7h2 graphing rational functions -...

7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)

1

Goals: Graph rational functions Find the zeros and y-intercepts Identify and understand the asymptotes Match graphs to their equations Vocabulary: Function notation: Domain: Asymptote: Intercepts: Zero: Table of values: Graph: Sketch: Function notation: Function f with input x can be written in several ways:

f(x) = equation rule f: x equation rule f: f(x)= equation rule In all cases, when you input a value for x, the output is the y coordinate.

Ex. ( x1 , y1) is the same as ( x1 , f(x1) ) | | on the x axis on the y axix

Ex. A. For each of the following: a. find f(0), f(-1), f(2) b. Graph the function y = f (x)

i. f(x) = 3x +1 ii. f : x→− x2


2

Graphing Simple Broken Rational Functions Definition of a Rational Function: A Rational Function is a function that has a variable, x, in the denominator.

For example: f : x→ 1x

f (x) = 2x − 5

f : x→ xx + 3

g :g(x) = x − 4x + 2

They all have a restriction on x. Therefore they have a restriction on their domain. That is why we call them broken. Ex. B

a. Fill the table of values for x = – 4 to x = 4 for g : x→ 3x + 2

x

y

b. Graph the function g At the “break” there is special behaviour. Let’s check out what happens when x is close to – 2, i.e. x =– 1.97, – 1.98, – 1.99, – 2.001,– 2.01,– 2.02,

x – 1.97

– 1.98 – 1.99 x = – 2 -2.001 – 2.01 – 2.02

y

error ! broken

This special behaviour is what defines an Asymptote.


3

Asymptotes An asymptote is a line which a graph comes closer and closer to, but never crosses.

Exploring Asymptotes: Activity:

Looking at the graph of f x( ) = 1x

using online graphing software (ex. geogebra), what do

we notice? Exploring graphing the broken rational function: Ex.C. Fill in the following table of values for each function. Plot the points, and graph the function.

a. f : x→ 1x

x

-100 -10 -2 -1 − 12

− 14

0 1

4

12

1 2 10 100

y


4

b. g : x→ 1x − 4

x

-10 -2 -1 0 1 2 3 3.5 4 4.5 5 6 10

y

c. h : x→ 3xx −1

x

-10 -2 -1 0 0.5 0.75 1 1.25 1.5 2 3 10

y


5

Finding the Asymptotes

Vertical Asymptotes: Vertical asymptotes occur at the restrictions on the domain. i.e. at the restriction on the x in the denominator. When x ≠ N , the equation of the vertical asymptote is x = N. Horizontal Asymptotes: The horizontal asymptote is a value that the output, y, will never reach, but will come closer and closer to. Here are two methods to find the horizontal asymptote.

Method 1: substitute in very large positive and negative values for x (extreme values) and see where the y value tends towards (comes closer to)

i.e. f(-10 000) f(-1000) f(1000) f(10 000)

or

Method 2: divide every term by x. Eliminate whatever will become insignificant as x becomes infinitely big.

Think: for example: 1x

becomes insignificant as x→∞ because 1∞

is very very small.

Also true for 5x

; as x→∞ , 5∞→ 0 or any number:

numberx

; as x→∞ , number

∞→ 0

Worked Example: Find the horizontal asymptote for f x( ) = x2x −1

by Method 1:

f −1000( ) = x2x −1

= −10002 −1000( )−1 =

−1000−2001

= 0.4997

close to 0.5 or 12

by Method 2:

x2x −1

⇒

xx

⎛⎝⎜

⎞⎠⎟

2xx

⎛⎝⎜

⎞⎠⎟ −

1x

⎛⎝⎜

⎞⎠⎟⇒ 1

2 − 1x

⎛⎝⎜

⎞⎠⎟

as x→∞ , 1∞→ 0 so

1

2 − 1x

⎛⎝⎜

⎞⎠⎟⇒ 12 − 0( ) =

12

the horizontal asymptote occurs at

y = 12

1

1-1-1

y = 1 2

f 1000( ) = x2x −1

= 10002 1000( )−1 =

10001999

= 0.5002


6

Ex. D Find the horizontal asymptotes. At first, use both methods. Then choose one method that you prefer.

