________ __,

\ To~c: I\~ c_la_ss_: _________ __,

Main ldeas/Questtons

StGps to Graph a QUADRATIC EQUATION

EXAMPLES

'{= \'2.= \

Y=12.='i

l= (-1)\1l-1)- \ ' - 1. - \ : --z.

~ ~ lo) ~ + 1. to) - I

O+-o-,=\

''/'= l,)\.1t,)-1 \+z-\-::"2..

~-:. -\(-3/·-s(-~)-11 - q 4-1.\.(- \j

= - '2.

Notes/Examples

• -\o Find the axis of symmetry. ➔)(-:: -5'7l

1

f) Find the vertex. -f? p' l,,l ~ l X ' 11) -to ~ ~ u_~ UY"')

O Put the vertex in the middle row of the table. Fill in a table of values using your calculator.

C, Plot the points and connect them into a smooth parabola!

Dlrecttons: Graph each quadratic equation using a table. Identify the axis of symmetry, vertex, domain, and range.

R~

Axis of Symmetry: - I

-~ Vertex: ( - \ > - 2)

D~

R~

Axis of Symmetry: - '1

- 2.

X y

\ 1. 0 -I

- I - l -2 - I

--~ -i.

-2 - 3 -1..

-y -s

\ 1

\

LJ' \

I

I ' 1

I

'

I

~-: -\ (-1.)-z._f(-'2.)-l l

Vertex: l -Y) - \)

~

~

Algebra 1 Name ----------------M2L l Graphing Quadratics Date Period -------Find the axis of symmetry, the vertex, and graph the equation.

y

--'

-3 -2 -I

1 I

-f

-4 __ , -2 -1

2 3 X

y

I

5. - '

- 1.5 -·- -

0.. -

2 X

2) y = -2x2 + 4x

Ii ~ _, µ ' _,

4) y"' -x2 - 4x - 8

-6 IS ~ p i,

'

-1-

y

I X

-

--.

y

I , X

-

-

-

--

-

5) y = 3x2 + 12x + 11

- 2 - 0 ~ k, ~

7) y = 2x2 + 8x + 4 y

k, ~ ~ ~ ~ I

-

9) y = -2x2 + 16x - 29 y

Ir

)'

~ X

-

X

X

6) y = 2x2 - 8x + 9

y

--6 -5 -l -3 - 2 - I I l 3 4 x

8) y = -2x2 - 4x - 3

y -;.

- ls 4 J 2 I X

-

-

-

-

-

-

-

-- ..

10) y = x2 + 6x + 8 . y

I I I ; 3.'.

I

i i I I ' ! I

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I ! I ' ! I ' I ..

I - '-· I - . i 1.: - 1-I I I

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l5 I I b ' I I X l - I ~o.: I

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1---I -

I I . , - --~J:! .• ! -1·-- . I I -

-2-

Mo ~U;lt, fl_ ll, ~ M5h &]_, ·. (vi.A. a cJ_ r ~17 l £.outs + 0 ,-! GV I (() ·, (\ CLn t-i Name: ll~oa=te: ===~' I Topic: II~ c_la_ss_: -------~'

Main Ideas/Questions Notes/Examples

QUADRATIC ""~ m,n+~ l.A) h LV'<., ---1'h I ~ Y-l'J/J_b

ROOTS I _J I

--to U 1,,h 1 .<. -1Vl ~ X-ttx'1s

also cal~d ... v-oo-+5 x -~~t1t-<X"" LLf'i:S , -.sv l u..-1\ Ot-'\ s 2 SOLUTIONS 1 SOLUTION NO SOLUTION

\ I \JI \l/ k"" G .

NUMBER Of ---SOLUTIONS ~ - tJ{.s n \'.

'

1\ l/\ -

1 ! -

iO'

' II.

