Power Function in Polynomial Context
A power function is a type of mathematical function that can be written in the form::
is a real number coefficient (non-zero), is the variable, n is a real number exponent, but in the context of polynomial functions, 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:Examples:
-
Linear power function:
-
Here,
- This is a power function and also a linear function (a degree-1 polynomial).
-
Here,
-
Quadratic power function:
-
Here,
- This is a power function and also a quadratic function (a degree-2 polynomial).
-
Here,
-
Cubic power function :
-
Here,
- This is a power function and also a cubic function (a degree-3 polynomial).
-
Here,
-
Constant function (special case):
-
Here,
- This is a power function and also a constant function (a degree-0 polynomial).
-
Here,
-
If 𝑛 is even (e.g., 0, 2, 4), the graph of
is symmetric about the y-axis.- Opens upward if
, downward if
- Opens upward if
-
If 𝑛 is odd (e.g., 1, 3, 5), the graph is symmetric about the origin.
- Increases in both directions if
, decreases if
- Increases in both directions if
A power function in the context of polynomial functions is a single-term expression of the form
Graphical Behavior
Power Function vs Polynomial Function:
Feather | Power function | Polynomial function |
---|---|---|
Form | |
|
Number of Terms | Single term | One or more terms |
Example |