VOCABULARY Completing the square The process tmt"I!QWS )101.1 to write an ¢l(pre$sion ofth~ form )(1 + bx as tho square of a binomIal
CA 8, 10: Solve quadratic equations by CQmpleting the §ouare
Make ajJerfect square trinomial Find the value of c that makes x 2 + 16x + c a perfect square trinomial. Then write the expression as the square of a binomial.
\;§9~Hj[!g~~t STEP 1 Find half the coefficient of x. 1 = 8 STEP 2 82 = 64 Square the result of Step 1. STEP 3 Replace c with the result of Step 2. x 2 + 16x + 64
g
3. x'+4x-5
t4~~l&~~~
(x + 5)(x - 1)
1
Make a perfect square trinomial
~~§W'§!{~
Find the value of c that makes the expression a perfect square trinomial.Then write the expression as the ..., 20.. ~ square of a binomial. ~ _
The trinomial x 2 + 16x + c is a perfect square when c = 64. Then x 2 + 16x + 64 = (x + 8)(x + 8) = (x + 8)2.
--z. ' .,
I
2. x 2 + 14x+c
;1~fi§~§tij
49; (x + 7)2
3. x 2 + 22x + c
4. x 2 -9x+c
'~Nlw~~'i
81; (x -
~)2.
'~,
Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
-~:\t.. '2.'l..~'44
-j!-:= S
S'l..= 2.'S"""
,)2-2J.!(_e
HW: P. 286 #'5 4 - 7,32 - 34
)(~-ax+7=O
)(l-.
e )r~e. ::-7-tc...
(.X-4) 1.-::=_1·1-/ h ~_
z L)("t;. .to l(
+-1;;' ) e
i>
~~'X-S)"&-:= _/""~
,-:-D
~5.iA'VI 'iite ""Tl"I€. STHN 1>1'7te-j) 'Fi:J-ar'h
~ -:: ('K-h ) ~ I<.
2
1
1. Solvex= 2"y 2 for y 2.
VOCABULARY Foous A fixed point that Hes on th$ axis of symmetry of a parabola Dlrettrlx A linl\il thllt is ~fPQrH;licJAl:ar to • hill Olxls of symmotry of" parabola
1
VOCABULARY Focus A fixed pOint tha~ lilll$ on the axis of symmetry of a parabola
VOCABULARY Directrix A Hne that is perpendiclIfiH tl) ihe l'ixii!< of $ymm(:!tryof a p?irabola
VOCABULARY Foc.us A fixed point 'that liJ1lis On the aXIS o.f symmetry of a parabola
VOCABULARY STANDARD EQUATION Or: A PARABOlA. WITH VERTEX
AT TtliORfCIH
The s.tandard fOfm Qf the equation of a parabola wlth at (0, 0) Is 3$ follows:
liertex
fqa(l\ll)/t
F()CU$
OI",etrt~
Ads at S\'lIlP~try
x2
'" 4AY
(0, p)
Y "" ..::::JL
Vertical (
y2 '" 4px
(p, OJ
x '"
_::.lL
x 'co ()
Horizontal ( y
=
)
IU
2
Graph an equation ofaparabola
t y2. Identify the focus, directrix, and axis
x =-
of symmetry. Then, make a sketch of the graph.
1§X>:~:qll:Q~~; STEP 1 Rewrite the equation in standard form.
t
x = -
-8x
=
y2
Graph an equation of a parabola STEP 2 Identify the focus, directrix, and axis of symmetry. The equation has the form y2 = 4px where p = -2. The focus is (p, 0), or (-2,0). The directrix is x = -p, or x = 2. Because y is squared, the axis of symmetry is the )C:" 1x-axis. ttl'
Write original equation. Multiply each side by -8.
Sketch Graph
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.
2. x2=2y
2. x2=2y
(0, ~), x ~o, y ~-~
2
r ::( 0,
o
>1;>
2
"l.- )
'1 ':" - '/"l-
3
Identify the focus, directrix, and axis of symmetry of the parabola. Then, make a sketch of the graph.
Identify the focus, directrix, and axis of symmetry of the parabola. Then, make a sketch of the graph.
1. y=-6x
3. y=-6x
~~§tYI!§~:; (-1.
2"
0) x~1.
2'
y~O
Identify the focus, directrix, and axis of symmetry of the parabola. Then, make a sketch of the graph.
