1

Unit 1 Transformations

General Outcome: • Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes:

1.1 Demonstrate an understanding of the effects of horizontal and vertical translations on

the graphs of functions and their related equations:

▪ ( )y f x h= −

▪ ( )y k f x− =

1.2 Demonstrate an understanding of the effects of horizontal and vertical stretches on the

graphs of functions and their related equations:

▪ ( )y af x=

▪ ( )y f bx=

1.3 Apply translations and stretches to the graphs and equations of functions:

1.4 Demonstrate an understanding of the effects of reflections on the graphs of functions

and their related equations, including reflections through the:

▪ x-axis ( )y f x= −

▪ y-axis ( )y f x= −

▪ line y x= 1( ) or ( )y f x x f y−= =

1.5 Demonstrate an understanding of inverse of relations.

Topics:

• Function Notation Page 2

• Translations (Outcome 1.1) Page 7

• Reflections (Outcomes 1.4 and 1.5) Page 18

• Stretches (Outcome 1.2) Page 29

• Combining Transformations (Outcomes 1.3 and 1.4) Page 39

2

Unit 1 Transformations

Function Notation:

( )f x → pronounced function “f” at x

→ means the function itself is

called “f” and the variable used

within the function is “x”

Ex) If 2( ) 5 8f x x x= + − , find the following.

a) (3)f b) ( 6)f − c) ( )12

f

3

Ex) If 2( ) 4 5f x x x= + + and ( ) 1 2g x x= + − find the

following.

a) (1) (8)f g− b) 3 ( 2)g x +

c) ( )(3)f g d) 2 ( 3) 11f x− − +

Ex) If ( )f x x= , write the following in function

notation.

a) 4x − b) 2x c) 3 2x + +

4

Ex) If 3( )f x x= , write the following in function

notation.

a) 3( 1)x + b) 32 3x + c) 37( 1) 4x− − +

Types of Functions:

Linear Quadratic

Basic Equation: Basic Equation:

5

Cubic Absolute Value

Basic Equation: Basic Equation:

Radical Rational

Basic Equation: Basic Equation:

6

Function Notation Assignment:

1) If 3( ) 5f x x= − , find the following in simplest form.

a) ( 1)f − b) 4 ( )f x c) (4 )f x

d) ( ) 5f x + e) ( 5)f x + f) ( )f x−

2) If ( )f x x= , write the following in terms of function f.

a) 1x − b) 3x + c) 2 1x − d) 3 x−

3) If 2( )f x x= , write the following in terms of function f.

a) 2 3x + b)

2( 3)x + c) 23x d)

2(3 )x

e) 24 7x − f) ( )24 7x − g)

42 1x− − h) 23( 2)x− − −

7

Vertical Translations:

The graph of 2y x= is shown below.

Sketch the graphs of 2 3y x= + and 2 4y x= − .

2y x=

Rule: Function Notation:

Mapping Notation:

8

Horizontal Translations:

The graph of 2y x= is shown below.

Sketch the graphs of 2( 3)y x= − and 2( 4)y x= + .

2y x=

Rule: Function Notation:

Mapping Notation:

9

In General if ( )y f x a b= + + then

0a the graph moves __________

0a the graph moves __________

0b the graph moves __________

0b the graph moves __________

Mapping Notation: ______________________

Ex) Describe how the graph of 4y x= − can be

obtained from the graph of y x= .

Ex) Describe how the graph of 4 3y x= + can be

obtained from the graph of 4y x= .

10

Ex) Describe in words and using mapping notation how

the graph of 6( 1) 5y x= − − can be obtained from the

graph of 6 3y x= − .

Ex) Write the equation of the graph produced when

the graph of y x= is translated 4 units to the

left and 7 units down.

Ex) Write an equation for each function represented

by the thick line

a) y x= b) 1y

x=

11

Ex) What vertical translation needs to be applied to

the graph of y x= so that is passes through the

point ( )16, 7 ?

Ex) What horizontal translation needs to be applied

to the graph of y x= so that is passes through

the point ( )17, 8 ?

