I can graph a rational function. 8-3 Rational Functions Unit Objectives: Graph a rational function Simplify rational expressions. Solve a rational functions Apply rational functions to real-world problems Today’s Objective: I can graph a rational function.

𝑓 𝑥 = 𝑃(𝑥) 𝑄(𝑥) Rational Function: 𝑃 𝑥 and 𝑄 𝑥 are polynomials 𝑦= 𝑥 2 𝑥 2 +1 𝑦= 𝑥+2 𝑥 2 −4 Hole Asymptote Continuous Graph: No breaks in graph Discontinuous Graph: Breaks in graph

Where the Denominator = zero Domain: Discontinuities: Where the Denominator = zero Domain: All real numbers (ℝ) except discontinuities Holes: Removable Same factor in numerator and denominator Vertical Asymptotes: Non-removable 𝑦= 𝑥+2 (𝑥+2)(𝑥−3) 𝑦= 𝑥+2 𝑥 2 −𝑥−6 = 1 (𝑥−3) 𝑦= 𝑥+1 (𝑥+1)(𝑥+3) = 1 (𝑥+3) Discontinuity: 𝑥=−1 or 𝑥=−3 𝑥=−2 or 𝑥=3 Domain: All reals but 𝑥≠−2, 3 All reals but 𝑥≠−1,−3 1 2 − 1 5 Holes: 𝑦= 𝑥=−2 𝑦= 𝑥=−1 V. Asymp: 𝑥=−3 𝑥=3

Horizontal Asymptotes: 𝑎𝑥 𝑚 𝑏𝑥 𝑛 𝑎𝑥 𝑚 𝑏𝑥 𝑛 Leading term of numerator and denominator (standard form) 𝑚<𝑛 𝑚>𝑛 𝑚=𝑛 No horizontal asymptote 𝑦=0 𝑦= 𝑎 𝑏 𝑦= 𝑥+1 𝑥 2 −4 𝑦= 𝑥 3 +6 𝑥+3 𝑦= 3 𝑥 2 +4 𝑥 2 +5 1 1 ? 2 < 2 ? 2 = 3 ? 1 > 𝑦= 3 1 𝑦=0 No horizontal asymptote =3 Range: All real numbers (ℝ) except horizontal asymptote & holes

Find and graph asymptotes & holes Find and graph additional points → each side of v. asymptote Sketch graph Discontinuities: 𝑥=3 Hole: None V. Asymp.: 𝑥=3 Additional Points x y H. Asymp.: 2(0) 0−3 =0 ℝ except 𝑥≠3 2 1 Domain: Range: 𝑦= =2 ℝ except 𝑦≠2 2(4) 4−3 4 =8

1 Graph: 𝑦= 𝑥−1 𝑥 2 −1 = 𝑥−1 (𝑥−1)(𝑥+1) Discontinuities: 𝑥=1 𝑥=−1 = 𝑥−1 (𝑥−1)(𝑥+1) Discontinuities: Additional Points x y 𝑥=1 𝑥=−1 Hole: 𝑥=1 1 −2 +1 −2 1 2 =−1 𝑦= V. Asymp.: 1 0+1 =1 𝑥=−1 H. Asymp.: Domain: Range: ℝ except 𝑥≠±1 ℝ except 𝑦≠0, 0.5 𝑦=

p.521: 13-31 odd = 𝑥(𝑥+3) (𝑥+2)(𝑥+3) Graph: 𝑦= 𝑥 2 +3𝑥 𝑥 2 +5𝑥+6 = 𝑥(𝑥+3) (𝑥+2)(𝑥+3) Discontinuities: Additional Points x y 𝑥=−2 𝑥=−3 Hole: 𝑥=−3 −4 −4+2 −4 =2 𝑦= 3 V. Asymp.: 0 0 +2 =0 𝑥=−2 H. Asymp.: p.521: 13-31 odd Domain: Range: ℝ except 𝑥≠−2, −3 1 1 𝑦= =1 ℝ except 𝑦≠0, 3