Identify the focus, directrix, and axis of symmetry of the parabola. Then, make a sketch of the graph.
4. y=_lX2
s.
4
i~§g:(ttL§~0
x=_ly2 3
?,§9lliQt:i9~;:
(0,-1 ),x~O,y~1
(-3
4"
0) x~~ ,y~O 4
3ff5bd#r ~
-4 -:;:'t.
1,..
4(' ::- -4
f
:-
-4
-7
1
-/ 4
• HW: P. 598 #'s 17 - 53 odd (*Two day assignment*)
5
so ...
CA 16: Graph and write equations of
VOCABULARY x 2 = -8y. Identify the focus, directrix, and axis of symmetry. Then, sketch the graph.
focus A fixed point that jiB'S on: the axis of symmetry of a parabola
4f ;;
..
Dlr~Qtrl)t A Iln9 ttmt 1$ p"fp~lldicutgr to {hi) ~)Ch, symm&trvof II parabola
-Cl'
p: _8/.,
or
-:-2-
r ~ (0,- '2.)
1
Graph an equation of a parabola
Graph an equationofa parabola x
=- t y2. Identify the focus, directrix, and axis
STEP 2 Identify the focus, directrix, and axis of symmetry. The equation has the form y2 = 4px where p = -2. The focus is (p, 0), or (-2,0). The directrix is x = -p, or x = 2. Because y is squared, the axis of symmetry is the x- axis.
of symmetry. Then, make a sketch of the graph.
"~~i1:q~19,~j; STEP 1 Rewrite the equation in standard form. x = Write original equation.
t
-8x
=
y2
Sketch Graph
Multiply each side by -8.
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 2.
y2=-6x ,,::;.-- "f'e~S '''"'
~A-nvl!l'
"l(
.~,
~§Q~ij~i§fj2 (-..1, O),x~..1 ,y~O 2 2
-'C'D~ wle: b.0-Aq\lS
M-~ ~'-'A.~.
~
v-A-1
\As. ~
-nTE
1\~
~"'\ V-~"\.,-o
~
~ ~l>
\V-JO
M~
-\
-
2...
'1:, ...1>
r..tJ.PR I+- ofI~ S
F I ~ ..
ftVJ. C) ""tt>
MA~
ANP
<1>. I toL~ VT"\.."~" IA-
~ ~
~~ ~O'M.'lS.
2
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. 3. x 2 =2y"::;- ~S" ~DSITtIiE'
'1
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. .,Pet"' $
4. y=_lx2 ~ 4
X t-", -4'1
;~PbiJI,,§:N~
4p;2-p:. 7/,+
,,"~ve-
Y_,o,
"
'P: -\ " Ill-
-I
-'Z...
'~.:
Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola. DP&'J".5 5
.
x=_ly2
3
'-12.;::-
I
:>S:9:~frt!§~' ( -.1 , 0), x =~
-3)c" tt!:- "Mf!!-W'vV~. -'/!ArC.-.,..,-,.. oToq
• HW: P. 598 #'5
17~
53 odd
,Y = 0
4
-(
-;1..
:J"{i ~ 1.1 3
Graph y "'.
_,;,,2, 4
Idl.mtlfy the focus, directrix, and
axis of symmetry.
xl. -= - 4'( ~A.lj H~6m1V#:'-
CA 16: Graph and write equations of dr(;J'~'---
_____________---j
'1
41' ;: -4 ..r f=--/ F' (~I -I) ,1>
I
'1
7
/
-(
VOCABULARY Cifel/! The set of all polllls i.,\; ;0 that are e
VOCABULARY SfANnARD EQUATION lHE.ORIGIN
os: A CIRCLE WITH CENTER AT
1l1e standard form of the equation of a drdl! wlU'! tef\t~ lit (0, 0) and fMIU$ r is t\$ foll()WS:
~111$ Th", di~hmce
r pejw",en the Q;!ntF.lr
1
Graph an equation of a circle Graph y2 =
-
x 2 + 36. Identify the radius of the circle.
~(SSig9t'P:~}, STEP 1
Rewrite the equation y2 = - x 2 + 36 in standard form as x 2 + y2 = 36.
Graph an equation of a circle STEP 3
Draw the circle. First plot several convenient pOints that are 6 units from the origin, such as (0, 6), (6, 0), (0, -6), and (-6, 0). Then draw the circle that passes through the points.
