Algebra 2: Graphing Rational Functions - Section 7-2, Day 1

Introduction to Rational Functions

• Definition: A rational function has the form ( f(x) = \frac{p(x)}{q(x)} ) where both ( p(x) ) and ( q(x) ) are polynomials and ( q(x) \neq 0 ).
• Standard Form: Typically includes a polynomial in the numerator and denominator.
• Example: Inverse variation function ( f(x) = \frac{a}{x} ) is a rational function.

Parent Graph of Rational Functions

• Parent Function: ( f(x) = \frac{1}{x} ).
• Graph Shape: A hyperbola consisting of two symmetrical parts called branches.
• Domain and Range: All non-zero real numbers.
  • Domain: ((-\infty, 0) \cup (0, \infty))
  • Range: ((-\infty, 0) \cup (0, \infty))

Asymptotes

• Asymptotes for ( g(x) = \frac{a}{x} ): Same as the parent function when ( a \neq 0 ).
• Labeling Asymptotes: Habit of labeling them; typically ( x = 0 ) and ( y = 0 ) for the parent function.

Graphing Without a Calculator

• Strategy: Start with critical points; avoid values that make the denominator zero.
• Example Points: ((-1, -2), (-2, -1), (-4, -0.5)).
• Symmetry: Recognize symmetry about ( y = x ) and ( y = -x ).
• Graphing Tips: Make sure branches hug but don't cross the asymptotes.

Translating Graphs

• Translation Concepts:
  • ( x - h ) controls horizontal shift.
  • ( +k ) controls vertical shift.
  • ( x - 3 ) means shift right 3; ( x + 4 ) means shift left 4.

Example Problems

Example 1: Basic Function

• T-Chart Points: ((-6, -2.5), (-3, -3), (1, 1), (3, -1)).
• Graph Characteristics: Asymptotes at ( x = 0 ) and ( y = -2 ).
• Domain: ((-\infty, 0) \cup (0, \infty))
• Range: ((-\infty, -2) \cup (-2, \infty))

Example 2: Complex Function

• T-Chart Points: ((-6, 0.5), (-3, -1), (-2, -0.5)).
• Asymptote Location: Vertical ( x = -4 ), horizontal ( y = 0 ).
• Domain: ((-\infty, -4) \cup (-4, \infty))
• Range: ((-\infty, 0) \cup (0, \infty))

Conclusion

• General Approach: Use T-charts for accurate plotting.
• Asymptotes: Key to understanding the graph behavior.
• Translations: Affect graph's position but not its basic shape.
• Practice: Essential for mastering graphing without calculators.