ICM ~Unit 4 ~ Day 2 (Extra Day for PSAT/Practice)

Section 1.2— Horizontal Asymptotes, &

Domain and Discontinuities Practice

Warm Up

Find the domain, x & y intercepts, and label any

discontinuities (including if they are removable

or non-removable).

2

3 2

42) ( )

4 32

x xf x

x x x

225

1.4

xh x

x

Warm Up ANSWERS

Find the domain, x & y intercepts, and label any

discontinuities (including if they are removable

or non-removable).

2

3 2

42) ( )

4 32

x xf x

x x x

1 1

( .) : (0, ) (4, )8 12

. . ( .) 8

Hole removable disc and

V A non removable disc x

: [ 5,4) (4,5]Domain

int : ( 5, 0)&(5,0)

5int : (0, )

4

x

y

2251.

4

xh x

x

Nonremovable

Discontinuity

(Vertical

Asymptote

at x = 4)

int :

int : ( )

x none

y none hole there

Homework Questions?

Tonight’s HomeworkFinish Rational Summary

& 5 Factoring Problems from the Puzzle

More Rational Fncns Handout Evens

• Finish Rational Summary & 5 Factoring Problems from the Puzzle• More Rational Fncns Handout

Evens

Homework

Notes Day 2 (Extra Day)

Section 1.2—Horizontal Asymptote and Domain and Discontinuities Practice

Definition of Degree• Degree of a polynomial in one variable:

the value of the greatest exponent

Ex:

Ex:

Ex:

2( ) 4 9 8f x x x

3 2( ) 5 6 4g x x x x

Degree: 2

Degree: 3

• Degree can help us with determining the horizontal asymptote of rational functions...

2 4( ) 5 3 2h x x x x

Watch out…the polynomial may not be in order!!

Degree: 4

Asymptote LabPacket p. 4-5

Let’s do one or two together!

Examine the table of values below. All of the following statements are true EXCEPT

A. x = -2 is a vertical asymptote in y1

B. x = -2 is an infinite discontinuity in y1

C. x = -2 is a removable discontinuity in y2

D. x = -2 is a vertical asymptote in y2

Looking at y2, the y’s are in order, but y = -4 was skipped,

so there is a Removable Discontinuity (Hole) at (-2, -4)

You’ll do an

Asymptote

Lab to learn

more about

this.

Horizontal AsymptotesFor horizontal asymptotes, think BOSTON for polynomials! Looking at the degree of top & bottom…

Bottom > Top

y=O

Same = ratio

Top > Bottom

O No HA.

N

2

2( )

3

xf x

x x

3

3 2

2( )

5 4

xg x

x x

2. . :

5H A y

. . : 0H A y

25( )

7 3

xh x

x

. .No H A

You Try! What is the EQUATION of the horizontal asymptote for the following functions?

2

2

3 9( )

7 4 11

xf x

x x

3

2

4( )

5 9

xg x

x

2

7 15( )

2

xh x

x

Bottom > Top

y=O

Same = ratioTop > BottomO No HA.N

You Try! What is the EQUATION of the horizontal asymptote for the following functions?

2

2

3 9( )

7 4 11

xf x

x x

3

2

4( )

5 9

xg x

x

2

7 15( )

2

xh x

x

Bottom > Top

y=O

Same = ratioTop > BottomO No HA.N

. . : 0H A y

. . :H A none

3. . :4

H A y

Let’s Summarize some rules and steps for discontinuities of Rational Functions

Rational Summary

• Rational Summary &• More Rational Fncns Handout

Practice – Finish for part of

Homework

Around the Room Activity (if time allows)

Domain Practice

You Try: True or False

1) The graph of function f is defined as the set of all points (x, f(x)) where x is in the domain of f. Justify your answer.

2) If a function is not continuous, then the domain cannot be all real numbers.

True or False ANSWERS

1) The graph of function f is defined as the set of all points (x, f(x)) where x is in the domain of f. Justify your answer.

2) If a function is not continuous, then the domain cannot be all real numbers.

True! This is the definition

of a function.

False! It could be a

piecewise function.

Is this continuous?

State whether the scenario is continuous or discontinuous.

• A) Outdoor temperature as a function of time.

• B) Number of soft drinks sold at a ballpark as a function of outdoor temp.

• C) Your hair length as a function of days in a year

Continuous

Discontinuous

Continuous

• Finish Rational Summary & 5 Factoring Problems from the Puzzle• More Rational Fncns Handout

Evens

Homework