Algebra 2: Parent Functions and Graph Analysis

Study Notes

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.

Description

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.