Section 1 LINEAR FUNCTION YI HIT, 0 = steepness Slope =m= = = m= & + m= m= m= f 0 uudefng %t f *xc%%, 1 a) Graph the points (4,) and (3,0) b) Find the slope using the formula Find the slope by counting $ = YI > XZX ', net Y = I SLOPEINTERCEPT FORM 7 Find the slope and yintercept for the following y=mx+b "( o 4), 1 ( 6%7) o, undefined none Slope Ytteoapf > ( 0,0 ) 1
Ee y GRAPHING AND X AND YINTERCEPTS Graph the following and label the x and yintercepts a) 3 y x 3 b) x 3y 6 c) x d) y 4 m Ko, ]) (3/0) ( 0, ) so @ WRITING AN EQUATION OF A LINE y mx b 1st find the slope nd find the yintercept Xy 1 Given a point (3,4) on the line and a slope of 3 Find the equation of the line 1 m= 3 3*+9, Or 4=9+5 ty ts Try: Given a point (1,5) on the line and a slope of Find the equation of the line Given two points on the line (3,) and (4,3) Find the equation of the line Try: Given two points on the line (1, 5) and (,4) Find the equation of the line Y H )=mcxx Y, ), y=mxtb 4=34 Hy,= 3) 4=313 )tb y=3#3 1 }=b 1 M= y=mxtb Et 1 m=, ytmxtb xz x, =16 )tb FI,= t mi4 uf Mt 3 ± y# yimxtb 34+3 4=6+6 tae= Y=3xtf b
LINEAR APPLICATIONS 1) Jimmy The Hands Bowman charges customer $5 monthly plus $1 per minute for backrubs Form a linear equation: c ltts ) Matt s door service charges customer $100 monthly plus 30 per hour Form a linear equation: ( =30t +10 3) The number of hot dogs I have eaten after a given number of days, t, is given by the formula A 5t 10 At what rate is the number of hot dogs I have eaten changing? $ : otckgslduy How many hot dogs did I initially eat? 10 hotdogs 4) The price in dollars of one share of Apple Computer stock over the span of 108 days from August 15 to December 1, 006, can be modeled by the function, where x is days afte August 15th a) At what rate was the price changing during that time period? Was the price going up or down? $03/Iay b) What was the price on August 15th (0 days)? on December 1st (108 days)? Do your results from parts A and B agree? 5) A geologist is alerted to a seismic disturbance at sea that causes a tsunami headed toward the coast of Japan At that time, the tsunami is moving at the rate of 35 miles per hour, and is 700 miles away Find the linear function describing the distance d of the tsunami from Japan h hours later If it would take 5 hours to evacuate the coastal communities, will they be able to accomplish this before the tsunami reaches Japan? 6) After four months of use, Biffs Spiffy SpamFree computer had dropped to $1100 in value After ten months, the value had declined to $60 Assuming the value of the computer is linear with respect to time, write an equation that expresses the value of the computer, V, in terms of time, t UP $6645 / P(w8)=oD( 1081+6645=49199 mt3smiyrdtr3sht70@b70cqmz3scs ) +700=11 smiles yes ( 4,1100 ) m= 110060 (10/60) # l = ytmxlb 80 EE?EEttsttoot# 9 3
LINEAR EQUATIONS AND MODELS Section SOLVING LINEAR EQUATIONS Algebraic 1) Simplify: i) Distribute/Multiply ii) Combine like term iii) Remove fractions Multiply by the LCD ) Isolate the x (pick an x side) 3) Divide Go T ) *E Is *s EI IE x I'a± kt *#(a*iy* :YxIfk 's : 35 a 4 # e SOLVING FOR A GIVEN VARIABLE (FORMULAS), for W b) A h B b, for b c) a b\ c\ a) P L W r±&= vw F =w # 1 w at #µ(f=gbhe FthyhB BBE#aa,=5taIBtfC=ba@ b a&=ab h±=b+$ 1 1 1, for c d) r ar 3y, for r rata ) = 3y b a*+abr=y+3# 4
' CONDITIONAL, IDENTITY, CONTRADICTION EQUATIONS Conditional Equation Has some true values/solutions and some false values Identity Equation True for all values Infinitely many solutions Contradiction Equation False for all values No Solution i) Determine if the following are Conditional, an Identity, or a Contradiction equation ii) If the equation is Conditional, then write the answer/s iii) If the equation is an identity, then write all real numbers iv) If the equation is a Contradiction, then write no solution v) look at the equations graphically and compare a) 4 3x 3 6x 16 ' tk x+3z K ' 1=3 Contradiction ' b) x Cy x 5 5 Kate 'I Kut # conditional x 6 *#yx * c) x 3 x 3 Ly 11=1 3+6 YEE : % ' Identity 5
