Study Notes on Rational Functions and Graph Characteristics

Introduction to Rational Functions

• Focus on understanding rational functions and their graphical characteristics.

• The lessons will explore how different components of rational equations influence the graph.

Understanding Graphs of Rational Functions

• The importance of grasping how altering parts of a rational equation affects its graph.

• Key characteristics are impacted when changing various parts of the equation:
    - Shape of the graph
    - Position of the graph in the Cartesian plane

Key Characteristics of Rational Functions

Vertical Asymptotes

• Definition: Lines that a graph approaches but never touches or crosses vertically.

• Occur when the denominator of the rational function becomes zero.

• Example: For $f(x) = \frac{1}{x-3}$, there is a vertical asymptote at $x=3$ since the function is undefined there.

Horizontal Asymptotes

• Definition: Horizontal lines that the graph approaches as $x$ approaches infinity (or negative infinity).

• Determined by the degrees of the numerator and denominator.
    - If the degree of the numerator is less than the degree of the denominator, $y = 0$ is a horizontal asymptote.
    - If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$ where $a$ and $b$ are the leading coefficients of the numerator and denominator respectively.
    - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Intercepts

• X-intercepts: Points where the graph crosses the x-axis (where $f(x) = 0$).

• Y-intercepts: Points where the graph crosses the y-axis (where $x = 0$).

Points of Discontinuity

• Points where the function is not continuous.

• May occur at vertical asymptotes or holes in the graph.

• A hole exists if a factor cancels out in both the numerator and denominator.

Direction of the Graph

• Indicates the behavior of the graph as $x$ approaches positive and negative infinity.

• Involves determining whether the graph approaches the horizontal asymptote from above or below.

Domain of the Function

• The set of all possible input values (x-values) for the function.

• Domain restrictions typically arise from vertical asymptotes or values that make the denominator zero.

• For instance, for $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2.$

Conclusion

• Understanding these characteristics is essential for analyzing and graphing rational functions accurately.

• The lessons will provide insights into how each component of the rational function equation informs the overall graph shape and traits.