Power Function in Polynomial Context

A power function is a type of mathematical function that can be written in the form:: \[ f(x) = ax^n \]

Where:

• \(a\) is a real number coefficient (non-zero),
• \(x\) is the variable,
• \(n\) n is a real number exponent, but in the context of polynomial functions, \(n\) is a non-negative integer (i.e., 0, 1, 2, 3, ...). In polynomial functions, a power function represents each individual term of a polynomial.

Relation to Polynomial Functions:

A polynomial function is a sum of multiple power functions: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1 x^1 + a_0 \] Each term like \(a_ix^i\) s a power function with a specific exponent \(i\)and coefficient \(a_i\) .

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: \[ f(x) = -2x^2 \]
   • Here, \(a = -2, n = 2\)
   • This is a power function and also a quadratic function (a degree-2 polynomial).
3. Cubic power function : \[ f(x) = 5x^3 \]
   • 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).

A power function in the context of polynomial functions is a single-term expression of the form \(ax^n\) 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 \(f(x) = a_nx^n \) 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	\( f(x) = ax^n \)	\(P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1 x^1 + a_0 \)
Number of Terms	Single term	One or more terms
Example	\(4x^2\)	\(4x^2 - 3x + 7\)