a. f : x→ 3x +1

b. g(x) = −1x

c. f x( ) = x2x +1

d. h : x→ 42 − x

e. g :g x( ) = 3x +1x − 2

f. h x( ) = x +1x(x −1)

Graphing Rational Functions Ex. E Graph the first three functions from Ex. D. (first find and mark the asymptotes, then create a table of values).

a. f : x→ 3x +1

b. g(x) = −1x

c. f x( ) = x2x +1


7

The Shape of Things (Remember: graphs ‘hug’ asymptotes) How to sketch a graph given:

1. The x and y intercepts (if there are any) 2. The vertical asymptote(s) 3. The horizontal asymptote

Ex. Sketch the graph with the given parameters:

Using the shape of things to sketch graphs of rational functions (no tables of values)

1. Find and mark the vertical asymptote on the graph

Vertical asymptotes occur at the restriction on x in the denominator.

2. Find and mark the horizontal asymptote on the graph Use either method 1 or 2 to find the horizontal asymptote (ie. f(1000) & f(-1000) or divide by x and eliminate insignificants)

3. Find x and y intercepts and mark them on the graph.

for y intercepts: let x = 0 find y. ( 0 , ) for x intercepts: let y = 0 find x. ( , 0 )


8

Ex.F Graph each of the following using the sketching method:

a. f x( ) = xx +1

b. f : x→ 32x −1


9

Match the Graph

Look for key features: Intercepts and Asymptotes Ex. G Match the graph to its equation:

a. y = 2x −1

b. f (x) = xx −1

c. f : x→ 4x −1( )2

d. f : x→ − 3x −1


10

Noticing Patterns: Use graphing software to graph each set of functions on the same set of axis and then discuss the differences and similarities:

A B C D

y = 3x

y = 6x

y = −2x

y = − 0.4x

y = xx +1

y = 3x −1

y = 3x+1

y = 3x−1

f x( ) = xx −1

f x( ) = 2xx −1

f x( ) = 4xx −1

f x( ) = 2x +1x −1

f : x→ xx +1( )2

f : x→ 3x2

f : x→ 6x2 + x

Practice Exercises: P108

Characteristics of broken rational functions

Example 1

Which numbers are not included in the definition

set?

a) 32

4:

�xxf �

b) 1

4:

2 �xxg �

Sketch the function graph of g.

Solution:

a) For x=1,5 the denominatorof the function

term ist zero. The number 1,5 can therefore not

be includes in the drfinition set.

b) As x2 is greater than or equal to 0, whatever

number is entered fpr the variable, x2+1

cannot be equal to 0 for any inserted value.

x -4 -2 -1 0 1 2 4

g(x)17

4 5

4 2 4 2 5

4 17

4

As no number can be excluded from the domain

no vertical asymptotes exist either

.

Example 2

The function 43

2:

�x

xxf � is given.

a) Which number cannot be included in the

domain? Indicate Df. Create a table of values

and draw te function graph

b) Indicate the equations of both asymptotes and

give reasons for them.

c) What percentage of the horizontal asymptote

is f(1000)?

Solution:

a) The denominator 3x+4 can equal 0 when

4

3��x . The number

3

4� can therefore not be

included in the domain.

