Rational Function Characteristics to Know for Honors Algebra II

Rational functions are ratios of polynomials, revealing unique behaviors based on their structure. Understanding their characteristics, like domain, asymptotes, and intercepts, is essential for graphing and analyzing these functions in Honors Algebra II.

1. Definition of a rational function

   • A rational function is a function expressed as the ratio of two polynomials, typically in the form f(x) = P(x)/Q(x).
   • The polynomial P(x) is the numerator, and Q(x) is the denominator.
   • Rational functions can exhibit various behaviors based on the degrees of P(x) and Q(x).

2. Domain and restrictions

   • The domain of a rational function includes all real numbers except where the denominator Q(x) equals zero.
   • To find restrictions, set Q(x) = 0 and solve for x; these values are excluded from the domain.
   • Domain can be expressed in interval notation, indicating valid x-values.

3. Vertical asymptotes

   • Vertical asymptotes occur at values of x where Q(x) = 0 and P(x) ≠ 0.
   • They represent values that the function approaches but never reaches, leading to infinite behavior.
   • Vertical asymptotes can be found by identifying the roots of the denominator.

4. Horizontal asymptotes

   • Horizontal asymptotes describe the behavior of a rational function as x approaches positive or negative infinity.
   • The position of horizontal asymptotes depends on the degrees of P(x) and Q(x):
     • If degree of P < degree of Q, y = 0 is the horizontal asymptote.
     • If degree of P = degree of Q, y = leading coefficient of P / leading coefficient of Q.
     • If degree of P > degree of Q, there is no horizontal asymptote.

5. Holes (removable discontinuities)

   • Holes occur at x-values where both P(x) and Q(x) equal zero, indicating a common factor.
   • To find holes, factor both P(x) and Q(x) and simplify the function.
   • The hole can be represented as a point on the graph where the function is undefined.

6. x-intercepts and y-intercept

   • x-intercepts are found by setting P(x) = 0 and solving for x; these are points where the graph crosses the x-axis.
   • The y-intercept is found by evaluating f(0) = P(0)/Q(0), provided Q(0) ≠ 0.
   • Both intercepts provide key points for graphing the function.

7. End behavior

   • End behavior describes how the function behaves as x approaches positive or negative infinity.
   • It is influenced by the leading terms of P(x) and Q(x).
   • Understanding end behavior helps predict the overall shape of the graph.

8. Slant asymptotes

   • Slant (or oblique) asymptotes occur when the degree of P(x) is exactly one more than the degree of Q(x).
   • To find a slant asymptote, perform polynomial long division of P(x) by Q(x).
   • The quotient (ignoring the remainder) represents the slant asymptote.

9. Symmetry

   • A rational function can exhibit symmetry about the y-axis (even function) or the origin (odd function).
   • To test for even symmetry, check if f(-x) = f(x); for odd symmetry, check if f(-x) = -f(x).
   • Symmetry can simplify graphing and understanding the function's behavior.

10. Graphing techniques

    • Start by identifying the domain, intercepts, and asymptotes to sketch the function accurately.
    • Use test points in intervals defined by vertical asymptotes to determine the function's behavior.
    • Combine all characteristics (intercepts, asymptotes, holes) to create a complete graph of the rational function.