12

Translations Assignment:

1) Given the graph of ( )y f x= below, sketch the graph of the transformed

function.

a) ( ) ( ) 3g x f x= + b) ( ) ( 2)g x f x= −

c) ( ) ( 4)g x f x= + d) ( ) ( ) 2g x f x= −

13

2) Given the function ( ) 2 3f x x= − + and ( ) 2 1g x x= + + , the

transformations that will transform ( )y f x= to become ( )y g x= are a

translation of

A] 4 units left and 2 units down

B] 4 units right and 2 units up

C] 1 unit left and 3 units up

D] 2 units left and 4 units down

3) Describe, using mapping notation, how the graphs of the following functions

can be obtained from the graph of ( )y f x= .

a) ( 10)y f x= + b) 6 ( )y f x+ =

c) ( 7) 4y f x= − + d) 3 ( 1)y f x− = −

4) Determine the equation of each transformed function.

a) 1

( )f xx

= is translated 5 units to the left and 4 units up.

b) 2( )f x x= is translated 8 units to the right and 6 units up.

14

c) ( )f x x= is translated 10 units to the right and 8 units down.

d) ( )y f x= is translated 7 units to the left and 12 units down.

5) Given the graph of ( )y f x= below, describe the transformations that can be

applied to the graph of ( )f x to obtain the transformed function, then sketch

the transformed graph.

a) ( ) ( 4) 3r x f x= + − b) ( ) ( 2) 4s x f x= − −

15

c) ( ) ( 2) 5t x f x= − + d) ( ) ( 3) 2v x f x= + +

6) The transformation of the function 3( )f x x= is described by the mapping

notation ( ) ( ), 4, 9x y x y→ − + . Describe the transformation on ( )y f x= .

7) What vertical translation is applied to the graph of 2y x= if the translated

image passes through the point ( )4, 19 ?

16

8) What horizontal translation is applied to the graph of 3y x= if the translated

image passes through the point ( )19, 125− ?

9) Determine the equation for each function represented by the thick line graph.

a) b)

10) Explain how the graph of ( ) 4g x x= + could be a vertical translation of 4

units up or a horizontal translation of 4 units to the left of the graph ( )f x x= .

17

11) The graph of the function 2y x= is translated so that the translated graph has

zeros of 7 and 1.

a) Determine the equation of the transformed graph.

b) Describe the translations of the graph 2y x= .

12) Describe how the graph of 1

yx

= could be transformed into the graph of

13

5y

x− =

−.

13) The roots of the quadratic equation 2 12 0x x− − = are 3− and 4. Determine

the roots of the equation 2( 5) ( 5) 12 0x x− − − − = .

18

Reflections:

Reflection over the x-axis:

The graph of 2( ) 10 25y f x x x= = − + is shown

below.

Write an equation that Sketch the graph of

represents ( )y f x= − ( )y f x= − on the grid

below. (Use your

calculator.)

Rule: Function Notation:

Mapping Notation:

19

Reflection over the y-axis:

The graph of 2( ) 10 25y f x x x= = − + is shown below.

Write an equation that Sketch the graph of

represents ( )y f x= − ( )y f x= − on the grid

below. (Use your

calculator.)

Rule: Function Notation:

Mapping Notation:

20

Reflection over the line y x= :

The graph of 2( ) ( 5)y f x x= = − is shown below.

Write an equation that Sketch the graph of

represents 1( )y f x−= 1( )y f x−= on the grid

or ( )x f y= (inverse) below. (Use your

calculator.)

Rule: Function Notation:

Mapping Notation:

**When to use: 1( )y f x−= ( )x f y=

21

In general:

If ( )y f x= is transformed into

1( )y f x−=

( )y f x= − ( )y f x= − or

( )x f y=

The graph is The graph is The graph is

reflected reflected reflected

over the over the over the

________ ________ ________

Ex) In each case below sketch the indicated

reflection given the graph of ( )y f x= .

a) ( )y f x= − b) ( )x f y= c) ( )y f x= −

22

Ex) Sketch the reflection indicated for each case

below given the graph of ( )y f x= .

a) ( )y f x= − b) ( )y f x= − c) ( )x f y=

Ex) If the graph of ( )y f x= is indicated by the thin

line graph, write an equation that represents

each transformation (thick line graph) in terms

of ( )f x .

a) b) c)

23

Ex) If 2

6( )

3f x

x=

+, write an equation for each of

the following.

a) ( )y f x= − b) ( )y f x= − c) ( )x f y=

Sketch the graphs for each of the following.

