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NON LI N EAR FU NCTION SPRING 2019 DEVIN C. SMITH, M.ED.
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Page 1: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

NONLINEAR FUNCTION

SPRING 2019 DEVIN C. SMITH, M.ED.

Page 2: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Unit 8- Nonlinear Functions Introduction Video and Warm-Up. https ://wc boe. d iscove ryed ucation .com/learn/player/ lle 1 lfda-cff7 -492c-a412-e8f0e042 b93d

l. Think about the varying speeds of racecars as they approach a turn or straight stretch. How might a racecar's speed change over the time of the race?

d~..--,_"'-~-' w '-<-v< 4p r r-,1,,- ~ J. a.. vJ ~. \-€.. OW-ìv~~ o~ 4- ~tr t= l6h-- ~¼,;J k~ S" ·\re+<>J.,,

2. If you plot a racecar's speed compared to time during a race, would the graph be linear? Explain.

•~ tltu~r's s~~J Cisra e: O.V\& ivic..re.~.s

µDIA..\~ d,A,e_o...,r 3. Classify each function as linear or nonlinear. y= 2x + 5 y= ✓s.6x Sx + 3y = O y= -ll y= xz 2x2 +X+ 3 = y 4x- 6 = y Lìv,..q_a, t-.)D V\. \: ---e ,v- l--ì~ Lìw.a..r ~O"'-kv<.•-r Noi,..,):"'-(__ ,.,r ~ÌH...~

4. Which of these inequalities can be used to identify the domain off (x) = ✓x + 5 - 2? a. X~ 0 b. x ~ 5 C. X - 2 ~ Ü

ßl. X+ 5 > Ü \ S. Use graph shown to the right that models the speed of a racecar during part of a lap.

a. Take on the role of a racing announcer and create a story to describe what the graph reveals about the race. What do the line segments R, S, T, U, and V reveal about the speed of the car?

202

200

Racecar Speed y

u J \ I \

\ T \V 'R I \ 'l.

\ I \ s \

X 2 4 6 8 10 12 14

Time (s)

Page 3: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 1- Introduction to Piecewise Functions Unit Essential Question: How are algebraic, numeric, and graphical representations of piecewise functions related?

Lesson Objective: You will analyze function that are a combination of two or more equations. You will represent these functions algebraically and with graphs. For real-world situations, you will interpret what the functions mean.

Activity #1: Graphing a Racecar's Changing Speed. Writing the Equations. l. For each of the segments on the graph to the right,

write the linear equation for each.

a. Segment R = ~ ::: -3 },. 1-.. +- \ q ~ b. Segment S= J ::. I q '5' c. Segment T = '1 "' 5 }.,. X. ~ \ ~S- d. Segment U= ~ " & óö e. Segment V= ':j - - .;l )( ~ ;;ld ~

2. Take another look at the graph. At what key points on the graph does the graph show changes in speed?

202

200

J u~.s 4 s.Q_~\s

~ ~ CJ>-..-6S

I 2. ~ e.ovJ&

] 196 Q. li)

194

Racecar Speed y

192

o

-- ,___ - - ,-- - --,- u J \ I \

Ì\ T \V ' R I \ i\

\ I \ s \

> X

2 4 6 8 Time (s)

10 12 14

3. Using the coordinate plane below to graph the equation you found for Segment R.

2

Page 4: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

4. Compare the graphed equation from above Racecar Speed and your graph, what is different about the two (2)

'- graphs? ·Th~s j'n. ~~ (c\~C('(«jf == ~--- ·-- __ ---=~ _.:= --=-~--~: ~-~=-~ ~ ~~,---r-- - ó,i..\i't.0-.)!:. DV'. ~ ôlA, ö..J e} _ __ _ __ _ _ _ _ >--·- 1- _

\,Jk(<.. 4--\.(__ c+\.~.r ÌV\ Va..rÌbv~ ¡:,icuS.