}3

4{\ ��QDf

b) Verical asymptote: 3

4��x

For relatively high x values the y-values

approach the number 3

2. The summand 4 in

the denomiator has a minor influence on the y-

value. The horizontal asymptote is therefor

described by 3

2�y

If one divides numerator and

denominator by x, one receives

the equation equivalent

(for x � 0)

x

xf4

3

2)(

��

By these it can easily be shown

that the y values of absolute

great x approximate

3

2

c) 3

2%87,99...6657,0)1000( off ��

x -100 -5 -4 -3 -2 -1,5 -1 0 1 2 5 100

y 0,68 0,9 1 1,2 2 6 -2 0 0,3 0,4 0,5 0,66

2 Which number(s) cannot be included in the

domain? Indicate a domain for each one.

a) x

xf32

4:

�� b)

xxf

2

3: �

c) 2

4:

xxf � d)

9

2:

2 �x

xxf �

e) 34: �zxf � f) )3()3(

2:

��

xx

xxf �

g) 52

13:

��

x

xxf � h)

tt

ttf

21

3:

��

�

i) xx

xf�2

1: � k)

4

12:

�ssf �

3 Exercises

Mixed (salad of) solutions

for 2:

For the following functions you cannot calculate

the restrictions on the domain directly. Why not?

Try to find the numbers using systematic

sampling.

a) 15,2

3:

2 ��xxf �

b) 8

3:

3 �x

xxg �

c) 483

27:

4

2

��

x

xxxh �

i.e. what are the restrictions on x ?

be included in the definition set (domain).

2for x, x + 1 will never become zero. Therefore, there are no definition

holes (no restrictions).

the function graph.

4

3

x

(x) =

(x) =

P 108


11

P109


4 �

Use a function plotter to draw the graphs of the

functions. Indicate the equations of horizontal

and ertical asymptotes, if they exist. Describe

how you could also have found these without th

graph by using the function rule.

a) x

xf3

1: � b)

13

2:

�xxf �

c) 13

2:

�xx

xf � d) s

sf�5,1

1: �

e) t

ttf

�5,1: �

f) )2()2(

1:

�� xxxf �

5

Given each function rule, ansider the graph

includes asymptotes and, if relevant, indicate the

equations of the asymptotes. Draw the function

graph (without using a table of values or a

function plotter).

a) x

xf41

1:

�� b)

)1(2

3:

�xxg �

c) 12

:�xx

xh � d) x

xk5,2

: �

e) x

xp5,02

1:

��

f) )3(

2:

��xx

xq �

6

Draw both graphs for the functions x

xf4

: �

and 2

4:

xxg � in a common coordinate

system. Describe the similarities and differences

of the two graphs in words.

7

a) Draw the graph of the function 3

1:

�xxf � .

Describe how the graph would change if the

number 3 was replaced with

- a larger number

- a smaller number

b) What would change in the graph of f if in the

numerator the variable x was included instead

of the number 1?

c) Using a function plotter draw the graphs of the

functions 3

:1 �xx

xf � ;32

:2 �xx

xf � and

33:3 �x

xxf � in a common coordinate

system and describe the differences between

the graphs.

8

Draw the graphs of the two functions x

xf3

: �

and xxg3

4

3

20: �� in a common coordinate

system. Read off the intersections of each graph

as precisely as possible. Determine the

approximate value the surface area of the areas

enclosed by the intersections of the two graphs.

9

Which numbers must be excluded from the

definition set? Indicate the definition set and

sketch the function graphs.

a) 2)2(

4:

�xxf �

b) )3(

1:

xxxg

��

10

Indicate each time a broken rational function

which has the line with the given equation as

asymptote.

a) x = 1 b) y = -2

c) x = 0 ; the graph should also go through the

point (2�4).

11

Indicate a broken rational function for which the

graph does not enter the yellow area.

vertical

the graph by using the function rule (equation).

consider if the graph

( x ) =

(Write the function rule for a graph).

For each, write the rule for a broken rational function

P 109


4 �






a) x

xf3

1: � b)

13

2:

�xxf �

c) 13

2:

�xx

xf � d) s

sf�5,1

1: �

e) t

ttf

�5,1: �

f) )2()2(

1:

�� xxxf �

5





function plotter).

a) x

xf41

1:

�� b)

)1(2

3:

�xxg �

c) 12

:�xx

xh � d) x

xk5,2

: �

e) x

xp5,02

1:

��

f) )3(

2:

��xx

xq �

6


xf4

: �

and 2

4:




7


1:

�xxf � .