2

6( )

3y f x

x= =

+

( )y f x= − ( )y f x= − ( )x f y=

24

Ex) Consider the function 2( ) ( 3) 5f x x= + − .

a) Graph the function ( )f x and its inverse.

b) Is the inverse of ( )f x a function? Explain how you

can tell.

c) If the inverse is not a function, how can the domain

of ( )f x be restricted so that the inverse of ( )f x is a

function?

25

Reflections Assignment:

1) Sketch the reflection of each graph in the x-axis and then determine the

equation of the reflected image.

a) ( ) 3f x x= b) 2( ) 1g x x= + c) 1

( )h xx

=

2) Sketch the reflection of each graph in the y-axis and then determine the

equation of the reflected image.

a) ( ) 3f x x= b) 2( ) 1g x x= + c)

1( )h x

x=

26

3) Use mapping notation to describe how the graph of each function can be found from

the graph of the function ( )y f x= .

a) ( )y f x= − b) ( )y f x= − − c) ( )x f y=

4) Determine the zeros of the function ( ) ( 4)( 3)f x x x= + − after each transformation.

a) ( )y f x= − b) ( )y f x= −

5) The graph of a function ( )y f x= is contained completely in the third quadrant.

Identify in which quadrant the graph will be in after each transformation.

a) ( )y f x= − b) ( )y f x= − c) ( )x f y= d) ( )y f x= − −

6) Given the graph of ( )y g x= below, identify the coordinates of the invariant point

after each transformation.

a) ( )y g x= − b) ( )x g y= c) ( )y g x= −

27

7) Given the graph of each relation below, sketch the graph of its inverse.

a) b)

8) For each function below, determine the equation of its inverse.

a) ( ) 2 3f x x= + b) 3( ) ( 4) 7g x x= − +

9) For each case below, identify a restriction that can be placed on the domain of ( )f x

so that its inverse is also a function.

a) 2( ) ( 21) 16f x x= + − b)

2

1( ) 20

( 18)f x

x= +

−

28

10) The function for converting the temperature from degrees Fahrenheit, x, to degrees

Celsius ,y, is 5

( 32)9

y x= − .

a) Determine the equivalent temperature in degrees Celsius for 90 F.

b) Determine the inverse of this function. What does it represent? What do the

variables represent?

c) Determine the equivalent temperature in degrees Fahrenheit for 32 C.

d) If both functions were to be graphed, what does the invariant point represent in

this situation?

29

Stretches:

Vertical Stretches:

The graph of 2( ) 4y f x x= = − is shown below.

Write the equation Write the equation

of 3 ( )y f x= of 1 ( )2

y f x=

Sketch the graph Sketch the graph

of 3 ( )y f x= on the of 1 ( )2

y f x= on the

grid below. grid below.

Rule: Function Notation:

Mapping Notation:

30

Horizontal Stretches:

The graph of 2( ) 4y f x x= = − is shown below.

Write the equation Write the equation

of (4 )y f x= of ( )13

y f x=

Sketch the graph Sketch the graph

of (4 )y f x= on the of 1( )3

y f x= on the

grid below. grid below.

Rule: Function Notation:

Mapping Notation:

31

Ex) Sketch the graph of the indicated transformation

if the graph of ( )y f x= is given.

a) 3 ( )y f x= b) (2 )y f x=

Ex) Write the equation of the image of ( )y f x= after

each transformation.

a) a horizontal b) a vertical c) a horizontal

stretch by a stretch by a stretch by a

factor of 3 factor of 14

factor of 12

about the y-axis. about the x-axis. about the y-axis,

and a vertical

stretch by a

factor of 37

about the x-axis.

32

Ex) How can the graph of 3 ( )y f x= be obtained

from the graph of ( )y f x= ?

Ex) What happens to the graph of ( )y f x=

( , ) (8 , )x y x y→ ?

Ex) Write the equation of the image of 2y x= after a

horizontal stretch about the y-axis by a factor of 3

4.

33

Ex) Write the equation of the image of 3 7y x= +

after a vertical stretch about the x-axis by a

factor of 4 and a horizontal stretch by a factor

of 15

about the y-axis.