S. How or what can you do to your equation to make them exact same?

4,v--to...>i,_,-l- of Ì S J eC{"lA.\~}

·-!- - t--- - - ·- . .. - -- - -- -- -- -·-- - •---~--- - - - •--- ·- - - """·- - ----~- -

).O l,

ioo l-l-+--+--+--+---+--+-+-+-1-l-+--+--+--+---+--+-+-~l-l-+-+--+-~

195 \ ,~, '--+-+-+--+--+---+--+--+-~--l--+-+--+--+--+--+--+-+--J--1---+-+-----l

\ (h, l--i c'r" -+--+-+--t--1--l--+--+--+--t--+--+--+--t--1--+-+--+--+--t--+-+-+--1 \

I C)1 -· -- \.""➔--t--+--+ - - - ~+--t--11--t--t - -

l'ÎZ.s--- -..-- ti:.. !:li

--

6. Find the intervals for each segment equation: /

a. Segment R = __,~,_::._··_3'-;;1_'/..._t-_lq_a Interval (x seconds):

b. Segment S= _'j_,___.::::_I '1.....:....:f>;__ Interval (x seconds):

c. Segment T= \f-= 5 JC}-- 'j.. .\.- \8°.5' Interval (x seconds):

d. Segment U= 4(-;:... WO Interval (x seconds):

/ö /2- ,...,

e. Segment V= l~ -c:2 'f.. -1-ól ~ 'i Interval (x seconds): /Q,¿~vi 7. Writing the composite function for the ra ceca r's speed:

o ~ ;)_St_,e_,~. J ANP t-/ k ~.$ L.J ~( t., U-<'-S & a~ ¡--i., s:¿c_¡ ;-i_ a-Á rs- ~s.

i = -{-;. -\- l q~ I

~= ¡qs

'f'';_ 1- ~ 1~5 I

~-= 2,00 j :_ -¿'f. t?-1-1.\ (

ô~)(.LZ,

lL. X LY '-/!=._,<L'-1,

(¿,f 'I-. L \-Z..

p_!::_x.51s--

3

Page 5: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Activity #2 Another Racecar's Speed ln the previous activity, you analyzed the graph of a racecar's speed and modeled the information algebraically. What if you are given an algebraic representation of a racecar's speed? How might you use the equation and the graph to interpret what is the happening during the race?

Consideration: The function defined below models the speed of another racecar on the track during a part of a lap.

{

2x+194 ifO~x<l 0.5x+195.5 ifl < x < 3 . f(x) = - , where x 1s the time in seconds 197 if 3 ~ X < 11 -0.5x + 202.5 if 11 ~ X ~ 15

l. Use the above piece-wise function, determine the value of f(O),f (1),f(3),f(ll), and f (15) .

.f(o ') = J (o)+- \'i~ :: l'îY ~( ,) :: e>, 5l 1) t- I q 5. 5 = ¡q (p

'tb,')::: ,9·1 :t ( Il) ::: - O , 5 ( \ \} \- ;z.o Z. 5 = ~ -=. l ~ 7 } l \ 5) :::. -o . 5 L \ ~) + 20.1.. 5 -= I e¡ 5'

2. At which of the following times is the racecar accelerating or decelerating?

~ x = o.5( ct. x = 9 • X= 1.5 ~:e. X = 1Î)

c. x=4 _f. x=13.5j

4

Page 6: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 2: Introduction to Step Functions A step function or staircase function is a piecewise function containing all constant pieces. The

constant pieces are observed across the adjacent intervals of the function, as they change value from one interval to the next. A step function is discontinuous cannot draw a step

function without removing your pencil from your paper.

.. •--o

Features (of step functions): • utilize open circles and/or closed circles on the graph open = point not on graph; closed = point is on graph

• horizontal "pieces"

----_¡.2--i.3---"4---"5➔x • discontinuous ( cannot be drawn without removing your pencil from the paper)

• notice the resemblance to a set of steps

5 4

3 o 2 1

-1 -1 -2

·3 -4 -5 Math81ts com

¡-3· f(x) = o;'

3;

x<-2 -2 ~ x ~ I x>I

• may, or may not, be a function. Check with the vertical line test. This example is a function.

• Step Function (Stairs)

Question #1: A. Graph the following step-function on a

coordinate plane. 4; -4~x~-2

f(x) = 2; -2 <X< 2 -3; 2 ~X~ 4

B. Using the step-function, evaluate the function at each the of the following: f(O),f(1),f(3),f(-4), and~-

t(t>) ·;:. ;i .ç (3) :: -3

}-(\) = 2- }(-c.\) ::: 1

4~

.,.

- ~ ~ ~

~ ,

5

Page 7: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Question #2: A. Graph the following step-function on a

coordinate plane.

{

-1; -4 ~ x < -I f(x) = 2; -I~ x ~ 2

-1; 2 <X~ 5

B. Using the step-function, evaluate the function at each the of the following: f (0),f (1),f (3),f (-4), and f(S).

Y.(;,) ·::. ;i. S<,):: J. ;(~)-=--l

~ (-t-(} ::.-1

S(s) =-I

.4 ~

.... ...

