Today in Pre-Calculus• No calculators needed• Notes:

– Rational Functions and Equations– Transformations of the reciprocal

function

• Go over quiz• Homework

Rational Functions

A rational function, f(x), is a ratio or quotient of polynomial functions p(x) and q(x) expressed as

* The domain of f(x) is all real numbers except where q(x) = 0

( )( )

( )

p xf x

q x

Rational FunctionsAre the following rational functions? If yes, state the domain.

2

3

2

2

3

1 5. ( ) . ( )

1

3 5. ( )

4

4 7 16 64. ( ) . ( )

10 5 3 1

xa f x d i x

x x

x xb g x

x

x xc j x e h x

x x x

yes. D:(-∞,0) υ(0,∞)

yes. D:(-∞,-2) υ (-2,2) υ (2,∞)

No, numerator not a polynomial

yes. D: (-∞,∞)

:

: , 2 2,0 0, 2 2,

yes

D

Transformations of the Reciprocal Function

The simplest rational function is the basic function, 1

( )f xx

Horizontal asymptote:y=0Vertical asymptote:x=0

Example 1Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 2 units to the right.

1( )f x

x

1( )

2g x

x

Example 2Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 5 units to the right, followed by a reflection across the x-axis

1( )f x

x

1( )

5g x

x

Example 3Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function, by a translation of 4 units to the left, followed by a vertical stretch by a factor of 3, and finally a translation 2 units down.

1( )f x

x

3( ) 2

4g x

x

Graphing Rational FunctionsThe graph of any rational function of the form can be obtained by transforming by using polynomial long division.

1( )f x

x

( )ax b

f xcx d

Example 13

( )2

f xx

Vertical stretch: 3Shift left 2

Example 21

( ) 2f xx

Reflect across x-axisShift up 2

2 1( )

xf x

x

Example 38

( ) 31

f xx

Vertical stretch: 8Shift left 1Shift down 3

5 3( )

1

xf x

x

Homework

• Pg. 245: 5-10 and 31-36

• Chapter 2 test: Tuesday, November 25