8-3Rational 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.

Rational Function:𝑓 (𝑥 )= 𝑃 (𝑥 )𝑄 (𝑥)

and are polynomials

𝑦= 𝑥2

𝑥2+1

-5 5-5

5

x

y

𝑦=𝑥+2

𝑥2−4

-5 5-5

5

x

y

HoleAsymptote

Continuous Graph:No breaks in graph

Discontinuous Graph:Breaks in graph

𝑦=𝑥+2

𝑥2−𝑥−6

V. Asymp:

Holes:

Domain:

Discontinuity:

Vertical Asymptotes:Non-removable

Discontinuities:

Holes:Removable Same factor in numerator and denominator

Domain:Where the Denominator = zero

All real numbers ( except discontinuities

or

All reals but

𝑥=−1𝑥=−3

or All reals but

𝑥=−2𝑥=3

𝑦=𝑥+1

(𝑥+1)(𝑥+3)𝑦=

𝑥+2(𝑥+2)(𝑥−3)

𝑦=¿ 12 𝑦=¿− 1

5

¿1

(𝑥+3)¿

1(𝑥−3)

Horizontal Asymptotes: 𝑎𝑥𝑚

𝑏𝑥𝑛

𝑚<𝑛𝑦=0

𝑚>𝑛No horizontalasymptote

𝑚=𝑛𝑦=

𝑎𝑏

Leading term of numerator and denominator (standard form)

1?2𝑦=0¿ 3? 1¿ 2?2

𝑦=31

¿

Range: All real numbers except horizontal asymptote & holes

No horizontalasymptote

1

¿3

𝑦=𝑥+1

𝑥2−4 𝑦= 𝑥3+6𝑥+3

𝑦=3 𝑥2+4𝑥2+5

1. Find and graph asymptotes & holes

2. Find and graph additional points → each side of v. asymptote

3. Sketch graphV. Asymp.:

Discontinuities:

Hole: None

𝑥=3

𝑥=3

H. Asymp.:

𝑦=¿ ¿221

Additional Points

x y

0 2 (0)0−3

¿0

2 (4 )4−3

¿84

Domain:Range:

ℝ except 𝑥≠3ℝ except 𝑦 ≠2

-10 10

-10

10

x

y

-10 10

-10

10

x

yGraph:

-10 10

-10

10

x

y

-10 10

-10

10

x

y

V. Asymp.:

Discontinuities:

Hole:

𝑥=−1

𝑥=1

H. Asymp.:

𝑦=¿0

Additional Points

x y

0 10+1¿1

1(−2 )+1¿−1−2

Domain:Range:

ℝ except 𝑥≠±1ℝ except 𝑦 ≠0 ,0.5

¿𝑥−1

(𝑥−1)(𝑥+1)

𝑥=−1𝑥=1

1

𝑦=¿12

-10 10

-10

10

x

y

-10 10

-10

10

x

y

-10 10

-10

10

x

y

-10 10

-10

10

x

yGraph:

V. Asymp.:

Discontinuities:

Hole:

𝑥=−2

𝑥=−2

H. Asymp.:

𝑦=¿ ¿1

Additional Points

x y

−4 −4−4+2¿2

0(0 )+2

¿00

Domain:Range:

ℝ except 𝑥≠−2 ,−3ℝ excep t 𝑦 ≠0 ,3

𝑥=−3𝑥=−3

11

Pg. 521#13-31odds

¿𝑥 (𝑥+3)

(𝑥+2)(𝑥+3)

𝑦=¿3

Graph:

-10 10

-10

10

x

y

-10 10

-10

10

x

y

-10 10

-10

10

x

y

-10 10

-10

10

x

y