1. 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.

2. 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

3. 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

4. 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 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

5. 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

6. 1 Graph: 𝑦= 𝑥−1 𝑥 2 −1 = 𝑥−1 (𝑥−1)(𝑥+1) Discontinuities: 𝑥=1 𝑥=−1
= 𝑥− (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=

7. 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
𝑥=−2
H. Asymp.:
p.521: odd
Domain:
Range:
ℝ except 𝑥≠−2, −3
1 1 𝑦= =1
ℝ except 𝑦≠0, 3