Power functions and polynomials are mathematical superheroes. They shape curves, predict behavior, and solve real-world problems. These functions are the building blocks of complex equations, helping us model everything from population growth to planetary orbits.

Understanding power and polynomial functions unlocks a world of mathematical possibilities. We'll explore their characteristics, end behavior, and how they differ. By mastering these concepts, you'll gain powerful tools for analyzing and predicting patterns in various fields.

Power Functions

Characteristics of power functions

• Power functions take the form $f(x) = kx^n$ where $k$ is a constant (vertical stretch or compression) and $n$ is a real number (determines shape and behavior)
• Domain includes all real numbers except when $n$ is a negative integer (excludes $x = 0$)
• Range depends on $n$ and sign of $k$
  • Even $n$ and $k > 0$: range is $[0, \infty)$
  • Even $n$ and $k < 0$: range is $(-\infty, 0]$
  • Odd $n$: range is $(-\infty, \infty)$
• Symmetry about y-axis for even exponents and origin for odd exponents
• At most one x-intercept ($x = 0$) and one y-intercept ($y = k$)
• Graphing power functions involves considering the exponent's effect on the curve's shape

End behavior of power functions

• End behavior describes function's behavior as $x$ approaches positive or negative infinity
• For $f(x) = kx^n$:
  • $n > 0$ and $k > 0$: as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to -\infty$ (odd $n$) or $f(x) \to \infty$ (even $n$)
  • $n > 0$ and $k < 0$: as $x \to \infty$, $f(x) \to -\infty$, and as $x \to -\infty$, $f(x) \to \infty$ (odd $n$) or $f(x) \to -\infty$ (even $n$)
  • $n < 0$: as $x \to 0$ from the right, $f(x) \to \infty$ ($k > 0$) or $f(x) \to -\infty$ ($k < 0$), and as $x \to 0$ from the left, $f(x) \to -\infty$ ($k > 0$ and odd $n$) or $f(x) \to \infty$ ($k < 0$ and odd $n$) or $f(x) \to \infty$ (even $n$)

Polynomial Functions

Components of polynomial functions

• Polynomial functions have the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_0, a_1, ..., a_n$ are constants (coefficients) and $n$ is a non-negative integer
  • Leading term $a_n x^n$ has leading coefficient $a_n$
• Degree is the highest power of $x$
  • Degree 0: constant function
  • Degree 1: linear function
  • Degree 2: quadratic function
  • Degree 3: cubic function
• End behavior determined by degree and sign of leading coefficient
  • Even degree and positive leading coefficient: as $x \to \infty$ or $x \to -\infty$, $f(x) \to \infty$
  • Even degree and negative leading coefficient: as $x \to \infty$ or $x \to -\infty$, $f(x) \to -\infty$
  • Odd degree and positive leading coefficient: as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to -\infty$
  • Odd degree and negative leading coefficient: as $x \to \infty$, $f(x) \to -\infty$, and as $x \to -\infty$, $f(x) \to \infty$

Power functions vs polynomial functions

• Similarities:
  • Both are continuous on their domains
  • End behavior determined by degree and sign of leading coefficient
• Differences:
  • Power functions have one term, polynomials can have multiple terms
  • Power function degree can be any real number, polynomial degree is always a non-negative integer
  • Power functions have at most one x-intercept ($x = 0$) and one y-intercept ($y = k$), polynomials can have multiple x-intercepts and one y-intercept
  • Power functions with non-integer exponents may have restricted domains, polynomials are defined for all real numbers

Analysis of Polynomial Functions

• Roots: The x-values where a polynomial function equals zero, also called zeros or solutions
• Factoring: A method to express polynomials as a product of simpler terms, often used to find roots
• Differentiation: The process of finding the derivative of a polynomial, which describes the function's rate of change