mat151

Mere som dette

CHAPTER 4: THE BEHAVIOR OF INTEREST RATE

af ANIS AZIZAN

MCR3M0

af Sebastian Acosta

FINANCIAL MATHEMATICS

af Maria Paula Hernadez Vergara

MAT.116 4.1

af David Kedrowski

Denice Ceniceros Algebra

3.5

Transformation of a graph

1. Great auntie- tries to stay near you, go the other direction (parenthesis)
2. Example: (x-4) Go to the right 4, opposite of what you are told
3. Elevator man- wants to get a tip so follows the same direction as in the equation
4. Example: x+5 Go up 5 like the equation says to
5. 
   

3.3

More with imaginary numbers

Note: i= Sqrt -1 , i^2=-1

If a>0 , then sqrt -a = i sqrt a

1. Simplify

Example: sqrt -16 Example: sqrt -3 X sqrt -3 | isqrt-3 X -isqrt3

=i sqrt 16 sqrt 9 | i^2sqr3 Xisqrt3

=4i =3WRONG | -1 (3) = -3 RIGHT

Example: 2 +_sqrt-24 Example: sqrt -2 X sqrt -8

2+i sqrt 24 /2 =i sqrt 2 X sqrt 8

2+isqrt 24 /2 i^2 sqrt 16

2+i sqrt 4 sqrt -1 (4)

=-4

i = sqrt -1 i^2=-1

• treat like a variable

Example: i+i=2i Example: 5i-3i=2i

Example: 5i X 3i= 15i^2 = 15(-1) =-15

Example: 5(-4i)= 20i Example: -6(2i-1)=12i+6

Example: 3i/21= 3/2 Example: (2i+1) (2i-1)= 4i^2-2i+2i-1

=4i^2 -1=4(-1)-1= -5

Complex Conjugates

1. (2-3i)^2 2. -1(5-2i)^2

Check Notebook

Complex Conjugates

3 . 4+i/5-i X 5+i/5+i = 20+5i+4i+i^2/ 25-i^2

= 20+9i-1/25+1 = 19+9i/26

Complex Numbers and Exponents

i= Sqrt -1 , i^2=-1 , i^3 = i^2 X i =-1 X i =-i

i^4 =1 , i^5=i^4 X i =i , i^6= i^4 X i^2= -1

i^12= (i^4)^3= 1^3=1 i^50=(i^5)^10= i^10

=(i^2)^25=(-1)^25 = -1

1. i^10 2. i^30 3.i^57 3.

3.2

Quadratic Formula

x= -b+- sqrt b^2-4 ac/2a

Graphically- X intercepts

(2 of these)

(1 of these)

Analysis

the Discriminant b^2-4ac

b^2-4ac > 0 2 real solutions

b^2-4ac=0 1 real solution

b^2-4ac <0 2 imaginary solutions

As long as the equation is of the form ax^2+bx+c=0 , I can use the quadratic formula.

Other ways to solve

1. Complete the square when in standard form. (changes to vertex form)

Example 1: x^2 -8x +9 =0 1. Move the constant to the other side

x^2 -8x=-9 2. (b/2)^2 (8/2)^2=16

(x^2-8x +16)-16 =-9 3. Add and Subtract (b/2)^2

(x-4)(x+4)=-9+16

Sqrt (x-4)^2 = Sqrt 7 4. Put in vertex form

x-4 = +_ sqrt 7 5. Take square root

Example 2: x^2 - 8x=7 IN NOTEBOOK

Example 3: 3x^2 -6x +z=0

When a=/ 1 ...

3x^2 - 6x= -2

3( x^2 -2x ) =-2

3( x^2 -2x +1 -1)=-2

3(x^2-2x+1)+3(-1) =2

3(x-1)^2 = -2+3

3(x-1)^2 =1

(x-1)^2 =1/3

x-1=+_ Sqrt 1/3

x= 1+_ Sqrt 1/3

Factoring when factorable

Example: 2x^2 +2x-11=1

2x=^2+2x-12=0

2(x^2+x-6)=0

2(x-2)(x+3)=0 Divide by 2

(x-2)(x+3)=0

x-2=0 x+3=0

x=3 x=-3

3.1

• Characteristics

1. Degree is 2
2. F(x)=$ax^2+bx+c$ (Standard Form)
3. Graph- shape is parabolic (parabola)

If a>0, then upwards parabola

If a<0, then upside down parabola

Ex: Desmos

D:(-infinity, infinity)

{x|-infin. <x<infin.} - all real #'s

R: [0,inin.)