..... ..... ó

H

Classwork For problems 1-2, evaluate each piecewise function at the given values of the independent variable.

{6x-l if x c O l. f(x) = a. f(-3) b. f(0) c. f(4)

7x+3 if x z O C,(-3)-ì ·¡(b),+3 7('1)t-3 \-=- -,i] [=- !>] (::. 3"\)

2+x if x <-4 2. f(x) = -x if-4s.xs.2 a. f(2) b. f(3)

1 if x > 2

-(z..) = ~ (3) -x ,~ -41 3 ED 3. When a diabetic takes long-acting insulin, the insulin reaches its peak effect on the blood sugar level in about three hours. This effect remains fairly constant for 5 hours, then declines, and is very low until the next injection. In a typical patient, the level of insulin might be modeled by the following function.

40t + 100 if0s.ts.3 220 if3<ts.8

f(t) = -80t+860 if8<ts.l0 60 if 1 O< ts. 24

Here, f(t)represents the blood sugar level at timet hours after the time of the injection. If a patient takes insulin at 6 am, find the blood sugar level at each of the following times.

a. 7 am b. 11 am c. 3 pm d. 5 pm I hrf 5 ~(5 q 'hrs il ~r:S- 6

£-10(1) + loo l= J:J.ól -~O('Ù \-~b @~8 l: ''-ifil l>=· ''1 §J

Page 8: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

For problems 4-7, graph each piecewise function.

{x+3 if x <-1

4· f(x) = 2x-1 if 1 l x¿-

{-1

6. f(x) = x-3

ifx <0 if x¿O

ifx5,3

if x > 3

{4-x if x 5, 2

7· f(x)= 3x-6 ifx>2

"\ .n. '4. I I

I I

.. J V I

.... I/ ) .... .... / I .....

/ J

/ { i/

./

, ,

H

JC /

7 ... 7 .... .... à V .....

~ ... / /

'-

H

Y:-5 H

V .. .... I/ - .... ,/ .....

/ V

V V

V ,I/ ·- ,,

ü ,

~ 1 f\. I ~

Î'\ I -

' I '\ J I I ... . - .... ...

,, 7

Page 9: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 3: Absolute Value Functions vs. Piecewise Functions

Objective: Mount Everest Expedition Have you ever felt "on top of the world"? If you want to stand on top of the world, you will have to climb to the top of Mount Everest, the world's tallest and most deadly mountain.

A film crew is documenting a hiking expedition to the top of Mount Everest. The expedition of the film crew begins at Kathmandu, with an elevation of 1,300 meters. The director will stay at the base camp, which has an altitude of about 5,400 meters above sea level. The summit of Mount Everest is at 8,850 meters.

Mount Everest

The table below shows the relationship between the film crew's current elevation and their current distance from the director at the base camp. Both distances are measured in meters.

Fill in the table by determining the vertical distance from the crew's position to the director's position at five locations: The crew's initial position in Kathmandu, Nepal A point on the way to the base camp At base camp A point on the way to the summit At the summit

Elevation (x) Vertical Distance from Base Camp (y) 1300 l-f 160 ~,coo I, l..\oO 5400 o (noôO ln hh

8850 '3 1-I Sb

l. What coordinate pair represents the film crew at base camp?

\ (s.LJoo >º}j 2. Using the graph from your graphing, calculator, what does the X-intercept represent on the graph?

Ti.e. i.-ìwk.re.-qt ìs -\-kro:.--+ wk.r<. +lt. ç::,- (Y(J...IJ ,s 'lt ¼- s~ l-<.ve-l (#.~ ~ J.·,-t'.(!4-. 3. Find the y-intercept on the graph. What might the y-intercept mean in the context of this problem?

8

Page 10: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

4. For what elevation(s) will the film crew be a vertical distance of 1,800 meters from the director? Explain how you determined you answer. 3

1 (pöO M qv-Ä ì I zoo ~, 4-\e., -h' \ IM (LV'l,v.J ; s C\ v'el h·to.,.,, (

d:ù-a..V\4.. .-,f \,~DIV\ -H't,-4,<,\, ~ d:.,..a .!.o.r I>--' k.vi +w..-y .:a-.--G I , ft)D M above b.ue.,

C.~ l 'í 1 '-loo -1 i J;,oo) cr i, fóO l'V'- ~Io.,.., fo tt4 {l_t> ,,,,,._f {

S. Does your graph on the calculator represent a function? Justify you answer.

~es I .f;r e\Jex'f 'J.. -J(l,\v¡¿ l ~ ~ S Ô\, unt¡ve. '{ -Va.,\~ ì"' ~-s Ve.-\ 0t- ·\--tov,.

