Power and Polynomial Functions

Power Functions

  • Definition: A power function is defined as a function of the form
    [ f(x) = kx^p ] where
  • [ k ] is a real number called the coefficient
  • [ p ] is a real number representing the exponent.
  • Examples:
  • Area of a circle: [ A(r) = \pi r^2 ] (coefficient [ \pi ], exponent 2)
  • Volume of a sphere: [ V(r) = \frac{4}{3} \pi r^3 ] (coefficient [ \frac{4}{3} \pi ], exponent 3)
Examples of Power Functions
  • [ y = -\frac{1}{2} x^5 ]
  • [ f(m) = \sqrt{2} m^{4} ]
  • [ p(x) = \sqrt[3]{x} = x^{\frac{1}{3}} ]
  • [ r(x) = -x ]
  • [ q(x) = 1 = x^0 ]
  • [ y = -7 = -7x^0 ]

Identifying Power Functions

Think-Pair-Share 1: Which are power functions?

  1. [ j(x) = \sqrt{x} ] (not a power function)
  2. [ f(x) = 2x ] (power function)
  3. [ g(x) = 3x^2 ] (power function)
  4. [ h(x) = x^2 + 3x + 1 ] (not a power function)
  5. [ q(x) = \frac{2x^5 + 1}{3x^2 + 4} ] (not a power function)
  6. [ k(x) = \frac{1}{x} ] (not a power function)
  7. Volume of a cube: [ V(s) = s^3 ] (power function)

End Behavior of Power Functions

Even Power Functions
  • Behavior: As [ x ] approaches [ \pm\infty ], [ f(x) ] increases without bound.
  • Example: Graphs of [ f(x) = x^2, g(x) = x^4, h(x) = x^6 ] all show similar behaviors where output becomes large positive values.
Odd Power Functions
  • Behavior: As [ x ] approaches [ -\infty ], [ f(x) ] decreases without bound, and as [ x ] approaches [ +\infty ], [ f(x) ] increases without bound.
  • Example: Graphs of [ f(x) = x^3, g(x) = x^5, h(x) = x^7 ] exhibit this odd function behavior.

Polynomial Functions

  • Definition: A polynomial function can be expressed in the form:
    [ f(x) = an x^n + a{n-1} x^{n-1} + \ldots + a1 x + a0 ]
  • Terms: Each term is of the form [ ai x^i ] where each [ ai ] is a coefficient and [ i ] is a non-negative integer.
Examples and Evaluation
  • Example of Polynomial Function:
  • [ f(x) = 2x^3 + 3x + 4 ] is polynomial (highest power of [ x ] is 3, coefficients are real).

Degree and Leading Coefficient of a Polynomial

  • Degree: The degree is the largest exponent in the polynomial.
  • Leading Term: The term with the highest degree.
  • Leading Coefficient: The coefficient of the leading term.
  • Example:
  • For [ f(x) = -4x^3 + 2x^2 + 3 ]:
    • Degree = 3, Leading Term = [ -4x^3 ], Leading Coefficient = -4

Identifying End Behavior of Polynomial Functions

  • General Rule: The end behavior is dictated by the leading term of a polynomial.
  • Example Configuration:
  1. Polynomial: [ f(x) = 3 + 2x^2 - 4x^3 ]
  2. Leading Coefficient: -4 (negative) reflects across the x-axis.
  3. Determine end behavior using the leading term.
Key Points to Remember
  • For Even Degrees: Ends will go to same direction (both up or both down).
  • For Odd Degrees: Ends will go in opposite directions.
  • Coefficient signs drastically influence the graph's behavior as it heads towards infinity.