Tut 4 Graphing Techniques

7/31/2019 Tut 4 Graphing Techniques

1/20

Jurong Junior CollegeJC1 Mathematics H2 (9740)

Tutorial 4: Graphing Techniques Solution Section A : Basic Questions

1(a) 121

y x

= +

Vertical Asymptotes: 1 x = Horizontal Asymptote: 2 y =

1(b)2 3 1 3

12 2

x x y x

x x

+ = = + + +

Vertical Asymptotes: 2 x = Oblique Asymptote: 1 y x= +

1(c) 2 0 ( 2)( 24 x x

y x x x

= = ++ )

Vertical Asymptotes: 2 and 2 x x= =Horizontal Asymptote: 0 y =

1(d)1

y x x

=

Vertical Asymptotes: 0 x =Oblique Asymptote: y x=

1(e) 2 21 1

04 4

y x x

= = ++ +

Vertical Asymptotes: 2[Note: Thers is no real root for whichNIL 4 0.] x + =Horizontal Asymptote: 0 y =

1(f) 1 e x y = +

Vertical Asymptotes: Note: is defined for all real vaNIL lues of ].[ xe x

[Note: As , 0, so 1.

Horizon

tal Asy

As , , s

mptote: 1

o .]

x

x

x e y

x e

y

y

+

+ +

=

1


2/20

2(a) 3 2 3 y x x= + y

( 0.816, 4.09)

3

(0.816, 1.91) x

1.89

y

2(b) 2e x y =

1

0 y =

x

y2(c) ln( 2) y x= +

ln 2

x1

2 x =

2


3/20

2(d) 2 1 y x=

y

x1

2(e)4 1 9

2 1 2 2(2 1) x x

x y

= =

+ +

2(f) ,ln y x x= 0 x >

y

x

12

y =

4

4

12

x =

y

x1

(0.368, 0.368)

3


4/20

2(g)2 2 8

23 3

x x y x

x x

+= = + +

y

2(h)2 4 1 2

31 1

x x y x

x x

+ += = + + +

(0.172, 0.657)

2 y x= +

3 x

(5.83, 10.7)

23

=

22 x

1 x =

y y 3

3

3 x

x= +

3.73

1

0.268

4


5/20

2(i)6 24

13 3

y x x

= + +

y

3(a)2 2

14 9

x y+ = - An ellipse 3(b)2 2

14 4

x y+ = - A circle

3(c)2 2

14 9

x y = - A hyperbola 3(e)2

2( 2) 125

x y

+ = - An ellipse

3(d) 2 2( 1) 16 4( 3) x y = + +

2 2

2 2

( 1) 4( 3) 16

( x 1) ( 3)1 - A

16 4

x y

y

+ =+

=

6

y x+ =

hyperbola

3(f) 2 2 2 6 x y x y+ + =

2 2

2 2 2 2

2 2

2 6 6

( 1) 1 ( 3) 3 6

( 1) ( 3) 16 - A circle

x x y y

x y

x y

+ + = + + = + + =

4(a) 2 24 162 2

14 16

x y + =

x(9, 0)

(1, 8)

1 y

=

3 x = 3 x =

9

x

y

4

4

22

5


6/20

4(b)2 2

125 4 x y =

4(c) 2 2( 2) ( 1) 5= x y +

4(d)2

2( 1)9

y x =1

x55

y

25

y x=

25

y x=

x

y

(2, 1)

5

2

4

3

0

y 3( 1) y x=

x

3( 1) y x

3 2

(1, 0)

(1, 3)

=

3 2

3 (1, -3)

6


7/20

5(a) 5(b)

5(c) 5(d)

5(e)

6(a) f ( ) f 2 x

y x y

= = Geometrical transformation: A stretch parallel to the x -axis with a scale factor of 2.

6(b) g( ) g( 2) y x y x= = +Geometrical transformation: A translation of 2 units in the direction of the x -axis .

6(c) h( ) 2h( ) y x y= = x

x

e

Geometrical transformation: A stretch parallel to the y-axis with a scale factor of 2.

6(d) k( ) k( ) 2 y x y x= = +Geometrical transformation: A translation of 2 units in the direction of the y-axis

6(e) m( ) m( ) y x y= = Geometrical transformation: A reflection about the x -axis.

