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:
    $f(x)=a<em>n x^n+a</em>{n-1}x^{n-1}+\dots+a<em>1x+a</em>0$
  • $a_n \neq 0$ → polynomial is said to be of degree $n$
  • $a<em>n,a</em>{n-1},\dots ,a_0$ 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 ($a<em>n>0$ 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.