Algebra 2: Parent Functions and Graph Analysis

Parent Function Characteristics

• A parent function that is symmetric across the y-axis with a zero at the origin is typically an even function, such as f(x) = x².
• The range of such a function is [0, ∞) since it includes all non-negative values.

Function Symmetry

• Functions symmetric to the y-axis are even functions. Examples include f(x) = x² and f(x) = cos(x).
• In contrast, odd functions exhibit symmetry about the origin.

Domain of f(x) = 1/x

• The function f(x) = 1/x is defined for all real numbers except x = 0.
• Therefore, its domain is (-∞, 0) ∪ (0, ∞).

Asymptotes

• Certain functions do not have horizontal or vertical asymptotes, often including polynomial functions of degree zero or higher, such as f(x) = x².
• Trigonometric functions like sin(x) are also examples of functions without vertical asymptotes.

Minimum Value of f(x) = |x|

• The absolute value function f(x) = |x| has a minimum value of 0, occurring at x = 0.

Absolute Value Function Characteristics

• The graph of the absolute value function is V-shaped and symmetrical about the y-axis.
• It is defined for all real numbers, having a minimum point at the origin.

Y-Intercept of a Function

• The y-intercept of a function is found by evaluating f(0).
• For example, f(x) = x² has a y-intercept of (0, 0).

Range of a Function

• The range indicates all possible output values.
• For instance, the range of the absolute value function f(x) = |x| is [0, ∞).

Cube Root Parent Function Characteristics

• The cube root parent function f(x) = ∛x has unique characteristics, including no asymptotes and a domain and range of all real numbers.
• Its graph is symmetric about the origin.

Additional Characteristics of Functions

• Functions can exhibit various properties, including continuity, limits, and points of inflection, which impact their behavior and graph.

Practice your Algebra 2 skills with these questions about parent functions and graph analysis. Identify the parent function that matches a given set of attributes and analyze the graph of a function to determine its domain and range.