Linear Applications 1) Suppose you are at a river resort and rent a motor boat for 5 hours starting at 7 am You are told that the boat will travel at 10 miles per hour in calm water You head north If the current of the stream is miles per hour from the north, then how far can you travel and return in the 5 hour time X= Speed Y= ofboatglakwuti I #" " speed wake = D = R * T up against current 8T 10=8 34ns up backwith current 10 D = R * T 60 1T 10+=1 5 T pt=60o 1T T=z hwsb= D=8(D=Z4 ) A chemist mixes distilled water with a 90% solution of sulfuric acid to produce a 50% solution If 5 liters of distilled water is used, how much 50% solution is produced? % 90% 50% Xqzs 50%9115<0 =So+so ( Stxazs tqox os so softy yhl %0= 665L Scx say 3) A radiator contains 16 quarts of fluid, 44% of which is antifreeze How much fluid should be drained and replaced with pure antifreeze so that the new mixture is 65% antifreeze? " Giotto :# 704 X X 16 44*+100 1040 Zoolytsaxtogg 56 =6qnar 6
Linear regression Price $/bushel Supply billion bushels Demand billion bushels 515 155 6 579 186 4 588 194 18 607 08 05 615 15 195 65 7 185 665 53 67 1) List: ) y=ax+b STAT, Edit, enter the data STAT, CALC, 4, nd QUIT Find a linear equation for the supply in terms of the price: 5=0667 1,94 Find a linear equation for the demand in terms of the price: D= 016 +30 Find the equilibrium price for Soybeans Otey 6 +1,94 0,667 1,94=016 +3,0 +016 HD4 X $6 7
Section 3 GRAPHING QUADRATICS Warm up: Fill in the table below f ( x) x f ( 3 ) f () 3 9 fat 4 tb # g QUADRATIC FUNCTIONS Graph and label the axis of symmetry : f ( x) x it :# to What is happening to the graph of f ( x) x f ( x) x in the following? f ( x) 3x 1 f ( x) x oz rµ? Two 3 ± KI to f ( x) a x h k a= h= k= up "a" over 1 lefttrizht uptduwy Vertex= (h,k) axis of symmetry is x=h Graph the following, label the vertex, and axis of symmetry f x 5x f x 3x 3 f x 1 x f x 1 x ( 0,0 ) (93) a=3 ( z a=l,, ) (/1) a= IV x=o µ xo rd xz few =z 8
, y=h Graph f x ( ) ( x 1) 5 Label 3+ points a) Compared to y x, we can conclude ( ) f x : i) has vertex: (, ) ii) has shifted units Left Right iii) has shifted units UP DOWN iv) is Stretched Normal Compressed v) opens: UP DOWN 8 It b) iii) Domain: iv) Range: v) The minimum value = when x = I toss ) day c) xintercept d) yintercept 0=615 's y=(o ts +5 D 's 's 5=6*15 4 ) y= FEINT co ;D 3 s FEE 'Ii±FD, 9
3 VERTEX FORMULA b, a y P x x x 3 P x x 6x a =l b=zc= AH b= 6 x=±a=s,= I x= a=' 5E y=cp± = 4 ' ' =lk±eszg 1 4 at ' 3 9 a=1 Vertex=(, ) Vertex=(, ) rat Id PGKICXHT 4 Pa )=Cx K P x x 1x 3 P x 3x 1x 1 Vertex=(, ) Vertex=(, ) 10
it hk ( 4, >) y BB =3 YFEEEYIEY 3= ag4 ) 't 7 3=16 at ) =,# x#*ah = ± a, EEYT a,* xht=6!heyy#e# 0=441+4,=16a±4y I a# ', 8,6
16 36 too VERTEX FORMULA APPLICATIONS b, y a 1) Suppose the marketing department of Samsung has found that, when a certain model of cellular telephone is sold at a price of p dollars, the daily revenue R ( in dollars) as a function fo the price p is R(p) = a) For what price will the daily revenue be maximized? f a= 6%07=1,805460 c) What is the maximum daily revenue? RCGD = SGD 't 60460 ) = $18,0 : ) FALLING OBJECT WITHOUT AIR RESISTANCE A forest ranger drops a coffee cup off a fire watchtower If the cup hits µ the ground 15 seconds later, how high is the tower? o =0=16+4 *# ho 164,51 't ho 0 = (5)+4 o no =3 8 ft U = th 0 3) A projectile is fired at an inclination of 45 to the horizontal The height of the projectile is given by the x function h x : x, where x is the horizontal distance from the firing point 500 b 1 a) Find the maximum height D= µa of the projectile zso To =Ia#f = hczs 1 ' +50=1576 b) How far from the firing point will the projectile strike the ground? no = = xfxtsod #t @f t o+ go =u x=s0o = xztsdx Sof c) When the height of the projectile is 100 feet above the ground, how far has it traveled horizontally? 138ft 561 KIT, 11