6. What piecewise- defined function can be used to model the relationship between the elevation of the film crew and the vertical distance that the film is from the director?

ln the previous courses and this year, you learned to represent distance between values using absolute value notation. 7. At the direction of your teacher, use the graphing calculator to graph y= lx - 54001.

Does the graph of y = lx - 54001 match the piecewise function you defined? Why or Why not?

~e_sl +lt,_ ~el:.L'-'-'1;s:.(....

r-c. r~ fu s--a..,..e e.i \J o..,+c"oYt tìV'-.J

fu~ovi ~.J4-- , a, ~So Gk -Ja, \ve.. e.1 u C\, ..\.; o v¡ ~ ~-\--- e{ --\\e_

yv'\öùl/\.\. t.\)~US4- -e.ï; F~c,\.·-~-tºV\ ÌV'c..l0c~ a res-tfc' C!.- kd

do<>AA.:""' V;~ a. r-es-J \ h c1 r-es- +e-,· e_ t-ï °"'1 < -tlc Î(l\/\.,j& ¡A-

lfJ h"~ e Ç"~OWV\ ì (À, +k- ~na.,1-r'o VJ ,s p 1' €.C..,~ L0; ~

9

Page 11: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Classwork: 8. Write the piecewise functions for the graphs shown.

a. b .

l~

I" i-.... V ' I ",

' - :¡ - J

I

1'

.. , -

fe I .,, ' ~ "' I V - I I - , l.

' / I ' - 11> ,,

1 ï',+3, ··LjL )(.f4)

J, Î;.-3 I X ~0

9. w· nte an a so ute va ue ec uation an ' ' , \

' I

\. ' , J

' .,. ' .4 , ,

b I d a piecewise function for the following graph:

Building New Graphs What effect does including different constants have on the graph of the absolute value parent function?

10. Parent function of an absolute equation: ----"~---\.....,Y-i"--\...__ _ 11. Possible effects eq~ation of an absolute equation: L-c..,f\-j fl ... tj "'-+ u-p \Jou,)\ S:J,c-<,. +eJ·/ ww--~N.55.

12. Observation #1- \ ,d ~ k. ~~.\j.¡c. ~ vr P~a+c''-<.. K.. cÍovJ ~ _ 13. Observation #2- \,;. -b I poS~\/:L h d3h+ "'e.dav.\t-.J-{__ V\. 1-d+. ( ~e,c,,re,f.;Ù 14. Observation #3- Q l ~ \ O-,;:;\ p-c-s,-w'(. :v+ a.::..-\ ~£\a-b j f°'-pk

tJVW 'l.-4,>~:s.

10

Page 12: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lessons 1 -3 Quiz Part A: Can a graph tell a story?

The bottom of the ocean is a mysterious place that is home to exotic creatures and plants. Scientists have discovered many fish and sea animals since the development of submersibles. Watch the video to get started.

https ://wc boe. d iscoveryed ucation. com/learn/player /3cbc6177-f66c-460a-b514-6a42b889 5817

1

) t ~- l

Write a story about what happening in the above graph after watching the video clip. Be sure to reference each section of the raph Thj wr_ite the piecewise function that mp~el~ ~he ~aP.,h,

1 [:\2.u~r-ic.._-~ìs ~.s kow "{Q1...TT ~ t~SCc-rt<-(f' \/<:,i..>_.; ~+oc~ ~ ... w,;HS Cl. s-+or'i G\.-t<lo--'4- ~ :)vt,y. ~ e.r;,'~ \~ bo..~J .,,V\

+\.A. ~ -r ~ \i' \,-t. r e.,,k("L V\-c: 1 e. a..c.,\,-\, ~('_J.,º i:-i.

3 ~ v-J l yv1 i lf\.or G<''fo t s,

J.. - bo es" '-+ t-.) o + ~ -kA -e.v\(!.,L e¿t <!~ s:.w.Ái 'o"'- (pî('<..Q., 4, \'/ I _. hc>~S..., '..\- 1o...__~ ~·+on-( off <l {.,.._f h N º f¼~-""''-~ +o e."-<!..,"'- ~~-\-{°"'-..