1. y = .l +4x-5

x~ -.£_ - -_«:! _-Y X y

EXAMPLES I 0 2 t'A - '2, l1) - 2. = -1. 0 -S

Find the solutions of the 'I~ l-1) '\4(-2.) - 5 foRowing quadratic - \ -,

equations by graphing. '-1-i - s ~ - 2 -&'\ -3 -8

Solutions: '(= l-\)1.+Y(-1)-5 -y -~ 1. X -= t,) -5~ \-'-t-6 -5 0

2. y = x2 -2x+l 2. X y

3.

·1/ ~-ro "--f Aj(, ~ ~ / Yl~+ ..... ' 3. y = - x2 + 2x - 3 / '

'y x/

/

I"-

4. y=x2 - 10x+16 /v X y

Solutions:

4.

5.

6.

/

5. y = - x2 +9

"' / X y

r\. / '\ /

'I'

/ '\. / '\

/ I"' ' t

6. y = -3x2 + 6x ~

X y

J\./ '? THE Formula:

~ DISCRIMINANT b 1. _ 1 °'- G If d > 0, then there are ---2._ soluttons.

If d = 0, then there are __L_ soluttons.

If d < 0, then there are __Q_ soluttons.

EXAMPLES Use the discriminant to determine the number

of solutions.

7. y=x2

+5x+4 C 2 solutions' 8,y = x2 - 3x+l0 2solutions

~= \ b:: s (..:"' - b~ -51 SOIUllOn a= I b-: -3 (,=IO -.b=: 3 1 solution

0 solutions ~olutiori't:

'51._ 4(t)(Y)-=_lq'J -3 2 - '1(1)(,0) =t 3cJ

9. y = x2

+ lOx + 25 2 salu.tions 10. y = 2x2 - 4x - 3 Msolutions- ~ CA.~\ b-::10 c~2s - b-=-1c'1"solutio~ 0.=2. \:):: - l( c.:-~ -t,= 4 1 solution

U miutions O solutions

,o~_ 4(1)(-is) = {Ql -'i~- ~( 2 )(-3) ~ Fol 11. y = 4x

2 - 12x + 9 2 solutions 12. y = - 3x2 + Sx -8 - 2 solutions

O.=~ 1:,-:,.\"l. e.,::q-b-=-12'f solulio~ CA."=-3 b-=S (._::-!{-b::-Slsolution

O solutions 1 ~ solulions'i

-11.1.-Yl ~)t q) -=@ S"2.-Y(-3)l-B) = 1-,,i

Algebra 1 Name _____________ _

M2L2 - The Discriminant + Quadratic Roots Date Period ------

Find the discriminant of each quadratic equation then state the number and type of solutions.

I) 9n2 - 6n - 5 = -6 2) -10r2 + Sr - 10 = -4

3) -10x2 + 3x - 12 = -3 4) 5n2 - 4n - 6 .. -5

5) 4x2 + 4x + 7 = -3 6) -8a2 - Ba - 9 = - 7

7) 4m2 - 9m + 11 = 9 8) -2x2 + Sx + 7 = 10

9) -n2

- 4n - 9 = -5 10) 8n2 + 8n + 6 = 4

11) 6n2 -4n+18=10 12) -x2 - 4x + 12 = 7

13) -10n2 + 9n + 5 = -4 14) 7n2 - n + 15 = 6

Mv~uk, ?_ US£on 3: Fa.otoY Cw-a.Jrafics wM17 a:. I \ Name: 11 Date: I

\ Topic: 11 Class: I

Main ldeas/QuesHons Notes/Examples

In many cases. we can find the solutions (or roots. zeros. x-intercepts)

SOLVING of a quadratic equation by factoring, rather than graphing. Follow the

steps below to find the solutions of the given equation by factoring.

QUADRATICS 0 Set the quadratic equation equal to 0. 1

~lven: y = x2 + 3x -10

By ractorin9 f) X 1. + tx -10 :o

.,.__.."'"'*imi t,rut,t, +ttb~

0 Set each factor equal to O and €) -·r,r .

5·-7. 5-2. l)(-+S) )(- 2.. )=o 0 solve each factor for x. X:!=O x;? -s-s ....

