V

Shifting, Reflecting, and Stretching Graphs of Functions

Graphical and Analytical Approaches

Many functions can be graphed by simply knowing the general shape of basic graphs of functions and understanding what effect adding, subtracting and negating certain parts of the function have on the graph. Graph each of the functions below on the graphing calculator and then state how the change in the

equations for Graphs B through I affected Graph A.

A. Basic Square Root Function-----:=�----·----,

j\x)- £)

D. Add 2 on the inside

X +'1�0 f(x)=�x+2 �:-2.

B. Add 2 on the outside

f(x) = --Jx + 2

E. Subtract 2 on the inside

f(x) = �x-2

C. Subtract 2 on the outside

f(x) = --Jx-2

F. Multiply the outside by -1

f(x) = ---Jx

eSTEM Pre-Calculus Function Basics 3

G. Multiply the o tside by 2 ANDMultiply the -1

fix)= 2h

i i ; i i

L._.L._ .. i-·-· � __ ... i ____ i _ I 1 I ! I I . 1 . . . . . . ! i

H. Multiply the x by -1 ANDSubtract 2 on the inside " _-')(_.A,

fix)= �-x-2 -�•�

I. Multiply the x by -1 AND -OAdd 2 on the inside. -x-t2.--

fix) = �-x+2

____ l ........ i ....... i-----!----1-i ____ .L_. __ l __ .J_ ! I ! i ·--·,

! ·---:--· ,:· I ; i ........ : .. --.!---4---· ! I ! i ; ; l ; ! J I I I ....... · .. --.·--· ·--···-····-·-·----..!

··---!-----!----!----·J ___ J ··--· � ·--· ! ___ ). .. __ . ··---!

i

Based on your observations, predict how each of the graphs of the functions below would be different

from the graph of fix)= fx. State the changes that would be made to the basic graph. Then, state the -

domain and range of the funct�on.

h(x) = -�x-l +3 (D�el-� ��s

@) ,x-\-=-0 LJ.,.:1;1(- �crU ()(..: \ _,...�' 0

© � '-13. i ·-·.··--··,·--· i--· i- ····,-···-· -·-··,

___ ; _____ ; _____ , ____ ! _____ , __ .. __ !_ -·-··I _J , ..... l , __ J __ J ___ J__J_ I

I I I ! I i i i i I I ; ----+ .. --+--.. 1·--·-:---:--.... :--­_..J, __ J.____ i l

! !

eSTEM Pre-Calculus Function Basics 4

Complete the table below based on the observations that you have seen in the previous

Equation with Transfonnati ons

y =fix)+ C

y = fix)- C

y =fix+ c)

y = fix- c)

y =-fix)

y =fi-x)

Y = ltcx)l

y = a·fix)

Describe the shift(s) and/or reflections that the graph of fix)

undergoes

"�� t,·,�°"' .. � 0...

� 0.

Describe what wou e done to the xand/or y coordinates to the graph off(x)

( 'X) y-tC...)

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( )( ) -y )

( )( , \y\;

In Algebra II, you learned how to graph several other functions. Graph the basic functions mentioned below.

I. Basic Quadratic Function: fix)

1111111 ..............................................

IL Basic Absolute Value Function: f(x)= I.Al

1111111 ...............................................

1111111 . ... : ...... : ...... : ...... = ...... : ...... : ...... :

eSTEM Pre-Calculus Function Basics 5

III. Basic Cubed Root Function: f(x) = Vx

lllil•••III••• .... i ...... 1 ...... 1 ...... 1 ...... 1 ..... 1 i

; ...... j ...... j ...... j ...... j ...... \ .... : ..

!ii!il!

IV. Basic Cubic Function: Ji x) = x3

i t

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Describe how the graphs of each of the following functions will be different from the basic function. Then, graph the given functions

g(x) = (x + 2)2 - 3

0 �-t.:2..:.0 � � "";,. - 2,.. "' I' ...... i",

··:······:·"···:

···"''1""":,·· .... .