Ex) In each case below, describe how the graph of

the second function compares to the graph of

the first function.

a) 2xy =

32 xy =

b) 3y x=

33 (2 )y x=

34

Ex) Given the thin line graph of the original

function, determine the equation of the

transformed thick line graph.

a) 2

6

1y

x=

+ transformed graph

b) ( 4)( 2)( 6)y x x x= + − − transformed graph

35

Stretches Assignment:

1) Given the graph of ( )y f x= below, sketch the graph of each given

transformation.

a) 2 ( )y f x= b) (2 )y f x=

c) 1

( )2

y f x= d) 1

2y f x

=

36

2) Determine the equation of ( )g x as a transformation of ( )f x for each case

below.

a) b)

c) d)

3) Use mapping notation to describe how the graph of ( )y f x= is transformed

into each of the following.

a) (8 )y f x= b) 6 3

7 2y f x

=

c)

15

4y f x

− = −

37

4) Describe how the graph of ( )y f x= can be transformed into the graph of

13

7y f x

=

.

5) Describe how the graph of 5log ( )y x= can be transformed into the graph of

56log (3 )y x= − .

6) Describe what happens to the graph of ( )y f x= after each of the following

changes are made to its equation.

a) Replace x with 4x .

b) Replace y with 5y .

c) Replace y with 3

y−.

38

7) Determine the zeros of the function ( ) ( 16)( 52)h x x x= − + after each

transformation given below.

a) 4 ( )y h x= b) (4 )y h x=

8) Determine the equation of the transformed function if the graph of

12 8y x= − is stretched vertically about the x-axis by a factor of 2 and then

stretched horizontally about the y-axis by a factor of 3.

9) Determine the equation of the transformed function if the graph of

( 5)( 25)( 45)y x x x= − + + is stretched vertically about the x-axis by a factor

of 3 and then stretched horizontally about the y-axis by a factor of 1

5.

39

Combining Transformations:

Rules:

Stretches Vertical

Stretch

y = stretch

factor

Horizontal

Stretch y = stretch

factor

Reflections Reflect in

x-axis y =

Reflect in

y-axis

y =

Reflect in

The line y x=

y =

or

x =

Translations Vertical

Translation y =

Horizontal

Translation

y =

Invariant Points:

40

Ex) Given the graph of ( )y f x= sketch its image

after the required transformations.

a) Stretch the graph of b) Translate the graph of

( )y f x= horizontally ( )y f x= 3 units up, and

by a factor of 2 about then stretch it

the y-axis, and then horizontally by a factor

translate it 3 units up. of 2 about the y-axis.

c) Translate the graph of d) Stretch the graph of

( )y f x= 2 units down, ( )y f x= vertically by

and then stretch it a factor of 3 about the

vertically by a factor of x-axis, and then translate

3 about the x-axis. 2 units down.

41

**Sometimes the order in which we apply the required

transformations matters.

To simplify the process, transformations should be

applied and described in the following order:

• Stretches

• Reflections

• Translations

Ex) Given the graph of ( )y f x= , sketch the graph

of the indicated transformed function. In each

case identify any invariant points.

a) 2 ( )y f x= − b) ( )1 12 2

y f x− −=

42

c) ( )12 ( 5)2

y f x= + d) 2 ( 3) 1y f x= − − +

e) ( )1 3 82

y f x= + −

43

Ex) Determine the equation of the graph of ( )y f x=

after it has been horizontally stretched by a

factor of 14

about the y-axis, then reflected

over the x-axis, and finally translated 5 units

down.

Ex) The function 3( )G x x= is transformed into a new

function ( )y P x= . To form the new function

( )y P x= , the graph of ( )y G x= is stretched

vertically about the x-axis by a factor of 0.2,

reflected about the y-axis, and then translated 3

units to the right. Determine the equation of the

new function ( )y P x= .

44

Ex) Describe the series of transformations required

to transform graph A into graph B.

Description 1:

Description 2:

45

Ex) Describe the transformations required to

transform:

a) graph A to graph B

b) graph B to graph C

46

Combining Transformations Assignment:

1) Given the graph of ( )y f x= below, sketch the graph of each transformation.

a) 3 ( ) 4y f x= − + b) ( )2( 3) 6y f x= + +

c) (3 15) 4y f x= − − − d) 1

( 3)2

y f x= − +

47

2) If the graph of 2y x= is stretched horizontally about the y-axis by a factor of

2 and then reflected about the x-axis, determine the equation of the

transformed image.

3) Describe how the graph of ( )y f x= could be transformed into the graph of

3 (4 16) 10y f x= − − − .

4) Given the graph of ( )y f x= below, sketch the transformed graph if ( )y f x=

is stretched vertically about the x-axis by a factor of 2, stretched horizontally

about the y-axis by a factor of 1

2, and then translated 3 units up.