- a larger number

- a smaller number



of the number 1?


functions 3

:1 �xx

xf � ;32

:2 �xx

xf � and

33:3 �x



the graphs.

8


xf3

: �

and xxg3

4

3






9




a) 2)2(

4:

�xxf �

b) )3(

1:

xxxg

��

10



asymptote.

a) x = 1 b) y = -2


point (2�4).

11



vertical



( x ) =



P 109


4 �






a) x

xf3

1: � b)

13

2:

�xxf �

c) 13

2:

�xx

xf � d) s

sf�5,1

1: �

e) t

ttf

�5,1: �

f) )2()2(

1:

�� xxxf �

5





function plotter).

a) x

xf41

1:

�� b)

)1(2

3:

�xxg �

c) 12

:�xx

xh � d) x

xk5,2

: �

e) x

xp5,02

1:

��

f) )3(

2:

��xx

xq �

6


xf4

: �

and 2

4:




7


1:

�xxf � .



- a larger number

- a smaller number



of the number 1?


functions 3

:1 �xx

xf � ;32

:2 �xx

xf � and

33:3 �x



the graphs.

8


xf3

: �

and xxg3

4

3






9




a) 2)2(

4:

�xxf �

b) )3(

1:

xxxg

��

10



asymptote.

a) x = 1 b) y = -2


point (2�4).

11



vertical



( x ) =



P 109


12

P110

P126


12

The function graphs shown should be matched to

the function terms given. In the order a) to h) the

correctly matched letters result in a solution

word. But first you have to >break> another code.

13

Indicate the broken rational functions for which

the graphs go through the point P (3�2).

14

Draw the graph of the function 12

2:

�xxf � ,

How would the graph change if the 2 was

replaced by a larger number? By a smaller

number?

15

The following functions each have a restriction

on the domain. The graphs of the function,

however, show four differenz behaviours near

these restrictions. Draw the function geaphs with

a function plotter and describe the four different

behaviours.

a)x

xf�3

2:1 � b)

12

3:2 �xxf �

c)23

4:

xxf � d)

1:4 �x

xxf �

e) 25)1(

2:

xxf

��

� f)

5,1

1:6 �xxf �

g)1

2:7 �x

xxf � h)

28

1:

xtf �

Could you determine which of these behaviours

will occur just by considering the function rule?

G16

The half-lines w1 and w2 are the bisectors of �

and �. The angle � is selected so that it is halved

by the line g.

a) Transfer the figure for �=40° into the notebook

(the points A and B are at random distances

away from S).

b) At wich angle � do the points A, S and B lie an

a line.

c) How should � be selected so that B ans S

create a line with w1? Give reasons.

3

3:

�xx

xf � |M 23: �xxf � |J

2)3(

4:

�xxf � |C

)2)(6(

1:

�� xxxf � |B

3

3:

�xxf � |W

3

)3(:

��

x

xxxf � |B

3(

2:

��x

xf � |S 2)3(: �xxf � |F

rulesa

the intersection of asymptotes occurs at the point

P( 3 | 2 )

t

2)

P 110


12





13



14


2:

�xxf � ,



number?

15






behaviours.

a)x

xf�3

2:1 � b)

12

3:2 �xxf �

c)23

4:

xxf � d)

1:4 �x

xxf �

e) 25)1(

2:

xxf

��

� f)

5,1

1:6 �xxf �

g)1

2:7 �x

xxf � h)

28

1:

xtf �



G16



by the line g.



away from S).


a line.



3

3:

�xx

xf � |M 23: �xxf � |J

2)3(

4:

�xxf � |C

)2)(6(

1:

�� xxxf � |B

3

3:

�xxf � |W

3

)3(:

��

x

xxxf � |B

3(

2:

��x

xf � |S 2)3(: �xxf � |F

rulesa


P( 3 | 2 )

t

2)

P 110


12





13



14


2:

�xxf � ,



number?