STEP 2
Identify the center and radius. From the equation, the graph is a circle centered at the origin with radius r = = 6.
-J36
4.
y2
=-x2 + 49
i~g~911[~:
Graph y2 '"
-x2 + 16. III.ntllY the radIUs of the elrel,,_
2
Graph the equation. Identify the radius of the circle.
[email protected]
2. x 2-IS =-yZ
['~g'~iji!C5N\
1. x2+yZ=9
3
J2
3
yl.~ ~~ /
J"
r~{/i &:~
-"'76 ~ 4.~
• HW: P.60S #'527- 37, 47 - 69 odd
3
What 1s thedlstonce between (-3, 5) and (4, -I)?
CA) vY3
© Y&1
,:'~g~gli,gi(
.t
'"'~~~'''
Let (X"YI)
=
(-3,5) and (x 2,Y2) = C4, - 1 ).
d =JCX2-X I )2+(Y2-YI)2
CA 16: Graph and write equations of circ.l. e",,_
.>ol.L-s._ _ _ _ _ _ _ _ _ _ _ _ _ _--j
= J 49
+ 36
= J(4-(-3))2+(-1-5)2
=
J85
j~~,~W~~~ The correct answer is C. ('f;) ~)
(ID
(Q)
VOCABULARY 1.
What is the distance between (3, -3) and (-1, 5)7
~A'~$WEij:, The distance between (3, -3) and (-1, 5) is
·······'·'4-!5
~\3-l-I));..+ l~Sy:-:~-&)l ~ ~ \\o;~{P4 "" ~:: -J
':
lttl 0 S-
Clre~ Tne set of ali pOil'lls ix.. }1 that are equl
Center The fixed point that is equidistant from ail rhe poirm:; ill) a clrcltl Radlu~ The dh,,11mc:e r between the o;:ent"r ",nd !lny point i..>;: on a circle
.»
~\fr::
1
vvrite an equation ofa, circle
VOCABULARY SfANI:JARD EQUATlON OF It CIRCLE WITtf CENTER AT
The point (2, -5) lies on a circle whose center is the origin. Write the standard form of the equation of the circle.
It!l; s!andafd form or tile equation 01. a clrele wlttt
~9~Yf!g.~.~
lH£ORI.G1N
.;ent1l1 at (0. 0) and radilis r Is M Mh.lws;
Because the point (2, -5) lies on the circle, the circle's radius r must be the distance between the center (0, 0) and (2, -5). Use the distance formula.
r
Write an equatiilii ilfa Circle Use the standard form with r equation of the circle. x2 +
y2 =
x2
+
y2
x2
+
y2 =
=
=59 to write an
r2
Standard ferm
(5 Y
Substitute.J29 fer r
29
Simplify
=
-J (2-0)2+ (-5-0)2
=-J4+25 =J29
IF I MAY GO OFF ON A TANGENT ...
2
2. Write the standard form of the equation of the circle that passes through (5, -1) and whose center is the origin.
• HW: P.60S #'5 27- 37,47 - 69 odd 3. Write an equation of the line tangent to the circle x 2 +y2=37 at (6, 1).
2. '(":;
~ 5"" +(:-'1 f-~ \J""·;u:... X 'G..-+- '-1~. :;2.,l?
.3
Slope OF
/2.44'l) rv.!;
-::
@ (v, I)
SWP E
Of'
rl~ O\JB'Q:. ~
-L ~
11+ N-C, 12"~
@ (Y'll)
l PEo<.PFN iJ Uu.. A-~ -ro I
'\
QA1>\"v~ @ irr-A:T pDlr~r' )
\ :: -~l~) +-'0 \ -:: -3<..p -I-
b
b ;; +3, 3
1. Write the standard form ofthe equation of the circle that passes through (5, -I) and whose center is the origin. ,
CA 16: Graph and write equations of
e.tlip_s.......,'--____________---1
SO IT'S OCCURRED TO ME ...
THE FOUR SEASONS
"'......