D - ]Des ¡..)o+ a. tk,1, ...... r-h

i-f- Wri'-1-es ~ f>;eQ.ew;u

3- W1i".kS 4. 1.): ef<-v-, ,· u....

J- wr.'ks

tuV\oJ101,,, ¼ ..... + ëo« í'C.¿+ ly f-1 ,,dt f s fi.<_. fv-.,-..@ ·\-r' º1-1 -C:, r o.,,.... l y '-{ f q__ r 6- h ~-J/ cr1,,,. /.Jr o- v- ( y "3 r + 5 does~'.\. M"J~ \ +t---- j Í°'t~

11

Page 13: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Part B: How Many T-Shirts Should Vou Purchase for a Fund raiser?

The student council is planning a fund raiser to help pay for field trips and school dances. They plan to purchase T-shirts online and sell them. They estimate that they can sell about 70 T-shirts. Below are price sheets from two different T-shirt companies.

T-Shirts D-Luxe Best Tees

Quanttty Cost Per Shirt ,

Fewer than 24 $5.64

24-35 $4.88

36-71 $4.64

72-142 $4.08

143+ $3.72

Quantity Cost Per Shirt

Fewer than 24

24-35

36-71

72-142

143+

$4.50 (plus a one-time print fee of $5.00 )

$4.25 (plus a one-time print fee of $10.00)

$3 .so (plus a one-time print fee of $15.00)

$3.00 (plus a one-time print fee of $18.00)

$2.90 (plus a one-time print fee of $20.00)

How many r.smns should they order to spend the least money?

Write piecewise functions to model the relationship between the number of T-shirts ordered and the total cost of the shirts in dollars for each company.

@ _,7 ~F\s ~\ e,ae.,I.A, Cor«-<!,\-'--'{ vx-:~ r ì ,e,'fik,v->,. se jp~ W\,:l'\.o-r ~«-oîS !6

\'() î .'o,~ 'J'í' ir\-:5 ~; J Y\ i Ç,-t--.....\. ~ro,..,S.. fº cil,)'<S ~' L frl "'k,~ Ip\- ~(>e\-~-\ v--,-y,' ~

D pl- f\Jn a,~.

Use the information from the functions you just wrote to determine how many T-shirts the student council representatives should purchase and from which company they should purchase the T-shirts.

t-1 r~s 3pfs

~pb

I ft~ [) p'ö

- CJ-lß.r'i .e_¿<_p\~ IA--S° ~ J l.J:>1--,'l.· d ~ o..v~~,.-N-i/ $

1:-''t.p(o,1~ ;· )vsl;+, 6 bv4- W,~ l'\('..12 e}.¡., <:I ~,'-í-7

.J._ V\~ k ·\t. CTT ,' n cor V'l£j_

fjê> ~~-

12

Page 14: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 4: Square Root & Cubic Functions The Square Root Function: The parent function is y= ✓x Let's look at its graph and table of values using our calculator:

X y=✓x

,.. I l'l lo. ~l"'\'~,c-

o ô I ' t./ ;;i

q 3

What is the domain of the square root parent function? x 2 o Why? (: o.-,,.. \ ..\--

How do we know that this is indeed a function?

1~ ~c,,,'v,..___ do ~4..;, \)o- \v-<- fu,<- , .s -e;<..o..<t.A- \y Ôl-'\.(.. ('a..,_,,,,J~" 1/41...£.

The Cube Root Function: The parent function is y= if;

Let's look at its graph and table of values using our calculator:

X y=¼

-5 - J. -I -, o o , ' 'l ~

-+----+--i..-.--t--t-+-_,_..,:-,,!~ ~-)11--+--+-+-+--t--+-+-1!-+-

.. ~ 8 . L.: I _, '"-·+-+--+-+ -l-- !-+-+- -1--+

-+-+--l--l-+--+-·-t--t--+-¡---t6 - ·-·· --+·-+--l--+--+--a>---+--+

I i i

.-1 .. ➔I I

I i ! f

j / lï

I ' i I t... -+--t--1-+--+--l .. ·+·t ·2• i- -- --+ .... · ... .,... +- - -+-J·-- •i-~,...~ __ ,__,__,_.....__¡ ,

-: ol..J8 r=k5-~~¿,¡,,_ -2-t4--L6---: -+rn - I ì I ¡2 I

. - - l : : 6' • -~- ·- -•-'---4----< ---+- -1--

I ¡ ¡ ·-· • ... l ¡ i 8+-l--1--+--+-+-+-+-+-t--+-

-···· --f- ··· ······ ····· -·-··+··1-~o ····· -··r··+···+--<-__,___, · .. ..1 ..... What is the domain of the square root parent function? A\\ r-eA.\ ü:-5

How do we know that this is indeed a function?

h:>r 13

Page 15: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Functions have the same transformations as the absolute value function y = a Ix - h I + k.