15






behaviours.

a)x

xf�3

2:1 � b)

12

3:2 �xxf �

c)23

4:

xxf � d)

1:4 �x

xxf �

e) 25)1(

2:

xxf

��

� f)

5,1

1:6 �xxf �

g)1

2:7 �x

xxf � h)

28

1:

xtf �



G16



by the line g.



away from S).


a line.



3

3:

�xx

xf � |M 23: �xxf � |J

2)3(

4:

�xxf � |C

)2)(6(

1:

�� xxxf � |B

3

3:

�xxf � |W

3

)3(:

��

x

xxxf � |B

3(

2:

��x

xf � |S 2)3(: �xxf � |F

rulesa


P( 3 | 2 )

t

2)

P 110

Fractional equations��S��

11

a) If you add the same number of the

numerator and denominator of the fraction

12

5 you get the fraction

5

4 . What is the

number?

b) A fraction when cancelled gives

3

2 . If you

add the number 5 to the uncancelled

numerator and denominator you get the

fraction .

7

5.

c) What is the uncancelled fraction?

12

Find the intersections of the graphs of

1

2:

2

��

x

xxf � and 3: �xxg �

mathematically.

Sketch both function graphs on the same axes.

13

Solve the following formulas for the variables

in the brackets.

a) )(VV

m��

b) )()(2

1chcaA ��

c) )(2

1h

h

caB

��

d) )(2

2

1

2

1 tt

t

F

F�

e) )(2

1 2 ggtF � f) )(

3b

b

bat

��

14

Karl learned the formula

v

sttvs

t

sv �� ,,

by heart for his homework.

Claudia says: I just need to know the formula

t

sv � . I can get both the other equations just

by transformation.

Discuss the advantages and disadvantages of

KarlOs and ClaudiaOs approaches. How would

you do it?

15

a) Is there a fraction with the same value as the

number 0.875 whose denominator is 3

greater than the numerator?

b) For which fractional equation for 0.875 do

the numerator and denominator add up to

300?

16

Write a fractional equation in which the

variable comes up at least twice and the

number 3.5 is the solution. Give the equation

to your neighbour to solve.

17

Describe the difference between the graphs of

the function

4:

xxf � and

xxg

4: � in

words. Calculate the coordinates of the

intersection of both graphs.

18

The Wittke family is going on holiday by car.

Ramona has calculated that they first need to

drive 645 km on the Autobahn and then 90 km

on the regional roads. Mr. Wittke says that

they can drive twice as fast on the Autobahn as

on the regional roads. Mrs. Wittke then says:

YSo we will need

2

17 hours for the journey.Z

What speeds does she base her calculation on?

19

A swimming pool is beeing emptied using two

pumps. One of the pumps on its own would

need 3 hours, the other 2 hours.

a) How long do both pumps need when they

are used at the same time?

b) How long would a third pump need alone if

the tree pumps together would manage it in

one hour?

c) Check if it is possible to use a third pump so

that three pumps together would manage the

job in 45 minutes.

20

a) The equation has two

solutions which can be quite easily

recognised without transformations. What

are they?

3)3)(1( �� xxx

b) Solve the equations and 11��x

2

1

2

1

��

��

xx

x

c) What transformations can you use to get the

equation in b) from the one in a)?

d) Compare the solutions of the three equations

and describe the effect of the

transformations you did.


13

Extra Practice Graphing: Graph each of the following functions. Indicate the intercepts and equations for the asymptotes in each case.

a. y = 2x

b. y = −1x − 2

c. f x( ) = xx − 3

d. f x( ) = −x2x + 4

e. f x( ) = 2xx +1

f. f x( ) = 0.5x +1x +1

g. f : x→ −x + 3x − 3

h. f : x→ 2x +1x

i. h :h(x) = 3x(x − 2)

Answers: S.108 S.109


14


15

S.110


16

S.126 Extra Practice Graphing: Answers:

a.

b.

c.

d.

e.

f.

g.

h.

i.

7h2 graphing rational functions -...

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