0 Write your answer using curly braces. ,x=~-sJ1.3J x:::.-s x= r2. "2.

DlrecHons: Solve the following quadratic equations by factoring.

YOUTRYI 1. x2 + 4x + 3 = 0 2. x2 + llx + 24 = 0 C b

' :., \; .. , (x+a) )(~1)::0 $ ()( + 8') X +-3)-= o x::r-o X:2t-::.o X)f.o ~~-:o -3 g 3

- I -4 - -3 )(:::-3 )(:: -1 X-=--rf X- 3

J{-:; i -~.)-,J 1 l ~ = ~ - &'J -31] , - -3. x2 + x-2 = 0 4. x2 + 6x- 27 = 0

-~ lx-2)r<+ l)~o -~ ( X +'l)I( )(- 3) "C -1.-1 i-, X;t:-1> X_;r:o '1·-3 q-3 X:t-1 X~~

+l -\

x~2 X:::-1 X _ ~ \ X ::: -"\ I X : 3 X::1_2)-15 \ - -th35 5. x2 -10x + 21 = 0 6. X

2 - X ~ 20 = 0

~ lx-1)\1<-3)=0 -u,r ()(-S)r+'-f)•O - -

3 X~=o X~=o S·'f -S+'-i X

~=o X~7'o +1 4-3 - t-S -\.(

X=}7)'3]j X=1 X-=3 X-=S ><=--L\ \ X==!s,-~~ 1

7. x2 + lOx + 25 = 0 8. x2- 8x + 16 = 0

~ (x + ~)1(>< +s) ::o ~ lX-4)\ x:-4)~0 S,s 5➔ 5 x~~o

)(~~ --41

-Y X'~o x~~o -~ )I::-~ X=-5 +~ "t'{

jX::. tY! \ )(::y X::y \X= t-s~l I

Algebra l

M2L3 - Factoring Quadratics when a= 1 Solve each equation by factoring.

1) x2 - 8x + 15 -0

3) n2 + 7n + 12 = 0

5) r2 + r - 42 = 0

7) x2 - 4x - 32 = 0

11) n2 +5n-14=0

13) p 2 - 9p + 8 = 0

Name ----------- ---

Date Period ------

2) p 2 -2p • O

4) x2 - 4x - 21 = 0

6) k2 + 3k - 4 = O

8) r2 + 4r = 0

10) x2 - 14x + 49 = 0

12) x2 - X - 30 = 0

14) x2 + X - 20 = 0

MO°'lLk 1 L!~JlJn 4:7¼, W,c0r@t Fr>rmtMll \ Name: ll~D=at=e :====::..::..-=..-=..-=..-=-.-=-.-=-.--:....--:....-_--:....-_-_-::::

\ Topic: I\._ c_1a_ss_: _______ _ _.,

Main Ideas/Questions Notes/Examples

The quadratic formula is another method to use to solve a quadratic

TII equation. Solve the equation below using the quadratic formula.

Steps Example

QUADRATIC Make sure the equation is ~ x2 -Sx-36 =0

FORMllA 0 set equal to O and written ~-=\ ~~-5 c~ -3l, - b= ~ 1.o..= I?_ in standard form.

©b~-Ya.c.i~ -si:."'fl1)l-3~)= \\· q

[ x= -b±./:-4ac] f) Identify a, b, and c.

®x=5:!"~\~'\~ s.-,?>_'\ -lo.) '2,rA.

Substitute these values '2. 2 -

~ s-1~ • into the formula and

IX=1GlJ-Yll SIMPLIFY! - = -'1 .., Directions: Solve each equation using the quadratic formula.

YOU TRY! ~ 1-~ x.1.-~x-1.0= o 2. ~ '2..x.~1-,x-"l=C 0..-= \ ~-=-i t:::-1.0 -b= 8' lo.-= 1..

