Rational Functions

Absolute Value Review:

Recall that for an absolute value graph, the _________________ cannot be ___________________ and are reflected over the ______________.

Example 1: Sketch: y x= .

Example 2: Sketch: 2 1y x= −

Example 3: Sketch: cosy x=

f(x)

x

Reciprocal Functions

Recall: When graphing the reciprocal function, the x-values will remain ________________ in your table of values but we take the _____________________ of the _______________.

Sketch 1yx

=

For reciprocal functions:

Vertical asymptote:

Horizontal asymptote: Steps in Graphing Reciprocal Functions:

1.) x-intercepts become __________________________________.

2.) y-coordinates of __________ remain the same.

3.) The graph approaches the _____________________.

4.) Plot the ____________________ of the ____________________.

Example 1: Given ( ) 3 6f x x= − , sketch 1( )f x

.

Example 2: Sketch: 2

14

yx

=−

Example 3: Given the graph of ( )f x , sketch 1( )f x

.

Reciprocal Trigonometric Functions: Example 4: Sketch each of the following:

a.) 1

siny

x= Note:

f(x)

x

b.) Sketch the graph of 1( )cos

f xx

= over the interval ]2,2[ ππ− . Note:

f(x)

x

c.) Sketch the graph of 1tan

yx

= . Note:

f(x)

x

You may want to use the grid below to graph tany x= first.

f(x)

x

Transformations of Rational Functions

To tranform the graph 1yx

= we can use the following equation:

a =

h =

k =

Vertical Stetches: 1 ay ax x

= =

Example 1: Graph each of the following:

a.) 1yx

= b.) 3yx

=

c.) 4yx

= −

Example 2: Sketch 6 32

yx

= −−

Example 3: Sketch:

a.) 2

1yx

=

b.) ( )2

1 23

yx

= ++

Converting to ay kx h

= +−

form.

Method 1 – Division Example 4: Convert each of the following to ay k

x h= +

− form.

a.) 4 52

xyx−

=−

b.) 2 23

xyx+

=+

c.) 5 4xyx+

= d.) 6

xyx

=+

Method 2 – Manipulate the equation algebraically.

Example 5: Convert each of the following to ay kx h

= +−

form.

a.) 4 52

xyx−

=−

b.) 2 23

xyx+

=+

c.) 5 4xyx+

= d.) 6

xyx

=+

Horizontal and Oblique Asymptotes Horizontal Aysmptotes:

• If the degree of the numerator is less than the degree of the denominator then the horiztonal asymptote is ________.

Ex: 21

yx

=+

; 2

31

xyx

=−

• If the degree of the numerator is equal to the degree of the denomintor, then the

horizontal asymptote is the leading coefficient of numerator divided by the leading coefficient of the denominotor.

Ex: 2

2

24

xyx

=−

horizontal asymptote:

• If the degree of the numerator is greather than the degree of the denominator

then there is _______ horizontal asymptote.

Ex: 3

2

21

xyx

=+

Oblique Asympote. An oblique asymptote is angled.

• If the degree in the numerator is _______________ than the degree in the denominator (when the expression is simplified) then there is an oblique

asympote. Ex: 3

2

21

xyx

=+

Ex: 3 12

y xx

= + −+

Oblique asymptote at________________

Identifying Horizontal and Oblique Asymptotes. Example 6: Use the equation to determine whether there is a horizontal or oblique asymptote and determine the equation of these asymptotes.

a.) 29xyx

=−

b.) 2 4

3xyx

− +=

−

c.)2 2

1x xy

x+ −

=+

Graphing Rational Functions

Example 1: Graph the function 4 52

xyx−

=−

. Idenitify any asymptotes and intercepts.

Example 2: Graph the function 2 31

xyx−

=−

. Idenitify any asymptotes and intercepts.

Example3 : Sketch the graph of the function: ( ) 2

62

f xx

=+

.

Example 4: Write the equation of the function in the form ay kx h

= +−

Analyzing Rational Functions

Graphs of rational functions can have a variety of shapes and different features. For

example a vertical asymptote corresponds to a ________________________________

in the equation of the function. However, not all ___________ result in vertical

asymptotes. Sometimes an ________ results in a

_____________________________________ in the graph.

Point of discontinuity:

Example 1: Sketch the graph of the function 2 5 6( )

3x xf x

x− +

=−

. Analyse its behaviour

near its non-permissible values.

Example 2: Sketch the graph of the function ( )2

2

xf xx x

=−

.

Example 3: Sketch the graph: 2 2( )

4 2x xf x

x−

=−

and compare it to the graph of 2 2( )

4 2x xf x

x+

=−

that is sketched using graphing technology.

The graph of 2 2( )

4 2x xf x

x+

=−

is shown below: 2 2( )

4 2x xf x

x−

=−

Example 4: Match the equation of each rational function with the most appropriate graph.

Example 5: Determine any vertical asymptotes, points of discontinuity, and intercepts

for 2

2

2 1512

x xyx x− −

=− −

Example 6: Write the equation of a possible rational function with a vertical asymptote

at 5x = − , a point of discontinuity of 5 19,2 5

− −

, and an x-intercept of 7

Connecting Graphs and Rational Functions

Relating Roots and x-intercepts. Example 1:

a.) Determine the roots of the rational equation 6 5 02

xx

+ − =+

algebraically.

b.) Examine the graph of 6 52

y xx

= + −+

and state the x-intercepts.

c.) What is the connection between the roots of the equation and the x-intercepts of the graph of the function?

Example 2: a.) Solve the equation: 2 3 7 13 2

x x xx

− −= −

−graphically. State approximate

solutions.

b.) Solve algebraically to check your answer.

Example 3: Solve the equation 8 1522 5 4 10

x xxx x

++ =

+ +algebraically and compare these

roots to the roots found on the graph of the function.

Example 4: In basektball, a player’s free-throw percentage is given by dividing the total number of successful free-throw baskets by the total number of attempts. So far this year, Larry has attempted 19 free-throws and has been successful on 12 of them. If he is successful on every attempt from now on, how many more free-throws does he need to attempt before his free-throw percentage is 80%?