5.2 Power Functions and Polynomial Functions

Power functions are the building blocks of polynomial math. They come in various forms, from simple squares to complex roots. Understanding their behavior is key to mastering more advanced algebraic concepts.

These functions have unique characteristics that affect their graphs and behaviors. By studying their end behavior, symmetry, and other properties, we gain insights into how polynomials work in general.

Power Functions

Characteristics of power functions

• Power functions have the form $f(x) = kx^n$, where $k$ is a constant and $n$ is a real number
  • Positive integer $n$ creates polynomial functions (quadratic, cubic)
  • Negative integer or rational $n$ creates rational functions ($\frac{1}{x}, \frac{1}{x^2}$)
  • Irrational $n$ creates irrational functions ($\sqrt{x}, \sqrt[3]{x}$)
• Graph passes through the point $(1, k)$
• Domain is all real numbers, except when $n$ is a negative integer or rational with even denominator, which excludes $x = 0$
• Range depends on $n$ and sign of $k$
  • Even $n$ and $k > 0$ has range $[0, \infty)$ (parabola opening upward)
  • Even $n$ and $k < 0$ has range $(-\infty, 0]$ (parabola opening downward)
  • Odd $n$ has range $(-\infty, \infty)$ (cubic, square root)
• Graph symmetry
  • Even integer $n$ is symmetric about y-axis (parabola)
  • Odd integer $n$ is symmetric about origin (cubic)

End behavior of functions

• End behavior describes the graph as $x$ approaches positive or negative infinity
• Degree determines end behavior
  1. Even degree, positive leading coefficient approaches positive infinity as $x$ approaches $\pm \infty$ (upward parabola)
  2. Even degree, negative leading coefficient approaches negative infinity as $x$ approaches $\pm \infty$ (downward parabola)
  3. Odd degree, positive leading coefficient approaches positive infinity as $x \to \infty$ and negative infinity as $x \to -\infty$ (increasing cubic)
  4. Odd degree, negative leading coefficient approaches negative infinity as $x \to \infty$ and positive infinity as $x \to -\infty$ (decreasing cubic)

Polynomial Functions

Classification 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 $n$ is a non-negative integer and $a_0, a_1, ..., a_n$ are constants with $a_n \neq 0$
• Degree is the highest power of the variable
  • Degree 0: constant functions ($f(x) = 5$)
  • Degree 1: linear functions ($f(x) = 3x + 2$)
  • Degree 2: quadratic functions ($f(x) = x^2 - 4x + 3$)
  • Degree 3: cubic functions ($f(x) = 2x^3 - x + 1$)
• Leading coefficient is the coefficient of the highest degree term

Manipulation of polynomials

• Evaluate by substituting $x$ value and simplifying
• Add, subtract, and multiply polynomials to create new polynomial functions (polynomial arithmetic)
• Factor polynomials into lower-degree polynomials
  • Factor out greatest common factor (GCF)
  • Factor by grouping
  • Factor using special patterns
    • Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
    • Sum and difference of cubes: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$ and $a^3 - b^3 = (a-b)(a^2+ab+b^2)$
• Find roots or zeros where function equals zero
  • Fundamental theorem of algebra: degree $n$ polynomial has $n$ complex roots (including repeated)
  • Rational root theorem lists possible rational roots
  • Use synthetic division or long division to find roots and factorize completely

Advanced Polynomial Concepts

• Function composition: Combining two or more functions to create a new function
• Graphing techniques: Using technology and analytical methods to sketch polynomial functions
• Complex numbers: Understanding the role of imaginary roots in polynomial equations
• Polynomial inequalities: Solving and graphing inequalities involving polynomial expressions