6(f) 2 4 2 4 2 4e e e x x x y y+ = = = +

Geometrical transformation: A stretch parallel to the y-axis with a scale factor of 4e .

OR [Replace x by ( x 2).]2 4 2 2( 2) 4e e e x x y y+= = = x +

Geometrical transformation: A translation of 2 units in the direction of the x -axis.

7


8/20

6(g) [Replace x by ( x).]ln(2 ) ln( 2) y x y x= = +

Geometrical transformation: A reflection about the y-axis.

6(h)2 2

1 2(

x x y y

x x

+ += =+ +1)

Geometrical transformation: A stretch parallel to the y-axis with a scale factor of 1

.2

.

7(a) f ( ) f ( 1) 2 y x y x= = + (i) A translation of 1 units in the direction of the x -axis.(ii) A translation of 2 units in the direction of the y-axis.

7(b) f ( ) f (2 1) y x y x= = +(i) A translation of 1 units in the direction of the x -axis.(ii) A stretch parallel to the x -axis with a scale factor of 1/2.

7(c) 1f ( ) f ( )2

y x y x= =

(i) A stretch parallel to the x -axis with a scale factor of 2.(ii) A reflection about the y-axis.

7(d) .f ( ) 1 f ( 1) y x y x= = +(i) A translation of 1 unit in the direction of the x -axis.(ii) A reflection about the x -axis.(iii) A translation of 1 unit in the direction of the y-axis.

.8(a) The equation of the resulting graph is f ( 2) y x

8(b) The equation of the resulting graph is f 2 x

y .

8(c) The equation of the resulting graph is 2f ( ). y x 8(d) The equation of the resulting graph is f ( ). y x 8(e) The equation of the resulting graph is f ( ) 1. y x

9(i) y( )2,3

( 4, 2) ( )3 y f x = +

x

( )1,0( )5,0 ( )3,0

8


9/20

y9(ii)

9iii)

9(iv)

9(v)

1( , 2

3 )

x

1,3

3

( )3 y f x=

( )0,0 2 ,03

2,0

3

y( 1, 3)

( )1, 2( ) y f x

=

x

( )0,0 ( )2,0( )2,0

( 1, 2)

y

x

( )1, 1

( ) 4 y f x=

( )0, 4

y( )1,12

( 1, 8) ( )4 y f x=

x

( )0,0 ( )2,0( )2,0

9


10/20

9(vi) y

9(vii)

9(viii)

9(ix)

( 1, 2)

x

( )1, 3

( ) y f x=

( )0,0 ( )2,0( )2,0

( 1, 3) y

( )1,3

( x

) y f x=

( )0,0 ( )2,0( )2,0

( 1, 2)

y( )1,3

( ) y f x=

x

( )0,0 ( )2,0( )2,0

y( )1, 3

( 1, 2) ( )2 y f x=

x

( )0,0 ( )2,0( )2,0

( )1, 3 ( 1, 2) y

10


11/20

9(x)

11,

2

10(i)

x1

1,3

( )

1 y f x

=

2 x =

0 y =

0 x 2 x = =

y

( )2 y f x=

x

2 x =

3 y =

( )1,0

3 y =

11


12/20

10(ii) y

11(i) The graph of could be mapped onto the graph of x y 3= 13 += x y by a stretch parallel tothe y-axis by a scale factor of 3ORa translation parallel to the x-axis in the negative direction of 1 unit.

(ii)ln 5

5 3ln 3

x px p= = .

The graph of 3 x y = could be mapped onto the graph of 5 x y = by a stretch parallel

to the x-axis by a scale factor of ln 3ln 5

.

x

1 x =

13 y =

( )1

y f x

=

( )2,0

(0, -2)

12


13/20

Standard Questions:12. f ( ) (2 )(4 ) x x x x= (i) 2 f ( ) y x=

x0 2 4

y

(ii) f ( ) y x= (iii) f ( ) y x=

(iv)1

f ( )

y

x

=

y

(0.845, 3.08)(0.845, 1.75)

x20 4

(0.845,-1.75)

(3.15, -3.08)

y

x2 40

x

y

2

(0.845, 3.08) (3.15, 3.08)

(-0.845, 3.08) (0.845, 3.08)

404 2

x

y (-3.15, -3.08) (3.15, -3.08)

(0.845, 0.325)

(3.15, -0.325)

2 x = 4 x =

13


14/20

13.

