Composition

A jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces produces and sold. How many necklaces must be produced and sold in order to break even?70 necklaces71 necklaces72 necklaces73 necklacesDont knowA jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?x=number of necklacesR(x)=number of $ earned for x necklacesC(x)=number of $ spend to make x necklacesx=?R(x)=C(x)90.75x=24.50x+4770 solve for x.66.25x=4770x=4770/66.25x=72CComplex NumbersTime to dance!

Multiply each number by -1

Multiply each number by -1 again

Think: *(-1)*(-1) is the same as *(-1)

Multiply each number by (-1)

Addition is shiftingMultiplication is stretchingLast time (Part of the truth)Addition is shiftingMultiplication is stretching & rotatingYou saw hints of this before:Multiply by a negative causes a flip.This time (The whole truth)

We need a new number line

i2i-i-2i

We need a new number line plane

i2i-i-2i3-2iComplex Numbers1 is a unit of rightward distance-1 is a unit of leftward distancei is a unit of forward distance-i is a unit of backward distance1 and -1 cancel each other out, they can be combined.i and i cancel each other out, they can be combined.1 and i do not cancel each other out. You need them both.Complex Numbers2+3i number that is 2 rights and 3 forwards.2-0.5i number that is rights and half a backwardGraphing becomes more complicatedWe dont have an x number line and a y number line anymore. We have an x number plane and a y number planeFour Axes:Re(x)Im(x)Re(y)Im(y)Arithmetic on complex numbers1 and i cannot be combined. They are on separate axes.1+i cant be simplified, just like x+y cant be simplified.Treat i like a variable and you will be ok.Remember that i2=-1 and (-1)=iThis can be simplifiedExamples

Examples(2+3i)+(1-2i)=2+1+3i-2i=3+i

(2+3i)-(1-2i)=(2+3i)+(-1+2i)=1+5iExamples

Examples

You are not done until you have the real and imaginary parts completely separateSimplify:

80-8080i-80iNone of the above

Simplify:

B

Powers of ii-i 1-124Powers of iMultiplying by i makes a quarter turnPowers of i 1=i0,i4,i8,i12,etc i=i1,i5,i9,i13,etc-1=i2,i6,i10,i14,etc-i=i3,i7,i11,i15,etcEach power of i is a quarter turn morePowers of i 1=i0,i4,i8,i12,etc i=i1,i5,i9,i13,etc-1=i2,i6,i10,i14,etc-i=i3,i7,i11,i15,etcTEST:Divide power by 4Number of turns is power/4Powers of i 1=i0,i4,i8,i12,etc i=i1,i5,i9,i13,etc-1=i2,i6,i10,i14,etc-i=i3,i7,i11,i15,etcTEST:Divide power by 4Powers of i 1=i0,i4,i8,i12,etc i=i1,i5,i9,i13,etc-1=i2,i6,i10,i14,etc-i=i3,i7,i11,i15,etcTEST:Divide power by 40,1,2,3,etc.25,1.25,2.25,3.25, etc..0.5,1.5,2.5,3.5,etc0.75,1.75,2.75,3.75,etc..Powers of i 1=i0,i4,i8,i12,etc i=i1,i5,i9,i13,etc-1=i2,i6,i10,i14,etc-i=i3,i7,i11,i15,etcTEST:Divide power by 4IntegerDecimal part 0.25Decimal part 0.5Decimal part 0.75Examplei1414/4=3.5Starting at 1:Integer part 3 three full turns (still at 1)Decimal part 0.5 half turn (wind up at -1)End result: -1i14=-1Simplify: i 2,789

i -1 -i 1None of theseSimplify: i 2,789

2789/4=697.25Integer part is 697 697 full turnsDecimal part is 0.25 one quarter turnEnd result: one quarter turn

A) i