BackKey Parent Functions: Cubic and Absolute Value Functions
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Cubic Function
Definition and Properties
The cubic function is a fundamental polynomial function in algebra, commonly written as:
General Form:
Domain: (all real numbers)
Range: (all real numbers)
The graph of the cubic function is an S-shaped curve that passes through the origin (0,0). It is symmetric with respect to the origin, making it an odd function (i.e., ).
Key Characteristics
Intercept: The function passes through the origin (0,0).
End Behavior: As , ; as , .
Increasing/Decreasing: The function is always increasing.
Example Table of Values
x | y = x^3 |
|---|---|
-2 | -8 |
-1 | -1 |
0 | 0 |
1 | 1 |
2 | 8 |
Example
For , .
For , .
Absolute Value Function
Definition and Properties
The absolute value function is another key parent function in algebra, defined as:
General Form:
Domain: (all real numbers)
Range: (all non-negative real numbers)
The graph of the absolute value function forms a "V" shape, with its vertex at the origin (0,0). The function is even, meaning .
Key Characteristics
Vertex: The lowest point is at (0,0).
Symmetry: The graph is symmetric about the y-axis.
Piecewise Definition:
For ,
For ,
Example Table of Values
x | y = |x| |
|---|---|
-3 | 3 |
-2 | 2 |
-1 | 1 |
0 | 0 |
1 | 1 |
2 | 2 |
3 | 3 |
Example
For , .
For , .