Given y = a • f ( x - h) + k

if lai> 1: Vertically Stretch the graph by a factor of lai if lai< 1 : Vertically Shrink the graph by a factor of lai

if a <O: Reflect the graph about the x-axis

(h, k): Translate the graph horizontally h units and vertically k units.

Let e be a positive real number. Let y= f(x).

Vertical shift e units upward: h(x) = f(x) + e

Vertical shift e units downward: h(x) = f(x)-c

Horizontal shift right e units: h(x) = f(x-c)

Horizontal shift left e units: h(x) = f(x+c)

Transformations of S uare Root Function:

h(x)=✓x+2

h(x) = ..¡x+Î_-4 (h, k): ( - I 1 - '-0

h(x) = ✓x +3

h(x)=-✓x+2

(h, k): (o I 3)

-

t~>,H ·.¡-;î'.·:t H.··

With SQUARE ROOT FUNCTIONS when you are completing the table of values ... you will have x-values on ONE side of the initial point (h,k). Why? v-te.... \.\£Lvt_ ô v,. \~ S"\..; t➔ 4..J ( +-n.u,"s \o.J-cJ) +,Lt,

14

rº.1"'ts ~ ~~ r~+ ~h·oV\.

Page 16: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Transformations of Cube Root Function:

h(x)=¼-4 (h, k):

J I I: t '

I y 8

6

4

2 X

-IO -8 ---ó -4 t

r - ~ ¡ ' .,,,~¡~~

¡ I V

I

-\0

y=3¼

h(x) = 1/x+2 +3 (h, k): l ~~ I ?>)

¡ I

-8¡---ó'-4'-2 +2 i ~2 i -4

-6

.l8

f 6 • 8 i 10 •·

' -I

(h, k): (D, o)

j(x)=1/x-2 Ch,k): la,b)

15

Page 17: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 5: Graphing Square & Cube Root Functions

In these notes we will ANALYZE the graphs of Square Root and Cube Root Functions

Domain Restrictions based on an equation:

1. Dividing by zero is undefined: a denominator can NEVER be equal to zero.

2. The square root of a negative number does not exist ... we NEVER put a negative number under a square root (unless we are dealìnc in complex numbers).

Case #2 above: No Negatives Under the Radical Sign!! x;:::-: O

Do you have a square root? Do you have a rational power that has a denominator of 2? If not, then you don't have to worry about this restriction.

f(x) = ✓x 1

f(x) = X 2 Domain: The set of all real numbers x ;:::-: O

y= .Jx-5 y= .Jx+3 f(x) = ✓2x+3 I

y=(x-4)2

Ds 'I ~ £ 5 \) . ' X)' -3 ì>o~:'"': xì -3 Ùo1,~;~ '. J( z t.-/ ""-O..• V\ . O~IN'.>.• Y\ • - z_ t-\: w\.. i )(.\-3=ö ól '#. t- 3 ·o:.{) f. - L-{ =-O

'(-5 :::. e S'o\vZJ 'Í~-3 ..¡, ""-3 }-z.. 'i.:=, '1 X. ::;.5" ;)

llS" "'° ~\.k_r ~a..A--....v- +-L.. rtu,,.,......h-<..,.- l,'.n) Ç.·ir<J. A\(ìµ<LiS h ~ ~ 'f Now let's go back and define our ctìaracteristics from Unit 2 with the square root and cube root function. The Square Root Function: The parent function is y= ✓x

(h, k): ~)

x - Intercepts: (ó10)

y - intercept: (o1o)

Domain: X 2:. o Range: I\- \I )qO. \ ~s

Increasing: X ~O

Decreasing: ')(~0

-· I_.,

-10 -+- \

-· t- i -;..

