lecture_recording_on_06_March_2025_at_16.04.27_PM

Horizontal Asymptotes

• Definition: A horizontal asymptote is a horizontal line y = b where b is a constant. It indicates that as x approaches infinity or negative infinity, the value of f(x) approaches b.

• Behavior of Functions:

  • As x approaches infinity (x increases without bound), f(x) approaches b.

  • Unlike vertical asymptotes, a graph can cross a horizontal asymptote.

Finding Horizontal Asymptotes of Rational Functions

• Overview: To find horizontal asymptotes for a rational function (f(x) = P(x)/Q(x), where P and Q are polynomials), compare the degrees of the numerator (n) and the denominator (m).

Rules for Finding Horizontal Asymptotes

1. Degree of Numerator < Degree of Denominator

   • Horizontal asymptote is y = 0.

2. Degree of Numerator = Degree of Denominator

   • Horizontal asymptote is y = (Leading coefficient of numerator) / (Leading coefficient of denominator).

3. Degree of Numerator > Degree of Denominator

   • No horizontal asymptote exists.

• Key Variables:

  • Leading coefficients: a_n (numerator), b_m (denominator)

  • Degrees: n (numerator), m (denominator)

Example Calculations for Horizontal Asymptotes

1. Example 1:

   • f(x) = (x^1) / (x^2)

   • Degree of numerator (1) < Degree of denominator (2)

   • Horizontal asymptote: y = 0.

2. Example 2:

   • f(x) = (16x^2) / (4x^2)

   • Degree of numerator (2) = Degree of denominator (2)

   • Horizontal asymptote: y = 16/4 = 4.

3. Example 3:

   • f(x) = (x^3) / (x^2)

   • Degree of numerator (3) > Degree of denominator (2)

   • No horizontal asymptote.

Graphing Rational Functions Without a Calculator

• Seven Step Method:

  1. Find Symmetry: Determine if the function is even, odd, or neither.

     • Even: f(x) = f(-x) (symmetric about y-axis)

     • Odd: f(x) = -f(-x) (symmetric about origin)

  2. Find y-intercept: Set x = 0 in f(x).

  3. Find x-intercepts: Set numerator equal to zero.

  4. Find vertical asymptotes: Set denominator equal to zero.

  5. Find horizontal asymptotes: Use the rules outlined above.

  6. Plot additional points: Choose points between and beyond asymptotes and intercepts.

  7. Sketch the graph: Connect all points and show asymptotes.

Polynomial and Rational Inequalities

• Understanding Inequalities:

  • f(x) < 0 (function is negative)

  • f(x) > 0 (function is positive)

  • These inequalities are crucial in determining the behavior of the graph.

Steps to Solve Polynomial Inequalities:

1. Set the inequality to zero (e.g., f(x) < 0).

2. Solve f(x) = 0 for boundary points (x-intercepts).

3. Place boundary points on a number line.

4. Determine intervals to test based on the boundary points.

5. Test each interval to see where the inequality holds true.

6. Report the solution using interval notation.

Notes on Open and Closed Circles:

• Open Circle: Used for points that are not included in the solution (e.g., vertical asymptotes).

• Closed Circle: Used when including points in the solution set (e.g., points defined by inequality).

Final Considerations for Graphing

• Combine all findings (intercepts, asymptotes, and symmetry) on the graph.

• Determine how the function behaves around vertical and horizontal asymptotes to finalize the graph.