Polynomials, Rational Functions & Exponential Models (Sections 2.2 – 2.3)

Terminology & Basic Definitions

• Term / Monomial
  • Any expression of the form a x^n
  • a = real‐number coefficient (can be $0$)
  • n \in {0,1,2,\dots} (natural number or zero)
• Polynomial
  • Finite sum of monomials
  • General shape:
    f(x)=an x^n+a{n-1}x^{n-1}+\dots+a1x+a0
  • a_n \neq 0 → polynomial is said to be of degree n
  • an,a{n-1},\dots ,a_0 are called coefficients

Polynomial Example: Volume of a Variable Box

• Geometry setup (visualized in class):
  • Width = x units
  • Length = x+2 units (the «+2» came from a fixed rim / radius)
  • Height = x^2 units
• Volume V = width × length × height
  V(x)=x\,(x+2)\,x^2 = x^4+2x^3
• Conclusions
  • V is a polynomial in x of degree 4
  • Demonstrates a highly non-linear relation between a single dimension (width) and volume
  • Motivates the need to graph polynomials to study growth behavior

Graphical Characteristics of Polynomials

• Turning points (local maxima/minima)

  • A degree-n polynomial has at most n-1 turning points
  • Example plotted in class: a quartic possessed exactly 3 turning points → matches rule

• End behavior depends only on

  1. Parity of degree (n even vs.
     odd)
  2. Sign of leading coefficient (an>0 or an<0)

  Degree n	a_n>0	a_n<0
  even	Up–Up	Down–Down
  odd	Down–Up	Up–Down

  (Left arrow indicates behavior as x\to-\infty, right arrow as x\to+\infty)

• Illustrative cases demonstrated on calculator:

  • x^4+2x^2 → even degree, a_n=1>0 → both ends ↑
  • -x^4-2x^2 → even degree, a_n<0 → both ends ↓
  • x^5 (or x^5+\ldots) → odd, a_n>0 → left ↓, right ↑
  • -2x^5 → odd, a_n<0 → left ↑, right ↓

Rational Functions

• Definition
  f(x)=\dfrac{P(x)}{Q(x)},\quad P,Q \text{ polynomials},\; Q(x)\neq0
• Everyday illustration: Speed over a fixed distance
  • Distance d=1 (e.g.
    1 m)
  • Time t varies
  • Speed v(t)=\dfrac{1}{t} → ratio of two degree-0 and degree-1 polynomials ⇒ rational
• Physical argument for undefined \tfrac{1}{0}
  • Reaching destination in zero time implies being in two places simultaneously ⇒ impossible ⇒ division by zero undefined

Asymptotes (introduced via 1⁄X)

• Vertical asymptote at x=k if |f(x)|\to\infty as x\to k
• Horizontal asymptote at y=L if f(x)\to L as |x|\to\infty
• Canonical example: f(x)=\tfrac{1}{x}
  • Vertical: x=0
  • Horizontal: y=0
• Practical model: Pollution-removal cost y=\frac{18}{106-x}
  • Vertical asymptote x=106
  • Interpretation: as pollutant concentration nears 106 %, cleanup cost skyrockets → stay well below that threshold

Exponential Functions

• Definition: f(x)=a\,b^x with
  • a\neq0 (initial/scale factor)
  • b>0 and b\neq1 (base)
  • Key distinction from polynomials: the variable is in the exponent

Example 1 – Exponential Growth

• f(x)=3\,2^x


• Computed table (values reproduced exactly from lecture):
  

• f(x)=\bigl(\tfrac12\bigr)^x (here a=1, b=\tfrac12<1)
  • Sample outputs: 8,4,2,1,\tfrac12,\tfrac14,\tfrac18 for x=-3\dots3
• Each +1 in x multiplies y by \tfrac12 → rapid decrease labelled “exponential decay.”

Exponential Growth & Decay: General Forms

• Growth: y=A\,B^{t} with B>1
• Decay: y=A\,B^{t} with 0<B<1
• The farther B is from 1, the faster the rise/fall.
  • B\gg1 → explosive growth
  • B\ll1 → precipitous decay

Financial Application: Monthly-Compounded Interest

• Formula for future value with compounding: A=P\Bigl(1+\frac{r}{m}\Bigr)^{m t}
  • P = principal (initial investment)
  • r = annual nominal rate (as decimal)
  • m = number of compounding periods per year
  • t = years elapsed
• Lecture numbers
  • P=2000
  • r=12.6\%\;(=0.126)
  • m=12 (monthly)
• Substitution (following instructor’s arithmetic):
  A=2000\Bigl(1+\frac{0.126}{12}\Bigr)^{12t}=2000\,(1.0512)^{t} (value inside parenthesis recorded exactly as on slide; mathematically, $1+0.126/12\approx1.0105$)
• Classification:
  • a=2000
  • b\approx1.01 (very close to 1) → slow, steady exponential growth typical of bank products

Conceptual & Practical Take-Aways

• Polynomial end-behavior and turning points can be predicted without graphing once degree & leading coefficient are known.
• Rational functions introduce natural “forbidden” $x$‐values, leading to vertical asymptotes that often have real-world significance (e.g., cost explosions, physical impossibilities).
• Exponential models are indispensable whenever data change multiplicatively (population, radioactivity, finance).
  • Growth/decay rates dwarf those of polynomials; choosing polynomials in such contexts yields poor models.
• Compound-interest formula is a direct, ubiquitous application of exponential growth; tweaking b (frequency, rate) controls the steepness of accumulation.