,zfsi+qy ( COMPLEX NUMBERS : a bi Section 4 i ± 1) 16 ) i 64 3) ia 9 4) 5 i 8 4 : if ) 6i is COMPLEX NUMBERS Identify the following complex numbers as real, imaginary, or pure imaginary 6 4i talpay 4 3i 3 0i 0 i Imaginary 3 zi Imaginary ztszi real Pure shag imaginary,!? Adding complex numbers: + 1 + 3 4 3 i 4 3i 7 i +8 : (3+) s Gi ) 3 4 5 3 4 3+ ; 5+6 ; Multiplying complex numbers: ftpt th imir 4 9 i 3 4i ziti ) Gi 3 ii7 3 i 1 4i 3i sp } as AHAB Gi 6it8 4 Giteitsil 3 Hi 8 4 Ri @ 8 5 1 9
' Dividing complex numbers: Conjugate of 3 i is 3 i Conjugate of 3 i is 3 i Give the conjugate of the following and Simplify: 3 ( HD 6+9 I 3i 4i ( Katz is Tz ' 1 3i t ' 4i ±IHE 13 7i 8 5i 4 i Try the following: 13
QUADRATIC EQUATIONS AND MODELS Section 5 SOLVING USING FACTORING x 6x 5 3x x 3 7x 1 QUADRATIC FORMULA ax bx c 0 x 3x 4 0 x b b 4ac a Solve 5x x 1 7 4x x 14
OTHER PROBLEMS THAT INVOLVE QUADRATICS 4 x 37 1) x ) x x 4 x 16 3) a c c for c DISCRIMINANT b 4ac 0 0 One real solution positive real negative imaginary x 8x 16 0 4x 6x 3 3x 4x 5 Find a number k such that the given equation has exactly one real solution kx 4x 9 0 15
APPLICATIONS OF QUADRATIC EQUATIONS 1) An architect is designing a small Aframe cottage for a resort area A crosssections of the cottage is an isosceles triangle with a base of 5 meters and an altitude of 4 meters The front wall of the cottage must accommodate a sliding door positioned as shown in the figure 4 m w 4 m w 4h h 5 m a) Express the area A(w) of the door as a function of the width w and state the domain of this function 5 m b) A provision of the building code requires that doorways must have an area of at least 4 square meters Find the width of the doorways that satisfy this provision c) What width will give us the maximum area? 16
ADDITIONAL EQUATION SOLVING TECHNIQUES Section 6 RADICALS True or false a) 4x 16x Why? b) 4 x 16 x Why? c) If x=9, then x 3 Why? ISOLATE the radical and square both sides 6 5x 1 3 5 x 0 4 x 5 Check your answer Check your answers Check your answers 17
Try these: 10 4 5x 4 3 x 1 3 1 x SPECIAL CASES x 5x 9 3x 4 x 4 3x 6 x 4 Check your answers: 18
USUBSTITUTION APPLICATIONS (part II) Get into groups and try the following: A water trough is constructed by bending a 6foot by 8foot rectangular sheet of metal down the middle and attaching triangular ends (see the figure) If the volume of the trough is 9 cubic feet, find the width/s correct to two decimal places w a) Possible domain for w: 8 ft 3 ft b) Height: Area of the base: 19
Section 7 SOLVING INEQUALITIES Linear Compound 3x 1 4 6 z 4 16 4 z 4 10 Absolute Value lessthand Greator x 4 x x 4 7 x 4 7 3 x 5 1 3 x 5 1 Special Cases: Use your calculator to Look at the following and think about the answer If there is no solution, then say no solution If there are infinitely many solutions, then use the proper interval notation x 5 0 x 5 0 5x 1 4 5x 1 4 x 5 0 x 5 0 0
Putting it all together Solve the equation algebraically, find the domain, and its related inequalities using your calculator 1) a) 15x x 6 0 b) 15x x 6 0 c) 15x x 6 0 Domain: ) a) 1/4 1/4 x 3x 10 b) 1/4 1/4 x 3x 10 c) x x 1/4 3 10 1/4 Domain: 3) a) x 3 x 7 b) x 3 x 7 c) x 3 x 7 Domain: 1