!{..)~ hH!\ "<':~~ .!~\<~~; ;e,;~I)Nn
~;,.qhl..'\f~l ..'Jl:
»: l~'<>f
NOTfuNG,
:.;),i!h~,;,;:,l)('n'~
~~ .h-er(2..€
'D\ S(.J.,L ~ Sol ,-0 G. \\-\--e
~~-r
~ e;:n-\Is,
1
VOCABULARY - ELLIPSES
VOCABULARY Ellipse Tn •••t Qf.1i pc.nts F.uch tit,t the £uO'. o11he distanc:es' bat:\v~€n Pana two fixed poim3, 6. (X)flSi:t€u"lt
¢alicdths f.ot'1" f~
Fod Two fix-ed points 1:1 an dflpsc 'f(ortlc~ Thtt ;')Qint~ ;tiwhich t!"t~ fo:::-ci inte-r:5('ct lh~ "!~ip~
!in~ thn:.n)~lh tt,~
----_._---._---_. CO--'lfIrttc:,es The· rminrs (}"( irr:ersr.crion of an (1!1iP!~~ aorlthe Hne ptrpandicuhu tC the Ma~j u:ds at 1he ~&rter
Mtnor ~is Tha ling. ·segment dlst joins th~ ~o-"e·rtk7-$
VOCABULARY
VOCABULARY
The major and minor alees are £If lengths 28 and 2:b, r'Q$f)<¥¢{flle1Y, Where it >- b ,. O. TM tC¢l·
STANDARD EqUATiON OF AN ELUP5S Wtffi eENTSR A.T mE (I/UQIN Equ~tklll K2: y~
;? + b'"~ ",:1 x~ . y2 .,•.,. "1" ••.. ~
a~
~N~~
M*"AAb
'1~rtjC%
¢¢'\f¢rtl¢C&
~llt'()l\tl!1
'!±_:'L'O)
{O,±..i2J
(O,:t:..E,)
(:t.JL, 0)
=1
Ver!lcal
~-h~
IF 1iN.. \()\~t+
-\'N... X?'" \ CJ (\~£[
--
f'i\-V-S-\-
'f\\A(y\
~~
= ,;.~" tr' .
\
\aQr
~ +\'-L, .Q.\ \ \ps,st...\f\
lie on th~ maj(lr a)(ls at a d!stance- of c units from the center. where c:2
x c\
,s
r~L+W(\
\r\CJq -=t:dY\ ~d
2
VOCABULARY Identify a and b. Is the major axis of the ellipse vertical or horizontal. \ , ex"2.::- 49;:D Q.::
-i",
CC·Viir-.ex:
(e, ~l '1QtT$'):
... ~r'.e;('.
I-~a
~~
b '--;: 2-S 1.
C(l·V~~X:
I~
-.1 Ellipse Ylith vertj~! I'I3Ajnr ..,.JlI
..£+£.1
f.~~=l
b
'
" 0" ,:S OtIUJQA.(S
\n
""'I~
9
C\.
-
4. 75;' + 36y2 '" 2700 2'l;>O
)12
lAr--I082--
d. 01.-:: LA ~
'v ....o
0.:
~ 'i$
4S----l:::> ''O-=-
-\f.45 -;
cls..i'\ .0('1'"\ ! {\ e),+ur
100",
-
2.?
= 1 Ct..,
~ \~~-t (\v.C'i\~
Identify a and b. Is the major axis of the ellipse vertical or horizontal. 3, o.'?--:;: t l.., ;b Q:; -14
2'(;>0
25
64+45""1
flQp;. W10 ~"1l0nllll ...j.r .,.;s
.z
:? + ;?
49
'0 :=- ±-'5 V l-~ \-l;':lN"~ +;;:v\
."t>
-::b
10:
-i
Identify a and b. Is the major axis of the ellipse vertical or horizontal. U-cc-. ?-I!r\..l.-et,1
S,
&:; ~O=D
:3
~'l.",
uf\cUr
-
(A;::!.G::""'..
.. 4~ \:;> :~q
~.C:;-
:c:t'Z-
--
6,...§li- + 63/ "" 5103
21vc>
'Slo~
c.'-~IS -;{)
f>.
:;J:;{1s-x 1:~,'
$1<>3
Slo,? ~. ()o..;c
\,0 :;
~\ =:1:''1
'\Z3 ""
~
-Vi
3
t:
If 9x2 + 4y2 = 36 is written in the form x~ + = 1, what are the values of a and b, is the graph vertical or
horizontal?
1§§t:(QIi'Q"~:
a = 3, b = 2
• HW:P.612 #'5 19-29,31 -41 odd
4
t:
If 9x2 + 4y2 = 36 is written in the form ~~ + = 1, what are the values of a and b, is the graph vertical or horizontal?
