Power Function in Polynomial Context

A power function is a type of mathematical function that can be written in the form:: f(x)=axn

Where:

Relation to Polynomial Functions:

A polynomial function is a sum of multiple power functions: P(x)=anxn+an1xn1+...+a1x1+a0 Each term like aixi s a power function with a specific exponent iand coefficient ai .

Examples:

  1. Linear power function: f(x)=3x
    • Here, a=3,n=1
    • This is a power function and also a linear function (a degree-1 polynomial).
  2. Quadratic power function: f(x)=2x2
    • Here, a=2,n=2
    • This is a power function and also a quadratic function (a degree-2 polynomial).
  3. Cubic power function : f(x)=5x3
    • Here, a=5,n=3
    • This is a power function and also a cubic function (a degree-3 polynomial).
  4. Constant function (special case): f(x)=7
    • Here, a=7,n=0
    • This is a power function and also a constant function (a degree-0 polynomial).
  5. A power function in the context of polynomial functions is a single-term expression of the form axn where 𝑛 is a non-negative integer. These functions form the building blocks of polynomial functions, and understanding their behavior helps in analyzing and graphing more complex polynomials.

    Graphical Behavior

    • If 𝑛 is even (e.g., 0, 2, 4), the graph of f(x)=anxn is symmetric about the y-axis.
      • Opens upward if a>0, downward if a<0
    • If 𝑛 is odd (e.g., 1, 3, 5), the graph is symmetric about the origin.
      • Increases in both directions if a>0, decreases if a<0

    Power Function vs Polynomial Function:

    Feather Power function Polynomial function
    Form f(x)=axn P(x)=anxn+an1xn1+...+a1x1+a0
    Number of Terms Single term One or more terms
    Example 4x2 4x23x+7

Even Power Function

f(x)=axn, where n is even

Odd Power Function

f(x)=axn, where n is odd