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f(x)=�-x+2-3

t!>�� � �-�Ci.i�-)C.�2.�0

!::::::!::::::!:::::!:::::::::::i::::::i::::-1-·······················"·'"···········, ....... : : : : .. - ')( ':. ·'2-

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h(x)= -lx+ll+4

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b. p(x) = (-x+4)3 +I

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eSTEM Pre-Calculus Function Basics 6

The table below shows ordered pairs on the graph of a function,fix), that consists of line segments connecting the points in the table. Use the table to create a table of values for each function below that is a transfonnation of the graph of.f{x).

-3 -1 1 3 5

5 1 -4 1 2

1. g(x) = -fix) + 2 State the shifts and/or reflections that fix) undergoes to obtain the graph of g(x), stating what changes are made to which coordinates of fix) to obt,ain

� the coordinates of point for g(x).

����

�-�� Coordinates on x coordinate

fix) on g(x)

@��!L (-3, 5)

-3(-1, 1)

_,

(1, -4) \ (3, 1)

3 (5, 2) 'S

y coordinate Ordered Pairs on g{x) on g(x)

-5�� � -3 (�., -3) -\-\-2_: \ l-\) ') 4 -t�: � ( ,, c.) -, -t.l. = \ (3� I) -A��:. 0 (s>o)

2. h(x) = 3fix + 2) - 3 State-the shifts and/or reflections that fix) undergoes to obtain the graph of h(x), stating what changes are made to which coordinates of fix) to obtain the coordinates of point for h(x).

u)\J�� ��� � ���!>

Coordinates on x coordinate y coordinate Ordered Pairs fix) on h(x) on h(x) on h(x)

(-3, 5) -�-�:. -s 3(s)��,:i. (-'i., ,,) @ )(-t:').. � 'X: -1

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(-1, 1) -, ... �:: -3 g(i)-�::- 0 (-3., 0J

(1, -4)l-.l::.-( �'U->=-tc; t -11 -1�)

®���

( ')(- .2.. , 3y- 3)

(3, 1) 3-;2:: \ 3(.i)-3-::. 0 (.,, o')

(5, 2) S-'-:.3 3(.2)-?.-= 3 ls ., !)

eSTEM Pre-Calculus Function Basics 7

3. q(x)=fi-x+3)-2 State the shifts and/or reflections that fix) undergoes to obtain the graph of q(x), stating what changes are made to which coordinates of fix) to obtain the coordinates of point for q(x).

(J)� � tt-"-l'(c;

® -'X-f'��o QA�lh. �kl-3-):.: -3 ---u·- u

,y.-:.. s

@G����

(:-x- -t"3 .) j - 2)

Coordinates on f(x)

(-3, 5)

(-1, 1)

(1, -4)

(3, 1)

(5, 2)

Graphs of Piece-wise Defined Functions:

x coordinate on q(x)

3-t;:;;. (,

1-t;::.t4

-H·?>:: ,_

.-Jt;._.o

-s +-3 -:. _.:,.

y coordinate Ordered Pairs on q(x) on q(x)

'5--?,. � � (bl g) ,-,._-:. ..... , L i�-1) -'1-l. -::- " ( 1.. -c.)

1-2,-:..-1 (o, -i)�-')...----0 L-:,.,o)

Graph the following piecewise defined functions on the provided grids.

{(X -2)2 -3, -1 < X :5: 4

1. g(x) = i

3x+2, X > 4

c.-\!

,'�) [-�� CD)

! Ix + 2j + 1, x < -1

2. h(x)= 3 x=-1

_.lx+S x>-1 2

'

!·-+--·»·'······'·-···''·····

(-coJ (l))

(- oo, �)

eSTEM Pre-Calculus Function Basics 8

{h + 3, -4 S X < 0 3. g(x) = 2, X - 0 4:- �\� to .. � Fx +3, X > 0

G�,,cr,) �:� �(�,a))

-?< -2.::. 0-v __ ,., .., .., " ... ":: -.