(0.368, 0.368) is the minimum point.

x

y

1

(0.368, 0.368)

(i) 2 ln= y x x . (ii) 1ln

y x x

=

x

y

1

x

y

1( )0.368, 2.72

14.

x

y

(4, 4.69)

4 x y x e =

0 y =

(0, 0) is the minimum point and is the maximum point.(4, 4.69)

x

14

y

(4, 2.17)

2 4 x y x e =

0 y =

(4, 2.17)


15/20

14(ii)

x

y

( 4, 4.67)

4 x y x e=

0 y =

x

y

(4, 0.213)

4 x y x e=

0 y = 0 x =

14(iii)

15. (i) 2( 2) 4 y x= + +

(ii)

2 2

4 42 4 x x

y

= + = +

Step 1: Stretch parallel to the x-axis by a scale factor of 1

2

Step 2: Translation parallel to the y-axis in the positive direction of 2 unitsStep 3: Reflection about the x-axis

OR

Step 1: Stretch parallel to the y-axis by a scale factor of 2Step 2: Translation parallel to the y-axis in the negative direction of 2 unitsStep 3: Reflection about the x-axis

16. Let 2f ( ) x y x= = .

After A: 2f ( 3) ( 3) y x x= =

After B: 22f ( 3) 2( 3) y x x= =

After C : 22f ( 3) 5 2( 3) 5 y x x= + = +

The equation of the resulting curve is 22( 3) 5 y x= + .

15


16/20

2

2

2

2f ( 3) 5

5f ( 3)

2

( 3) 5f ( ) 2

x x

x x

x x

+ =

=

+ =

The equation of the resulting curve is2 6 4

2 x x

y+ += .

17 (i) f (2 2 )= y x y

17 (ii)

(2, 2)

x 5

( , 0)2

(1, 0)

1 y

( )2 2 y f x= =

3 x =

y

x

( 2, 5)

(0,1)

4 y =

( )1 3 y f x=

4 x =

16


17/20

y 17(iii)

x

2 y = ( )2 y f x=

17(iv) 17(v) y

y ( 2, 2)

( 2, 2)

17(vi)

3

( ) y f x= x 3

x

( ) y f x=

( )3,0

y

( )2 y 1= y f x=

x

1 y =

4 x =

17


18/20

y 17(vii)

1( 2, )

2

18(i) )(f x y =f ( 2) f ( 2 2) f( ) y x x y x [ 2 units in the direction of x-axis]

( , ) ( 2, ) x y x y

1 x =

02

1

x

y

= y

( 1, 5)

3 x =

( )4,0

1 y

=

( )

x

1 y

f x=

0 x =

18


19/20

(ii) )2

1(f x

y =

f( ) f(1 ) f(1 ) f (1 )2 2

x x y x y x y y= = + = + =

n o p

[ translation -1 units in the direction of x-axis.stretch // x-axis, factor 2 ; Reflection in y-axis. ]

( , ) ( 1, ) (2( 1), ) ( 2( 1), ) ( 2 2, x y x y x y x y x ) y

( , ) (2 2 , )

(0, 0) (2, 0)

( 2, 0) (6, 0)

( 1, 5) (4, 5)

x y x

y

y

0 x =

1 0 x x= =

19(a) .f ( ) y x=

(b) .f ( ) when 0

f (| |)f ( ) when 0

x x y x

x x

= =


20/20

20.2 7

22( 2) 3

23

22

A = 2, B = 3

x y

x x

y x

y x

+=+

+ += +

= ++

1 1 3 32

2 2 x x x x +

+ + 2+

Following the formula for composite transformations, y = d + cf(b x + a)

( ) ( )( ) ( 2) 3 2 2 3 2 f x f x f x f x + + + +

Sequence of transformations will be1) Translation of -2 units along the x - axis2) Scaling of 3 along the y - axis3) Translation of 2 units along the y - axis

-2

2

7/2

y

x-7/2

20

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Tut 4 Graphing Techniques