48

5) If the point ( )12, 18− is on the graph of ( )y f x= , determine the coordinates

of its image under each of the following transformations.

a) 6 ( 4)y f x+ = − b) 4 (3 )y f x= c) 2 ( 6) 4y f x= − − +

d) 2

3 6 53

y f x−

= − −

e) ( )1

3 2( 6)3

y f x−

+ = +

6) Given the graph of ( )y f x= below, determine the equation of each

transformed graph.

a) b)

49

7) If the graph of ( )y f x= is stretched vertically about the x-axis by a factor of

3, reflected in the x-axis, and then translated 4 units to the left and 5 units

down, determine the equation of the transformed function.

8) If the graph of ( )y f x= is stretched horizontally about the y-axis by a factor

of 1

3, stretched vertically about the x-axis by a factor of

3

4, reflected in both

the x and y-axis, and then translated 6 units to the right and 2 units up,

determine the equation of the transformed function.

9) Describe using mapping notation the transformations on ( )y f x= for each of

the following.

a) 2 ( 3) 4y f x= − + b) (3 ) 2y f x= − −

c) ( )1

( 2)4

y f x−

= − + d) ( )3 4( 2)y f x− = − −

e) 2 3

3 4y f x

− − =

f) 3 6 ( 2 12)y f x− = − +

50

10) Describe how the graph of y x= could be transformed into the graph of

14 10

2y x

−= − +

11) Given the thin line graph of the original function, determine the equation of

the transformed thick line graph.

a)

b)

51

c)

12) If the domain and range of function ( )h x are such that

D : 12 18,x x x R− and R : 36,y y y R , determine the domain

and range of each transformation on ( )h x .

a) ( )3 2( 3) 5y h x= − + + b) 1

( 3 ) 82

y h x= − −

13) The graph of the function 22 1y x x= + + is stretched vertically about the

x-axis by a factor of 2, stretched horizontally about the y-axis by a factor of

1

3, and translated 2 units to the right and 4 units down. Determine the

equation of the transformed function.

52

14) If the x-intercept of the graph of ( )y f x= is located at ( ), 0a and the

y-intercept is located at ( )0, b , determine the x-intercept and y-intercept after

each of the following transformations on the graph of ( )y f x= .

a) ( )y f x= − − b) 1

22

y f x

=

c) 3 ( 4)y f x+ = − d) 1 1

6 ( 4)2 4

y f x

+ = −

e) 3 ( 7) 4y f x= − + + f) 2 (3 18) 34

yf x= − +

53

Answers

Function Notation Assignment:

1. a) 6− b) 34 20x − c) 364 5x − d) 3x e) 3( 5) 5x + −

f) 3 5x− −

2. a) ( ) 1f x − b) ( 3)f x + c) (2 1)f x − d) 3 ( )f x−

3. a) ( ) 3f x + b) ( 3)f x + c) 3 ( )f x d) (3 )f x e) 4 ( ) 7f x −

f) 4 ( ) 28f x − g) ( )22 1f x− − h) 3 ( 2)f x− − −

Transformations Assignment:

1. a) b)

c) d)

2. A

3. a) ( ) ( ), 10, x y x y→ − b) ( ) ( ), , 6x y x y→ −

c) ( ) ( ), 7, 4x y x y→ + + d) ( ) ( ), 1, 3x y x y→ + +

54

4. a) 1

45

yx

= ++

b) ( )2

8 6y x= − + c) 10 8y x= − −

d) ( 7) 12y f x= + −

5. a) The graph of ( )f x is translated b) The graph of ( )f x is translated

4 units to the left and 3 units 2 units to the right and 4 units

down. down.

c) The graph of ( )f x is translated d) The graph of ( )f x is translated

2 units to the right and 5 units up. 3 units to the left and 2 units up.

6. The graph of

3( )f x x= is translated 4 units to the left and 9 units up.

7. 3 units up

8. 24 units to the left

9. a) 2( 7) 2y x= + + b)

3 2( 12) 17( 12) 90( 12) 147y x x x= − + − + − +

10. If ( )g x x= , then ( ) ( ) 4 ( 4) 4f x g x g x x= + = + = +

11. a) 2( 4) 9y x= − − b) The graph of

2y x= is translated 4 units to

the right and 9 units down.

12. The graph of 1

yx

= is translated 5 units to the right and 3 units up.

