Graphs of Common Functions in College Algebra

Graphs of Common Functions

Understanding the basic graphs of common functions is essential in College Algebra. These functions serve as foundational models for more complex mathematical concepts and problem-solving. Below are the most frequently encountered functions, their properties, and graphical representations.

Constant Function

The constant function outputs the same value regardless of the input. Its graph is a horizontal line.

• Definition: $f(x) = c$ where $c$ is a constant

• Domain: $(-, )$

• Range: $y = c$

• Graph: Horizontal line at $y = c$

Identity Function

The identity function outputs the input value itself. Its graph is a straight line passing through the origin with a slope of 1.

• Definition: $f(x) = x$

• Domain: $(-, )$

• Range: $(-, )$

• Graph: Diagonal line through the origin

Square Function

The square function outputs the square of the input. Its graph is a parabola opening upwards.

• Definition: $f(x) = x^2$

• Domain: $(-, )$

• Range: $[0, )$

• Shape: Parabola

Cube Function

The cube function outputs the cube of the input. Its graph is an S-shaped curve passing through the origin.

• Definition: $f(x) = x^3$

• Domain: $(-, )$

• Range: $(-, )$

Square Root Function

The square root function outputs the non-negative square root of the input. Its graph starts at the origin and increases slowly.

• Definition: $f(x) = x$

• Domain: $[0, )$

• Range: $[0, )$

• Note: Only defined for $x  0$

Cube Root Function

The cube root function outputs the cube root of the input. Its graph passes through the origin and is defined for all real numbers.

• Definition: $f(x) = x$

• Domain: $(-, )$

• Range: $(-, )$

• Note: Defined for both positive and negative $x$

Summary Table: Properties of Common Functions

Function	Equation	Domain	Range	Graph Shape
Constant	$f(x) = c$	$(-, )$	$y = c$	Horizontal line
Identity	$f(x) = x$	$(-, )$	$(-, )$	Diagonal line
Square	$f(x) = x^2$	$(-, )$	$[0, )$	Parabola
Cube	$f(x) = x^3$	$(-, )$	$(-, )$	S-curve
Square Root	$f(x) = x$	$[0, )$	$[0, )$	Curve starting at origin
Cube Root	$f(x) = x$	$(-, )$	$(-, )$	Curve through origin

Example Applications

• Constant Function: Used to model fixed costs in economics.

• Identity Function: Used in linear transformations and as a basis for other functions.

• Square Function: Models area calculations and projectile motion.

• Cube Function: Used in volume calculations and certain physics applications.

• Square Root Function: Appears in geometry (e.g., Pythagorean theorem) and statistics.

• Cube Root Function: Used in scaling laws and engineering.