Increasing: (-infin.,0) Decreasing: (0,infin.)

4 . Verex Form $f(x)=a(x+h)^2+k$

(h,k) vertex

Ex: Find the equation given the vertex and a point on the curve.

V= (0,3) (2,1)

Need to find a and the vertex

1=a(2-0)^2+3

$-2=a(2-0)^2$ Answer: f(x)=-1/2(x-0)^2+3

$-2=a(4)$ f(x)= -1/2 x^2+3

$-1/4 =a$

Find the equation given

V=(-2,4) and (3,6) Answer: F(x)= 2/25(x+2)^2

6=a(3+2)^2+4

2=a(5)^2

2/25=a

Find the function form given standard form

Ex: f(x)=2x^2-4x+1

• x-coordinate of the vertex is -b/2a
• y-coordinate F is (-b/2a)

Ex: b=4 a=2 x= --4/2(2)=1

f(1)= 2(1)^2-4(1)+1= -1 V=(1,-1)

Inequalities

1. One variable $x>_-3$

x+3x-2<4 --> 4x-2<4

4x<6 --> x<6/4=3/2 I.N = (-3,infinity)

2 . Two variables

Graph to find the solution set

y<3x+2 Test Point (-2,3) --> 3<3(-2)+2__ FALSE

3 . 2x+3y>_ -4

1. Test point--(0,0) Answer on Desmos

4 . Absolute Value

|2x|=3

2x=3 or 2x=-3

5 . -2 |5x+10|=-4

-2/-2|5x+10| =-4/-2

|5x+10|<_ 2

5x+10=2 or 5x+10=-2

x<_ -8/5 x>_ -12/5

2.1

2.1 Linear Functions

Types of lines

• y=mx+b - slope-intercept form
• Ax+By=C - standard form
• y-y1=m(x-x1) - point slope

Graphing

y=-2x+3

Parallel Lines-

• Have same slope, different Y intercept

Perpendicular-

• slopes are the opposite reciprocals

Find the equation Given...

1. 2 points
2. (2,5) (1,0)

m= 0-5/1-2= -5/-1 =5

y-0= 5(x-1)

y=5x-5

2 . A point and the slope

Ex: (3,8) m=4/3

y-8= 4/3 (x-3)

y= 4/3x +4

3 . Line parallel to given line

• same slope m= -1/2 gos through (0,0)

y-0= -1/2 (x-0)

y=-1/2x

4 . Line perpendicular

• negative reciprocal of given slope

m= 3/4 m2= -4/3 through (0,-3)

y--3= -4/3 (x-0)

y+3= -4/3x

y=-4/3 x-3

Chapter 2

4.4

Real zeros of higher degree polynomials

Example: $f(x)=x^3-61^2+839x+4221$ reps a

Small country's bird population , x days after May 31st.

Find the date(s) when the population was 5000. Find X when when f(x)= 5000 See Notes

Factoring to find zeros f(x)=ax^2+bx+c

F(x)= x^2-3x+2

0=(x-1)(x-2)

x-1=0 x-2=0

x=1 x=2

Zeros with Multiplicity (factored form) Graph

Example: f(x)=(x+2)^3 Example: F(x)= (x-4)^3(x+1)

=(x+2)(x+2) x=2 =4 Multiplicity

x=-1

Complete Factored Form

a) f(x)= 13x^2 -13x-26 b) f(x)=7x^3-2|x^2-7x+2|

GCF = 13 13(x^2-x-2) GCF= 7 (USe grouping method)

=13(x-2)(x+1) 7(x^3-3x^2)-(x-3)

0=13(x-2)(x+1) x=2 x=-1 7 [(x^3-3x^2)-(x-3)]

7[x^2(x-3)-(x-3)]

7(x-3)(x^2-1)

7(x-3)(x-1)(x+1) X=3,1,-1

C) F(x)=2x^3-4x^2-10x+12 (Use Graph)