{.J-x-2 +2 4. p(x) = -2x-2,, -�-4,

X < -2 -2<x<lx>l

···-···· ····...l. ....... .L. .... i . .i ... J._.LL: _:: .. L_L�L .. J.L.. .. _i __ j __ .. L .... i1> � (-�J-��(-2., ,_,0(,., 00)

� � (-CO, --',)\)( �) 2.) l) (2.. I �

eSTEM Pre-Calculus Function Basics 9

(

•

Name f1\'\Sww- K� Date ���������-Period -��- Day #1 Homework

Write the transformations, in words, that exist in the equation of the function that cause the graph to be different from the basic function. Then, graph the function. (Calculator NOT Permitted) l. ... o 2 -�-t ... ,.1. f ( X) = -( X -3) + 5 2. f ( X) = .J- X + 2 + 3 - )C.

::, "' .. ..,. ,. ,.

. . . : . : : : :

111 1

3. f(x) =Ix+ �l-2. ....

5. f(x)=�x+3-l

: .. i.�1······1·"'

1 ...... j ...... j ...... 1 ...... \ ..... .L..) .. : : -+-,·""'� '-f-+-4-l-+-+-+-+-+--l

4. f(x) = -�x-2 +3

6. f ( x) = -Ix - Ii + 3

• i : ... ..l.. . ...\. ... J .. ...l. .... L ..!

eSTEM Pre-Calculus Function Basics 10

7. The graph ofj{x) is formed by connecting the points in the table in exercise #9 below with straight linesegments. Graphj{x) on the grid below.

8. The function p(x) is such that p(x) = 3j{-x - 2) + 2. State the shifts, reflections, and/or stretches thatj{x) would undergo to obtain the graph of p(x). Write the x - y rule that would transform points on

0· the graph off{x) into points onp(x).

-1.. ... )..� d> �et � d - �� s

-;::,.. @v�n� ,;.� ... -1;1,-- "'j O\.,r-�l

- ©�-�;,)_®s�et Uf �

9. Perform the appropriate operations to determine the corresponding 9rdered paii:s on the graph of p(x).Complete the table to the right. Then, graphp(x) on the same coordinate grid asj{x).

Coordinates X y Ordered onj{x) coordinate coordinate Pairs on

onp(x) on p(x) p(x) (-4, -3) +-�= � -'l-tl.:-7

('l.) -,) (-2, 1) �-:2. :Q 3-t:)..::. S (o,c;') . (1, -4) ;.-\-). =-3 -1l.....-l.= -•o

(-3,, -\0) • . (3, -1) ...�-,. :-5 -3-tl-:-I (-sj -i)(5, 2) -s-2:-7 ,�=&-

(_-1, 1,)

-9<x:s;-l i:tl!:! =iJT1

, , , , ,, , , , IHI!! 11 ! / _ � I '\ ( � ( ...... ( ...... / .•... ( ...... ( ...... ( ...... ( ..... ,:...) ....

-jx+4j +5,

10. GraphF(x)= -.Jx+i-2, -l<x<3

_lx-3 3

' x>3

�· L-9 g. v -:, O'/i ... ) .. ) .... ) .... ) .... ) ...... / .... ) .... , . ..

J ., l····+··+··+··+··+"+"+"+" ){,/, A/IIJ.O., '. ( i) I '\ ( � ( ;i ; ...... ; ...... : ...... : ...... : ...... : ...... : ...... ; ..... ,; ....: -- -0

-00 ) � '1 V - � ) -%) U O 1 5 ..J i· .. -+ .. + .. + .. + .. -+ .. + .. + .. + ..

!L!JLJ!JieSTEM Pre-Calculus Function Basics 11