Key Parent Functions: Cubic and Absolute Value Functions

Cubic Function

Definition and Properties

The cubic function is a fundamental polynomial function in algebra, commonly written as:

• General Form: $y = x^3$

• Domain: $(-\infty, \infty)$ (all real numbers)

• Range: $(-\infty, \infty)$ (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., $f(-x) = -f(x)$).

Key Characteristics

• Intercept: The function passes through the origin (0,0).

• End Behavior: As $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$.

• 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 $x = 2$, $y = 2^3 = 8$.

• For $x = -1$, $y = (-1)^3 = -1$.

Absolute Value Function

Definition and Properties

The absolute value function is another key parent function in algebra, defined as:

• General Form: $y = |x|$

• Domain: $(-\infty, \infty)$ (all real numbers)

• Range: $[0, \infty)$ (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 $f(-x) = f(x)$.

Key Characteristics

• Vertex: The lowest point is at (0,0).

• Symmetry: The graph is symmetric about the y-axis.

• Piecewise Definition:

  • For $x \geq 0$, $y = x$

  • For $x < 0$, $y = -x$

Example Table of Values

x	y = |x|
-3	3
-2	2
-1	1
0	0
1	1
2	2
3	3

Example

• For $x = -4$, $y = |-4| = 4$.

• For $x = 2$, $y = |2| = 2$.