Polynomial Functions and Graph Extrema Study Guide

Polynomial Functions and Fundamental Concepts

• Definition of a Polynomial Function:

  • A polynomial function is defined algebraically in general form as:     $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_2 x^2 + a_1 x + a_0$
  • Structural constraints and component definitions:
  • The exponent $n$ must be a positive integer.
  • The value $a_i$ represents a real number coefficient for each index $i$ from $1$ to $n$.
  • The leading coefficient $a_n$ must be nonzero ($a_n \neq 0$).

• Key Structural Components:

  • Leading Term: The complete term containing the highest power of $x$, written as $a_n x^n$
  • Degree: The highest exponent $n$ in the polynomial function.
  • Leading Coefficient: The numerical coefficient $a_n$ multiplying the variable of highest degree.

• Examples: Determining Leading Coefficient and Degree:

  • General Rule: Functions must always be rearranged into standard form (descending powers of $x$) before identifying coefficients and degrees.
  • Example 1a:
  • Given function: $f(x) = 3x^4 + 2x - 7$
  • Leading Coefficient: $3$
  • Degree: $4$
  • Example 1b:
  • Given function: $y = 10x - 7x^3 + 11$
  • Standard form rearrangement: $y = -7x^3 + 10x + 11$
  • Leading Coefficient: $-7$
  • Degree: $3$
  • Example 1c:
  • Given function: $g(x) = 4$
  • Standard form rearrangement: $g(x) = 4x^0$
  • Leading Coefficient: $4$
  • Degree: $0$

Extrema of Polynomial Functions

• Overview of Extrema:
  • The extrema of a graph are the minimums and maximums of a function.
  • There are $2$ distinct categories of extrema: Relative (Local) Extrema and Absolute (Global) Extrema.

• Relative Extrema (Local):
  • Definition: A polynomial has a relative minimum or relative maximum where the polynomial switches between decreasing and increasing (or at an endpoint if the polynomial has a restricted domain).
  • Local Minimum: Occurs at point locations where the function changes from decreasing to increasing.
  • Example graph values: Local minimum at $x = -1$ and $x = 4$ (and endpoint $x = -4$).
  • Local Maximum: Occurs at point locations where the function changes from increasing to decreasing.
  • Example graph values: Local maximum at $x = -2.5$ and x = 3$.\n\n* Absolute Extrema (Global):\n * Absolute Maximum Definition: Of all local maxima, the greatest value is called the absolute maximum.\n * Example graph value: Absolute maximum = 2$at$x = 3$.
  • Absolute Minimum Definition: The least of all local minima is called the absolute minimum.
  • Example graph value: Absolute minimum $= \text{none}$ (due to end behavior where function approaches negative infinity as $x \rightarrow \infty$).