Precalculus 1 Nov25

Definition: A rational function is the ratio of two polynomials.
Polynomial Powers: Only positive whole number powers are allowed in polynomials.
- Whole numbers include: 0, 1, 2, 3, 4, 5, 6.
- Excluded: fractions, decimals, negative numbers, and radicals.
A quadratic equation can be a rational function but it cannot be irrational or have negative powers.

Vertical Asymptotes: Represent boundaries where a function is undefined (denominator = 0).
Identification: Set denominator equal to zero and solve for x to find vertical asymptotes.
- Example: If the denominator is quadratic and factored as (x + 2)(x + 1), then vertical asymptotes occur at x = -2 and x = -1.
A function can have multiple vertical asymptotes, leading to multiple separate branches of the graph.
Holes: If a solution also makes the numerator zero, it indicates a hole in the graph instead of a vertical asymptote.
- If both numerator and denominator have a common factor, that factor results in a hole.

Graph Behavior: The presence of vertical asymptotes breaks the graph into various sections depending on the number of asymptotes.
- 1 vertical asymptote: 2 separate branches.
- 2 vertical asymptotes: 3 separate branches.
- A hole in the graph indicates a point of discontinuity without creating separate branches.
As x approaches a vertical asymptote, the function tends towards positive or negative infinity.

Domain Restrictions: Exclude values from the domain where the function is undefined (where the denominator equals zero).
- For example, if vertical asymptotes occur at x = -2 and x = -1, they are excluded from the domain: (−∞, -2) U (-2, -1) U (-1, +∞).
Domain notation indicates which values x can or cannot take.

When comparing the degrees of the numerator and denominator:
- If degree of numerator < degree of denominator, the horizontal asymptote is y = 0.
- If degree of numerator = degree of denominator, the horizontal asymptote is determined by the leading coefficients of the numerator and denominator.
- If degree of numerator > degree of denominator, there is no horizontal asymptote.

Example Process: When encountering a rational function, always:
- Identify vertical asymptotes through factoring.
- Determine if the function results in holes or vertical asymptotes.
Check if substituting x-values near the vertical asymptotes gives valid function behaviors (approaching infinity, etc.).
A complete understanding relies on testing values and practicing with different rational functions to reinforce concepts.