**RATIONAL FUNCTIONS!**

Navnit, Puneet, Ishween, and Manroop!

Navnit, Puneet, Ishween, and Manroop!

**Chapter 3.1 Reciprocal of a Linear Function**

**Chapter 3.2 : Reciprocals of Quadratics**

A rational function - defined by rational fraction

Rational Functions of the Form :

(ax + b)/(cx + d)

KEY FEATURES :

Vertical Asymptote : -d/c

Horizontal Asymptote: a/c

Y – intercept : b/d

X – intercept: -b/a

Chapter 3.3

How To Graph Rational Functions :

Domain

Plot V.A. and H.A.

Calculate/Investigate Limits

Perform a 'End Behavior' Chart

Calculate Y-intercept

DRAW GRAPH !!!!

Range

3 possibilities of quadratic expressions;

x + 3x - 4

2

x + 21x + 1

2

factorable

perfect square factor

not factorable

(x - 1) (x + 4)

(x + 8)

x + 21x + 1

2

x² + 16x + 64

2

**What's a Reciprocal of Quadratic Function?**

function in the form of

f(x)=

g/ Quadratic

*g can represent any number

A Quadratic?

"Solve" the denominator by

factoring

An example...

perfect square factor

GRAPHING THE FUNCTION...

5 STEPS:

Find the

Domain

of the function

Include restrictions on x that makes f(x)=0

Solve for the

LIMITS

As x -> 5 from +/- side, f(x) -> -/+ infinity

Determine the

End Behaviours

Substitute x=1000,10000,100000 to analyze if f(x)-> 0 from +/- side of x-axis

Use the

"Zmoofab Method"

Legitimize

the graph!

State the

Range

of the function!

*TARGET*

Aim to finish the question in 10 minutes!

KEY FACTS!

Factorable Quadratic Rational Function ALWAYS has

"Line Symmetry"

Function may cross the Horizontal Asymptote

An Example:

Our original equation

Factor the equation

State restrictions

1. State the Domain!

2. Find Limits

This trend will be seen on the graph!

g(x)

**Any Questions?**

Thank You for listening! :)

Thank You for listening! :)

a function of the form;

a

______

(mx - n)

2

only one factor, so only one asymptote.

in this case, the asymptote would lie at n

_

m

unfactorable expression

assess x towards the asymptote (1.5)

assess x towards the infinities

Domain = {x e R | x =/= 1.5 }

Range = {y e R | y > 0}

f(x) =

a function of the form

a

________

px + qx + r

f(x) =

Domain = {x e R} since the discriminant < 0

"the speed bump graph"

H.A. : y = 0

V.A. : x = 1.5

Range = {y e R | 0 > y >= 1/5}

Only assess x towards the infinities

Find your maximum value by making the denominator the smallest possible value (for this instance, it would be at x = 0), making your maximum value 1/5 (0.2).

H.A.: y = 0

Forms

f(x)= P(x)

Q(x)

Example

g(x)= 1

x+7

VA: x= -7

HA: y= 0

Reciprocal of a Linear Function

Restrictions: x= c

k

Vertical Asymptote: x= c

k

Horizontal Asymptote: y= 0

f(x)= 1

kx-c

As x g(x)

-7^+ +inf

-7^- -inf

+inf 0 from above

-inf 0 from below

2

hi