13. 2 and 9

55

Reflections Assignment:

1. a) 3y x= − b) 2 1y x= − − c) 1

yx

−=

2. a) 3y x= − b) 2 1y x= + c)

1y

x

−=

3. a) ( ) ( ), , x y x y→ − b) ( ) ( ), , x y x y→ − − c) ( ) ( ), , x y y x→

4. a) 4− , 3 b) 3− , 4

5. a) 2 b) 4 c) 3 d) 1

6. a) ( )0, 4− b) ( )4, 4− − c) ( )4, 0

7. a) b)

8. a) 3

2

xy

−= b) 3 7 4y x= − +

56

9. a) 21 or 21x x − − b) 18 or 18x x

10. a) 32.2 C b) 9

325

y x= + y represents the temperature in Fahrenheit

and x represents the temperature in Celsius c) 89.6 F

d) The invariant point represents where the temperature in Celsius means

the same thing in Fahrenheit.

Stretches Assignment:

1. a) b)

c) d)

2. a) ( ) 4 ( )g x f x= b) ( ) (5 )g x f x= c) 1

( ) (4 )2

g x f x=

d) 1

( )3

g x f x

= −

3. a) ( )1

, , 8

x y x y

→

b) ( )2 6

, , 3 7

x y x y

→

c) ( ) ( ), 4 , 5x y x y→ − −

4. The graph of ( )y f x= is stretched vertically about the x-axis by a factor of 3

and then stretched horizontally about the y-axis by a factor of 7.

57

5. The graph of 5log ( )y x= is stretched vertically about the x-axis by a factor of

6, then it is reflected about the x-axis, and finally it is stretched vertically about

the y-axis by a factor of 1

3.

6. a) The graph of ( )y f x= is stretched horizontally about the y-axis by a factor

of 1

4.

b) The graph of ( )y f x= is stretched vertically about the x-axis by a factor

of 1

5.

c) The graph of ( )y f x= is stretched vertically about the x-axis by a factor

of 3 and then it is reflected about the x-axis.

7. a) 52− and 16 b) 13− and 4

8. 8 16y x= −

9. 375( 1)( 5)( 9)y x x x= − + +

Combining Transformations Assignment:

1. a) b)

58

c) d)

2.

21

2y x

= −

3. The graph of ( )y f x= is stretched vertically about the x-axis by a factor of 3,

then it is stretched horizontally about the y-axis by a factor of 1

4, next it is

reflected about the x-axis, and finally it is translated 4 units to the right and

10 units down.

4.

5. a) ( )8, 12− b) ( )4, 72− c) ( )6, 32− − d) ( )1, 49−

e) ( )12, 9− −

6. a) ( )( 2) 2y f x= − + − or ( 2) 2y f x= − − −

b) ( )2( 1) 4y f x= + − or (2 2) 4y f x= + −

7. 3 ( 4) 5y f x= − + −

8. ( )3

3( 6) 24

y f x−

= − − + or 3

( 3 18) 24

y f x−

= − + +

59

9. a) ( ) ( ), 3, 2 4x y x y→ + + b) ( )1

, , 23

x y x y

→ − −

c) ( )1

, 2, 4

x y x y−

→ − −

d) ( )1

, 2, 34

x y x y

→ + − +

e) ( )4 2

, , 3 3

x y x y− −

→

f) ( )1 1

, 6, 22 3

x y x y−

→ + +

10. The graph of y x= is stretched horizontally about the y-axis by a factor of

2, then it is reflected about the y-axis, and finally it is translated 8 units to the

left and 10 units up.

11. a) 4 2 2y x= − + − or 4 8 2y x= − + − b) 22( 4) 6y x= − + +

c) 2

25

( 3) 1y

x= −

− − + or

2

25

( 3) 1y

x= −

+ +

12. a) D : 9 6,x x x R− , R : 103,y y y R −

b) D : 6 4,x x x R− , R : 10,y y y R

13. ( )22 2(3 6) (3 6) 1 4y x x= − + − + − or 236 138 130y x x= − +

14. a) ( ), 0a− and ( )0, b− b) ( )2 , 0a and ( )0, 2b

c) ( )4, 0a + and ( )0, 3b − d) ( )4 4, 0a + and 1

0, 62

b

−

e) ( )7, 0a− + and ( )0, 3 4b + f) 1

6, 03

a

+

and ( )0, 8 12b +