~~~~~rJ a = 3, b = 2 ~1"'ItYi~
CA 16: Graph and write equations of
~
Lks\'
~ut:>'I.?
O~
p~\CV S.
Uc:iSo!::.c1N.
eUips""""''-----_____________-;
VOCABULARY
VOCABULARY
STANDA.RD EQUA.TION OF AN EL.UPSli
ATTH£ ()1l1~IN
wrm CENTER
EquatlOIi
MlIJOt,ws
VlIrt:kes
Co.vilrtJees
x; y2 -+-=:1 <'12 b2
Hor1%1!fltlll
(:t....:L 0)
co, =...£..)
({i,±~)
(±~,Ol
.~
r
~+~"'1
Vertical
The major end minor (.I)(es are o.f lengths 2<* and2b. respectlvel)',. where 8 :> b
;lJ
O. Thetocl 01 the elllps@
lie on the: major axl$ at a dlSotanee of ¢ units from the cQnter, Wb~r~ c~ "" ifI2 ".,. Ji4 •
1
VOCABULARY
, '1Cf'liJx.:iO,af
Graph the equation. Identify the vertices, co-vertices, and ~~Ci o~he ellipse. C. "l:: ! \ttl _ , :- -; C;:::::f
J=i
8·16+"9 'JbI2--;
=1
(:f 4, 0 ')
Cov~( 0, :t:z,)
Y'/}~
(= -.Fi ,0"")
().. ':<
-.:tT~ ~ a...;:,:i: -4
b; {q""
Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse.
x2
x2
9.
y2
8·16+"9 =1
i~:Q[~I!R~;: (± 4, 0), (0, ± 3), ( ± 17, 0).
0;;
~ h·~-:f ~
y2
36 + 49
±J41 : :II
b ~ :lJ"31...
-0-
j: f.t;
2
9.
t~ + -£
= 1
C::=
i:~g!sY!lq:~ll
~4<1-3(.., ="-{i3
10. 25x2+ 9y2 = 225
m-
""2"~
(0, ±7), (±6, 0), ( 0, ±m).
10.
25x2+ 9y2 = 225
;~9~~UJ!QN: (0, ±5), (±3, 0), (0, ±4)
Graph the equation 4x2 + 25y2 = 100. Identify the vertices, co-vertices, and foci of the ellipse.
:'§Q~QTig~:i STEP 1 Rewrite the equation in standard form.
4x2 + 25y2
= 100
4x2 + 25x2 = 100 100 100 100 x2 y2 25 +""4 = 1
Write original equation. Divide each side by 100.
Simplify.
3
Graph an equation of an ellipse
Graph an equation of an ellipse
STEP 2
STEP 3
Identify the vertices, co-vertices, and foci. Note that a2 = 25 and b 2 = 4, so a = 5 and b = 2. The denominator of the x2 - term is greater than that of the .0- term, so the major axis is horizontal. The vertices of the ellipse are at (±a,O) = (±5, 0). The co-vertices are at (0, ±b) = (0, ±2). Find the foci. c2 = a2 - b 2 = 52 - 22 = 21,
Draw the ellipse that passes through each vertex and co-vertex.
soc
=51
The foci are at ( ±
51 , 0), or about ( ± 4.6, 0).
• HW: P.612 #'5 19 - 29, 31 - 41 odd
4
Identify the foci of each ellipse.
1.
.E.+Z_I 15
16-
C~ ~\"'-lC:;"'" !-....[\.:o -± \
:Aij~W~ffi~ (0. -I) and (0. I) x2y2 2. 49+43"=1
Graph and write equations of I
I-
:=-
;~i:'§~[~~ (2.0) and ( -2. 0)
VOCABULARY
VOCABULARY ~Yp#b
f\
v -; '-I~-4S
Tho '.01 of.1I
po,~ts Psuc~ tl\'llh~
difference of the
fix,"" poims, 0.11.<1
t~"
roci. ;, a ,"<)".Ia"\
STANDARD EQUATION Of A HYPERBOLA WITH CENTm AT THE ORIGIN Equation x'Z y3
lIelUces Ttl. points of InwfsO
bn~;\f$tA:lI$
V~rt~
M)'l!IpiQtes
-1 $.4 - -1)-(:-
lforlZollttll
y=±
h aX
(±."tL 0)
¥~-~~-1 b2-
V<1rtJe'i'1
Y = ,±
."k". X
(O.±2:,.1
,,2
Transverse JUIS T<')e li!18 segment fna<:: :cnl~r.eC':s rhS'
ven.ices
0'(
a }'\Yf"HHbo;G:
\
1
VOCABULARY
VOCABULARY
TIro f\1C! lie 00 tbe Itans¥ers,e ads.,c units flOffl tile center. wl'i&re c2 "'" /)1 + tY _
flvper/ltll.:a wltb vartiell' tr.weru axis ,:z , ..... _L .. , il'
,~ ,?i::;:".?t'1f~l>}~f1!
