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Chapter 4Polynomial and Rational

Functions

Section 2Properties of Rational Functions

DAY 1

A rational function has the form

R x =𝑝(𝑥)

𝑞(𝑋)

Where p(x) and q(x) are polynomials and q(x) ≠ 0.

Domain: all real numbers except any x-values that would make q(x) = 0

Example:Find the domain of each rational function

(set denominator = 0, solve for x domain excludes the solution)

R(x) = 4𝑥

𝑥−3

R(x) = 𝑥2−𝑥−6

4(𝑥2−9)

R(x) = 3𝑥2+𝑥

𝑥2+4

Graphing Using Transformations of 1) 1

𝑥2and 2)

1

𝑥

What x-values would make each equation undefined?

What number could 1) and 2) not equal?b/c numerator is 1, the fraction willnever reduce to 0

(y-value)

Before we would use -1, 0, 1 to find basic points. Now we can only use -1 and 1 because x ≠ 0

Basics:*Vertical Asymptotes = x’s that make the function undefined*Horizontal Asymptotes = what graph approaches for large x’s

1

𝑥21

𝑥

(-1, 1) (1, 1) (-1, -1) (1, 1)V.A. x = 0; H.A. y = 0 V.A. x = 0; H.A. y = 0

Example: Graphing Using Transformations

R(x) = 1

(𝑥−2)2+ 1

1/x^2 (-1, 1) (1, 1); VA x = 0, HA y = 0

-2 right 2 (1, 1) (3, 1); VA x = 2, HA y = 0

1 (numerator) NO STRETCH/COMPRESS/REFLECT

+1 up 1 (1, 2) (3, 2); VA x = 2, HA y = 1

Example: Graphing Using Transformations

R(x) = −2

(𝑥+1)− 3

1/x (-1, -1) (1, 1); VA x = 0, HA y = 0

+1 left 1 (-2, -1) (0, 1); VA x = -1, HA y = 0

-2 S/R (-2, 2) (0, -2); VA x = -1, HA y = 0

-3 down 3 (-2, -1) (0, -5); VA x = -1, HA y = -3

DAY 2

Finding Vertical Asymptotes:

Vertical asymptotes represent x-values that would make the equation in lowest terms undefined

simplify = cancel like terms from top to bottom then set denominator = 0 and solve for x

**any terms that cancel out creates a “hole”

The graph can never cross or touch a vertical asymptote

Example:

R(x) = 𝑥

𝑥2−4R(x) =

𝑥2

𝑥2+9R(x) =

𝑥2−9

𝑥2+4𝑥−21

Horizontal or oblique (a line on a diagonal y = mx + b)

Lines that represent where the graph is approaching for large values of x

It is possible to cross these asymptotes

Can’t have a horizontal and oblique at the same time

To find them you have to compare the degree of the top polynomial to the degree of the bottom.

For each part, assume R(x) = 𝑝(𝑥)

𝑞(𝑥)

1) If degree of p is less than degree of q H.A. y = 0

2) If degree of p is equal to degree of q H.A. y = 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓.𝑜𝑓 𝑡𝑜𝑝

𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓.𝑜𝑓 𝑏𝑜𝑡𝑡𝑜𝑚

3) If degree on top is exactly one bigger than degree on bottom O.A. y = quotient

4) If none of the first 3 are true, there are no oblique or horizontal

Examples:

1) degree of p < degree of q

R(x) = 𝑥

𝑥3+1

2) degree of p = degree of q

R(x) = 6𝑥2+𝑥+2

4𝑥2+1

Examples:3) Degree of p > degree of q (by exactly one)

R(x) = 3𝑥4−𝑥2

𝑥3−𝑥2+1NEED TO DO LONG DIVISON!

CAN STOP WHEN DEGREE OF DIVIDEND IS SMALLER THAN DIVISOR

Example:

4) None of the first 3 types are true

R(x) = 𝑥5+3𝑥

𝑥3+2𝑥+1

EXIT SLIP