.. t-

. I

16

Page 18: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

The Cube Root Function: The parent function is y=¼

(h,k):~

x - Intercepts: (0,0·2

y - intercept: (Dl e;, 2 Domain: A-\\ ~\ ~ ',:5 Range: /+I\ ~e..c...\ \l. 's Increasing: 'i. 2..ô

Decreasing: X LO

l e l- t· . ' i . I • t ! :-

+- .\ • ..Z..-+ . ..¡..

i l I f f : I l

I I t t r

I •

X

6 8 IO

¡

·!1 : . . 1

I •. ; I l

Complete the following. Graph without a calculator. Then verify with your calculator and use to find your intercepts if necessary. Round to the nearest tenth.

y=-2✓x-3

(h, k): X-int: y-int: Domain: Range:

(310) (:3,c,) ~ I"-

X

r(x) = ✓x +4

.,. ' I ·2 j .;. • ; X

-IO -8 -6'-4 -2 • • •. ;. - ·~2

(h, k): (o I L.\J x-int: b) IV\ y-int: lo, :\J Domain: 'l.. Z O Range: ,¡ t. ':I

-io -1l ;-6 -4, -2 . =2

2 4 6 8 IO !· !·. ¡

t.

y=✓x+2-5 y=;¿/x+4-l

(h, k): (-;;z ,-s) x-Int: {à3, o) . y-int: { l) I - 3. S'ß!P) Domain: ')( 2: -;;. Range: ,, '7 -Ç

1

-I O -8 •-6 . -4 ··-2 • ¡. ' !· ¡ .. ; I a..2 i ....

; ' '-6 - - -8

1' • ' -I

(h,k): (-4,-l] X-int: • O

y-int: o , SSi-1 Domain: A--11 1?,u; Range: 4-B e,,tl,\~

17

Page 19: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 4 & 5 Quiz Directions: Describe how the graph of the parent function changes based on the transformation below.

l. Parent: ✓x Transformation: '!-..~

2

3. Parent: 1/x

Transformation: 51/x + 3

S. Parent: y = ✓x, Transformation: y = ✓x - 3 + 1 ,

2. Parent: Vx Transformation: -Vx + 3 - 7

4. Parent: 21/x

Transformation: -421/x + 1- 7

Directions: Graph each parent function below. Then graph the transformed function by moving the parent function. DO NOT USE DESMOS! USE YOUR BRAIN!

6. Parent: y = 1/x Transformation: y = -½ 1/x ,

10

9

• 1

' s 4 )

2

t • •

10

' • • ' . . 1

' s 4 )

2

·10 -t -t · -, ·s ·4 ·J -2 ·1 1 2 l 4 s , 1 • , 10 " •10 -t -t 1

•2 •)

., -S ·4 ·) ·2 ., 1 2 J 4 S 6 1 I t 10 x 1

' I

' . .. . . . I ~ ♦ . .. • ·7 .. .,

·10

18

Page 20: NONLINEAR FUNCTIONsmh314.weebly.com/uploads/4/8/9/2/48920851/nonlinear...algebraically and with graphs. For real-world situations, you will interpret what the functions mean. Activity

Lesson 6: Investigating Rational Exponents

Have you ever heard of "happy" numbers? Examples of happy numbers are 263 and 19. Take a look at these two numbers.

263

22 + 62 + 32 = 49 42 +92 = 97 92 +72 = 130 12 + 32 + 02 = 10 12 + 02 = 1

19 12 + 92 = 82 82 + 22 = 68 62 +82 = 100 12 + 02 +02 = 1

Do you see a pattern? What do you think might be the definition of a happy number?

Example #1 Write in the numbers below to show why 32 is a happy number.

32

32 + d- '=~ 12 + 2

3 = 10

8'+ 2 =1 o

Exploring the digits of numbers such as happy numbers and looking for relationships between them is one way to think about the structure and properties of numbers.

Rewriting numbers in different but equivalent forms can give your insight into the properties of the numbers themselves. For example, in elementary school you learned to rewrite the number 512 as 500+10+2 to help you understand the place value of each digit in the number 512. ln middle school, you used your understanding of numbers to rewrite numbers in scientific notation, which is more convenient for representing very small or very large numbers.

Take a look at the number 32 again, but this time, look for a way to rewrite it so that the expression is equivalent in value. Think of other ways to represent 32 using only multiplication, division, exponents, square roots, and cube roots.

Example #2: Equivalent Expressions Using mental math, write three (3) different ways to represent the value of the number 32. Representation #1 Representation #2 Representation #3

19


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