b'
'
Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola.
1.
L
16
K _
- 49 - 1
Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola.
1.
x2
y2
16-"49
=1
~qgq!19ii~ (±4, 0), (±J65 . 0), y
=
± ~x
2
2.
=it-
- X2
=
1
2.
L
36
-
X2
=
1
~~~Q,IIQffij (0,±6), (O,±ffi ), y=±6x
3.
4y 2_9x2 =36
;!§§~!;!I!§lj);J (0, ±3) , ( 0, ±
m ), y
=
±~ x
3
Graph 25y2- 4x 2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola.
i[~g@Ji9~~j STEP 1
Rewrite the equation in standard form. 25y2 - 4x 2
12Y
=
100
4x2 = 100
100 - 100
L
4
100
..i _
- 25
- 1
Write original equation. Divide each side by 100.
STEP 2
Identify the vertices, foci, and asymptotes. Note that a2 = 4 and b 2 = 25, so a = 2 and b = 5. The y2 - term is positive, so the transverse axis is vertical and the vertices are at (0, ±2). Find the foci. c2 = a2
-
b2
= 22 - 52
=
29.
soc =f29. The foci are at (0, ±f29.) = (O,± 5.4).
Simplify.
The asymptotes are y = ± ~ x or y = ± ~ x
STEP 3
Draw the hyperbola. First draw a rectangle centered at the origin that is 2a = 4 units high and 2b = 10 units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.
• HW: P.618 #'5 IS - 41 odd
4
t Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square
1 7", ""l'l
\ L\-
of a binomial.
~ '" 7
1. X2 + 14x + c
C"'" ...q0
2. X2 + 22x + c
CA 16 and 17: Graph and classify _ _ _ _ _ _ _ _ _ _ _---l
cQnk~s,
-
STANDARD FORM of EQUATIONS OF tRANsLATE!) CONICS In the followiNg
~
the po,nt (h,
P_h
(y - k)'i: = 4p(,. - It) (~- hf- "" ilp(y - 1<)
1'2\
3. X2 - 9x + c /
U(~
0("\ ' '2--)
-
ill is. the vertex
parabola and the e~ml"t or Ihe eth". Circle (x - h'Y- + (y - k)2 = ,2 of the
L:;;.
¢I)1>ics.
Gener~lseC(Jf1d4fi'!groe ~u.atlon An
formA.~
equarlon of the
+ B.<:v+ (.~ + Dx''- E;v+ F= 0
\;oritontol ""is Vert/oal ".I.~
Discriminant The expression .tf1 - 4ACto[ tha
equation Ax2 + B.;y+- c;A -+ D!(~ £y+ F= 0,
Ve,1Jcal axls
used t<:> f:clentify
{I
conk section
Hori'~lIt.l..ws
Verfical axit
1I i !
Pl4-ie
A
t~
,(
PrN'f;>
C-
ll,., i U:... ''''l...l.. l../ tV\,
A$f'{;f:l·'T!~{';''2
i'tl.-l
'1 etA
N
k \...!c;v-J,
1
1. Classify the conic given by 2x2 + 2y2_ 5x + 3y-l = O.
ClASSIfYING CONICS USING THEIR EqUATIONS Any conic can be desllfilllld by a &eneral equ~t!(>1!1
Ax-!l
SOOOI'\d-d~
In I( \'tlld y:
- 4 C:z--) (2--")
+ B>;y + Cy~ + /)11. + Ey + F = O.
The e~pr(!S$lotI EP - 4AC Is the d!scrlmlrtalll d the C(lnlc equallQI1 lOnd can be ~ to Idenlffy 1\, l>l:!cilmlr;3I1t lIlre 111 Collie Slt -
-Me _
0, B"" O. a~d A '" C
*'
~ - \ k,
A:=-
c...