Factoring Continued

a) f(x)= 2x^3-2x^2-34x-30 with given zero See Notes

If remainder = 0 ... TRUE FACTOR

Rational Zeros (Not all whole # zeros)

Let f(x)=

4.2

Polynomial Functions and Models

1. Turning Point - Point when it goes from increasing to decreasing/decreasing to increasing _Cubic Function_
2. End Behavior - f(x)--> - infin, x --> - infin

Example: See note book

Intercepts (x and y intercepts)

Example: See notebook x- intercepts (y=0)

zeros

Example: see notebook and desmos

• All intercepts
• End behavior
• Increasing/ Decreasing intervals
• Concavity *Concave up / Concave down*

3 . Piece wise Functions

F(x)= {x x>2 Slope=1

{2 x<_2 y-int=0

^ See notebook

{2x x>0

{x+2 x<0

Project - Compound Interest

FV=PV(1+i)^n

F+P(1+r)^t

#1 and 2# YOU TRY IT

F(x)+ax^2+bx+c F(1+r)^2+b(1+r)+c

4.1 Review

1. Transformations of graphs
2. Translations/ Reflections
3. Minimum
4. Maximum
5. End behavior
6. Increasing/ Decreasing Intervals
7. Domain/ Range
8. Even/ odd functions

4.1

Non Linear Functions

Symbolic Form for a polynomial

Example: y=mx+b OR f(x)=mx+b

F(x)= ax^m+bx^m-1+cx^m-2+. . . .

Graphs

Min and Max , Absolute Min , Absolute Max , Local Min , Local Max

• End Behavior - Tails
• Increasing/ Decreasing
• Domain/Range

Example: $2x^3+3x^2+x-1$ See Desmos

F(x)--> -infinity as x-->- infinity

F(x)--> infinity as x--> infinity

Example: $2x^3+5x^2+x-2$ See Desmos

Minimum - (-1.404,-3.557) Maximum - (0.5,0)

Increasing: ( Decreasing:

Domain: Range:

1.4 Types of functions

Types of Functions

Linear

Symbolic Form- y=mx+b (Slope intercept form)Highest exponent degree is 1.Ex: y=3x+1(Vertical Line)x=2 (a line that is NOT function)(Horizontal Line) y=2 (is a function)

Quadratic y=ax^2+bx+cHighest Degree is 2.Ex: $y=x^2+2x+1$1a. $f(x)=3x^2+4x+2$1b. a>02a. a<02b. $f(x)=-x^2+2x+1$

Cubic$f(x)=ax^3+bx^2+cx+d$(a,b,c,d are constants)Highest exponent is 3.$y=x^3$ $f(x)=6x^3$a>0 -- Regular parabolaa<0 -- Upside down parabola

Quartic$f(x)=ax^4+bx^cx^2+dx+e$Degree is 4.Ex:a>0 --a<0 --

Rational$f(x)=a/x-h +k$x-h DOES NOT EQUAL 0Ex:$f(x)=(3/x+2)$

Exponential$f(x)=(a)(b^x)$(a,b are constants)Ex: $a=1$ , $f(x)=3^3$Ex: $f(x)= e^x$( e is a constant)Ex: $a= 1/2$ $f(x)=1/2(2^x)$Logarithmic Absolute Value Functions$f(x)=|x-h|+k$Ex: $f(x)=|x|$Average rate of change- (Slope)$y2-y1/x2-x1=f(x2)-f(x1)/x2-x1$Ex: $f(x)=2x^2$ (1,3) find average{interval} rate of change$f(-2)-f(-3)/-2--3$(-3,-2)Difference Quotient$f(x-h)-f(x)/(x+h)-x$ = $f(x+h)-f(x)/h$ Ex: $f(t)=-16t^2+16t+32$(0,2) $-16t^2+16t+32$f(0)=32f(2)= 0$f(2)-f(0)/2-0 = 0-32/2-0=-32/2=-16$

1.3 Functions

1.3 Functions and their representations

Definitions:

1. Variable- a symbol /letter that represents an unknown number
2. Dependent Variable- "y" output
3. Independent Variable- "x" input
4. Equation- two equal algebraic expressions
5. Relation- comparing at least two things (usually x and y)
6. Function- relation where each input has only one output

1. $y=x^2$
2. () Not equal
3. [] Equal to

Chapter 1