9o.~ 2. l?=; (..: - q - b-= -7 J4.:lf

~ -g"2._ Y{1)l-1.o) = \Y4 ()1 '2. - 4 ('2.)l-'1) = I l. \

( D \:= 8 :~ .,,>, g+\,z\O @ x. = - lt m\.,,:,,,, -~'- \ 'l. 2.

~ '1 -

I ~ a--12 ~ ~2.:J.1_ • X-= l \Ol -25\ z = -2. X =- i , )- Y. s j \ ½ --

. s 3. 3x2 -12)f=O 4. ,,~ ~ 2. - 'x : O ~ ~=~ b-=-12 t::.o -b'=n la=" ~:: I b=-q t = 0 -= -b = • ~ 2a ~2

~ 'l. 't1.- Y(1)lo) = 81 -11. - 4 ( 3 X o) ~ , 'i4

X.:: '1±~f ~qt(t=" \1 +\l '-© X-= '1. !' ✓ 1'1~ ~ ~:. '2. 2.

(, \\ ~- c; ~ q-~

X=1'-l,oj\ v - Ix-:: t 1 ) o !J 2 = 0

5. x2 10x 21 0 6. t.tx 1._ 3x -= o ~-=-\ b.:-10c-::-1\-1o=101a=-] Cl= '-I b-::.-3 C.=o -b-=3 2a.=1l..

-\O~-Y(-l)(-21)-= '" - 3 '- y ( '-f) ( 0) = ~ X = ,o=. .fi; _,,, ~=-, ~ ~ 3 ± ~ "1 _.>, ~ + 3~ 0

-2 -2 11- 12- .c; ~ \O-Y_

3 ~3-3 I ~ --- o lx=~-7,3~1 ~- A -:: l 0. S) o J I 2 -

Algebra l

M2L4 - The Quadratic Formula

Solve each equation with the quadratic formula.

I) ,,,2 + 6m - 7 = 0

3) m2 - 3m - 130 = 0

5) 4v2 - V + 8 = Q

Name -------------Date Period ------

4) 4x2 - 16 = 0

6) 2b2 - llb + 9 = 0

-1-

7) 9n2 + 10n + 3 = 0 8) x 2 - X - 132 = 0

9) 2n2 + 7n - 130 = 0 10) 5b2 - 11b - 124 = 0

12) 2v2 - 32 = 0

-2-

Algebra 1

1? You must 6 h DLO

wo rk_ -fvy- Of'~f I District Formative Assessment

Name: ____________ _ Date: -----

DFAl.Algl.M.A.APR.B.03 Identify zeros of polynomials when suitable factorizations are

available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.

t/ Students may use a graphing/scientific calculator.

I . What are the real solutions of the equation y = x2 - 81 ?

a) 9, 1

b) -9, 1 c) -9, 9 d) No real solutions

. 2. Which of the following is the factorization of 2x2 + x - 3 ?

a) (x - 3)(x + 1)

b) (2x - 3)(x - 1)

c) (2x-3)(x+l)

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

3. A quadratic function is graphed. What are the factors of this polynomial?

a) (x+6)(x+2)

b) (x-6)(x-2)

c) (x+6)(x-2)

d) (x-6)(x+2)

_...I . -- ! I . ' - • ( • -

d

Algebra 1 District Formative Assessment

4. Which quadratic function has zeroes of -2 and 6?

a) f(x) = xL 4x - 12

b) f(x) = x2 + 4x - 12 c) f(x) = -x2 - 4x - 12

d) f(x) = 2x2 + 4x - 12

5. What can be determined by the factored form of the polynomial f(x) = (x-l)(x+9)?

a) The x-intercepts of the function are 1 and -9

b) The x-intercepts of the function are -1 and 9

c) They-intercepts of the function are 1 and -9

d) The vertex of the function is (-1, 9)