CA ruz.L~
Try- it. 5x2 - 3y2 -IOx- 12y- 22 = O.
CIrcle
8 2 - 4AC
0 and either B i' 0 or A
8 2 - 4AC
(l
Parabola
8 2 - ..lAC
0
Itwl>fhola
C
Blip"
If B 0, each axis of the coni<:e Is hOfiZ:Olltai or vertical.
\.
"Sc:Pf\a~
2.. ~f'~
Xs I+-tE
I "/.,,
~ MOV~ ~ ::)::l:
SQ~
-t
mp<~
"8..A~
h
'£?QUo.-l'\4loi'-\
-;0
~-~
-4(.\)(,,) -=-.-"'\U~ 2. Classify the conic given by x 2 + y2 - 2x + 4y + 1 = O. Then graph the equation.
for Examples 6 and 7 10. Classify the conic given by x' + y' - 2x + 4y + 1 = O. Then graph the equation.
(x-l)2+(y+2),=4
2
for Examples 6 and 7 11. Classify the conic given by 2x2 + y2 - 4x - 4 = O. Then graph the equation.
for Examples 6 and 7 11. Classify the conic given by 2x 2+ y2 - 4x - 4 = O. Then graph the equation.
Ellipse (x-l)2 -3-+
y2
(5
for Examples 6 and 7 12. Classify the conic given by y2 - 4y - 2x + 6 = O. Then graph the equation.
=1
_ for Examples 6 and 7 12. Classify the conic given by y2 - 4y - 2x + 6 = O. Then graph the equation.
Parabola (Horizontal) (y- 2)2 = 2(x-l)
_ CJy: .,.~
f...5/
-;:;;Lv
C'1
'2. -
4\--~ ~ - ~
+-C'1L--~+.4J=--~+-4
-dY ~(';J-2),k;;:: -d-
e"j -0)
L..
~
-;:;J
-+d X ~ cJx-~
l ~ -d ) ~ -::- d (Yr'/)
3
for Examples 6 and 7
for Examples 6 and 7
13. Classify the conic given by 4x2 - y2 - 16x - 4y - 4 = O. Then graph the equation.
13. Classify the conic given by 4x2- y2 - 16x - 4y - 4 = O. Then graph the equation.
Hyperbola (Horizontal) (x-2)2
(y+2)2
- - - --=1 4 16
~.L~ ,
-0 "" 4
1'4, '--' J\
(i) C4l~(1'_<+YT"<\ -=sj -t-( \("i1.-1-,,\,( +4)1 o
'1L )(-2Y" - l\:) -\--iY-'" \ \0
Classify it
:c
~
4 +- 4(-
4 (X-
coniC
Classify the conic given by 4x2 + y2 - 8x - 8 = O. Then graph the equation.
;~9,~P,~!g~'~ Note that A = 4, B = 0, and C = I, so the value of the discriminant is: B2- 4AC= 02 - 4(4)(1)=-16 Because B2- 4AC < 0 and A F C, the conic is an ellipse. To graph the ellipse, first complete the square in x.
Classifya conic 4(X2_2x + I) + yh 8 + 4(1) 4(x-I)2 + y2 = 12 (x-l)2 y2 -3-+12=1
From the e,9.!!ation, you can see that (h, k) = (1,0), = 2 ~ 3 , and b = Use these facts to draw the ellipse.
a=
Jl2
13.
4X2+y2-8x-8=0 (4X2- 8x)+ y2 = 8 4(x2- 2x) + y2 = 8 4(x2 - 2x +? ) + y2 = 8 + 4( ? )
4
• HW: P.628 #'s 25 - 390dd ,47 - 57 odd
5
Tell which conic section the equation describes.
1. y=tx2
CA 16 and 17: Graph and classify cQni~cs''--______________---j
VOCABULARY Tell which conic section the equation describes.
3. f-~=I t[~§~R1
Conic sections The intersection of a plane and a hyperbola
double-napped (one
2 X9 =1 . fs+
4
1
STANDARD FORM OF £QUATIONS OF mANSLAT£D
I.Graph(x+ 1)2+(y_3)2=4.
CONICS
In the following equations. the potnt (h. K) is the vertex p.~rIlMla and tile teoter lit mi!> t,tllet t"'l\i~~.
