Fundamental Math

IT100 Session 6: Polynomial and Rational Functions
5 Sept 2017

Agenda

Agenda: Chapter 3

  1. Complex Numbers
  2. Quadratic Functions
  3. Power Functions and Polynomial Functions
  4. Graphs of Polynomial Functions
  5. Dividing Polynomials
  1. Zeros of Polynomial Functions
  2. Rational Functions
  3. Inverses and Radical Functions
  4. Modeling Using Variation

3.1 Complex Numbers

Graphing imaginary numbers

imaginary

\[ \large 3-4\sqrt{-1} = 3-4i \]

imaginary

Basic arithmetic with imaginary numbers

\[ \large\begin{array}{rcl} (4 + 3i) + (2 - i) & = & (4+2) + (3-1)i= 6 +2i\\ \\ (4 + 3i) \times (2 - i) & = & 4 \times 2 + (6-4)i -3i^2 \\ &=& 8 + 2i - (-3) = 11 + 2i\\ \\ (4 + 3i) - (2 - i) & = & (4-2) + (3+1)i= 2 +4i\\ \\ {5\over 2-i} &=& {4 - i^2 \over 2-i} = {(2 - i)(2+i)\over 2-i} = 2 + i \\ \end{array} \]

Challenge

\[ \large\begin{array}{rclr} (5 + 3i)(5 - 3i) &=&........ &(1)\\ \\ {10\over 3-i} &=&........ &(2)\\ \\ (5 + 3i)-(2-2i)&=&........ &(3)\\ \end{array} \]

3.2 Quadratic Functions

Basic Parabola

\[ \large\begin{array}{lcr} y &=& (x+1)(x-3)\\ & =& x^2 -2x -3\\ \end{array} \]

equation

Parabolic Solar Cooker

equation

Old CDs on satellite dish \[ \large 400^{\circ}C \hbox{ or } 720^{\circ}F \]

eq

Different quadratic curves

plot of chunk unnamed-chunk-1

Roots

Quadratic Equation

Formula:

\[ \large\begin{array}{rcl} 0 &=& ax^2 + bx + c\\ \\ x &=& {-b \pm \sqrt{b^2 - 4ac}\over 2a}\\ \end{array} \]

Example:

\[ \large\begin{array}{rcl} 0 &=& 6x^2 + 5x + 1\\ \\ x &=& {-5 \pm \sqrt{5^2 - 24}\over 12}\\ &=& {-5 \pm 1\over12}\\ &=& -{1\over 2}, -{1\over 3}\\ \end{array} \]

Challenge

Find the roots of these equations:

\[ \large\begin{array}{rclr} 0&=&x^2-7x+10&....... (1)\\ \\ 21&=&x^2-4x&....... (2)\\ \\ -3 &=& 4x^2 + -22x +7&....... (3)\\ \end{array} \]

3.3 Power Functions and Polynomial Functions

Effects of power

plot of chunk unnamed-chunk-2

3.4 Graphs of Polynomial Functions

Type of functions

Challenge

Identify Odd vs Even and direction

\[ \large\begin{array}{rclr} y&=&x^3-x^2-7x+10&....... (1)\\ \\ 21&=&x^4+x^3-x^2+x+yx&....... (2)\\ \\ y-3 &=& 4x^2 + -22x +7&....... (1)\\ \end{array} \]

3.5 Dividing Polynomials

Division of polynomials

\[ \large {2x^3 − 3x^2 + 4x + 5 \over x+2} \]

\[ \large\begin{array}{cccccccccc} \hbox{Ans:} & & 2x^2 &+& -7x &+& 18\\ \\ (x + 2)& &2x^3 &−& 3x^2 &+& 4x &+& 5\\ \\ &−(&2x^3 &+& 4x^2&)\\ && & &−7x^2 &+& 4x\\ && & −(&−7x^2& -& 14x&)\\ && & & & & 18x &+& 5\\ && & & & −(&18x &+& 36&)\\ && & & & & & & −31\\ \end{array} \]

Challenge

\[ \large\begin{array}{rclr} {x^4+6x^3-12x^2+10+5\over x-1} &=& .......& (1)\\ \\ {x^4 + 4x^3 + 3x^2 -4x -4\over x^2 + 4x + 4} &=& ........ & (2)\\ \end{array} \]

3.9 Modeling

Data modelling

xx

Data modelling

xx

Dow Jones Industrial Average

plot of chunk unnamed-chunk-3

plot of chunk unnamed-chunk-4

Preparation for Unit 7

Reading

Chapter 4: Exponential and Logarithmic Functions