Q-N"'\Bt- @
Of the
Ch<;k
(x -I,,,? + (V
P_bc»a
(y - k)t '" 4p(x - III (x - 1l)2 "" 4p(y - k)
£lfip""
(X - h)~
........(ii .......
(x - 11)2
"~
Hypemo'a
(X - II)~
- k,):l '"
(y- k)2
+~i
+
r"
=1
jy _1<)2
'-;,T"" =
')I - k)1
(:.1,3.)
HolitQut~1 ~~ Velti.~1
l
a~ -"ii"· "" 1 tr..::~): _ (.~. ::.~~ = 1 a. tJZ
axis
""rilantal rods Vertical axl.
IloriZOllt.1 rode Vertlcal 21ls
1. Graph (x + 1)2 + (y - 3) 2 = 4.
2.Graph(x-2)2=8 (y+3)
:~9~~!19iji circle with center at (h, k) =
C.B, ~)
y't.~ ~j (-
1,3) and radius r
=
2
YfW-fI-€;o\....A-
"'\
6~::' !,...Lf' l--l-j
)t'-+-+-1-+-+- I...4_.;.-!...I....;.•.:
4P~~"''P P~::L Sl~~ 0i"1::''N.S \J..{>
0>ClA3. IS ;;2
4<
"'-NITS
i
Aeov~ 'J~e;? @lPl,~'3::)
2
2. Graph (x - 2)2 = 8 (y + 3)
3. Grap h
(y _4)2 (x+ 3)2- -
=
9
1
!~Q:~gI!st~~ parabola with vertex at (2, -3) ,focus (2, -1) and directrixy =-5
~Tf\-~Le
c;;,.oes
\ \J..NIT i"~(r\ ~
~ 'D ~ -: ~ Vvt-\.l-rs
\ ~ ,,~ '-\ - 'DI ~or-..\
"?~~';':d~,~;,,~;:,X~ 3. Graph
"
(x+3)2- (y-4)2
9
i)~Q;!lijj;igti~ hyperbola with vertices (-4,4) and (-2,4), asymtotes y = -2x - 2 and y=2x+l0
=
4.Graph (X_2)2 + (y-I)2 = 16 9
1\0
<1
~~N.~ ~I(~ :!&r
Ul'ii~ >( "\lea-neG'S Pr<2.e ~ =-
L\ tJ-N'\S. ~ \'A-Ii::
Got rc;;Y2... ~.i'
.,p:;;;;- 3
\), """'(.:cp 0.-:>)
UN rfS
C.CNQ-Q...
Fa-<>.fY'
(':1 i:>1~n.;o-.\)
3
4. Graph (x - 2)2 + (y-I)2 16 9
Graph (x - 2)2 + (y + 3) 2 = 9.
1~'gg9I!Q~
:;~qbg[!g;B';~
STEP 1
Ellipse with center (2, I), vertices (6, I) and (-2,1) and
co-vertices (2, 4) and (2, -2)
Compare the given equation to the standard form of an equation of a circle. You can see that the graph is a circle with center at (h, k) = (2, -3) and radius r = [9 = 3.
T STEP 2 Plot the center. Then plot several points that are each 3 units from the center:
(2 + 3, -3) = (5, -3) (2, -3 + 3) = (2,0)
(2 - 3, -3) = (-I, -3) (2, -3 - 3) = (2,-6)
STEP 3 Draw a circle through the pOints.
Graph (y- 3)2 4
~§QgQI!~N~ STEP 1 Compare the given equation to the standard forms of equations of hyperbolas. The equation's form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (-I, 3). Because 0 2 = 4 and b2 = 9, you know that 0 = 2 and b = 3.
4
STEP 2 Plot the center, vertices, and foci. The vertices lie a = 2 units above and below the center, at (-1, 5) and (-1,1). Because c2 = a2 + b 2 = 13, the foci lie c =.JI3 "" 3.6 units above and (".T•.T ..., •..r ..., .....,....,..".,.....,....., .....,....., below the center, at (-1, 6.6) and (-1,-0.6).
STEP 3 Draw the hyperbola. Draw a rectangle centered at (-1,3) that is 2a = 4 units high and 2b = 6 units wide. Draw the asymptotes through the opposite corners ofthe rectangle. Then draw the hyperbola passing through the vertices and approaching the asymptotes.
• HW: P.628 #'s 25 - 390dd ,47 - 57 odd
5
