Chapter 11 II Graphs of Functions ENHANCE

8/3/2019 Chapter 11 II Graphs of Functions ENHANCE

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Graphs of Functions 1


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CHAPTER 11 : GRAPHS OF FUNCTIONS II

Steps involved in solving an equation graphically:

1. Find they value by substitutingx valueinto the given function.

2. Draw the graph by using the given scale for x-axis and y- axis.

3. Find the value of y or x when given value of x or y from the graph.

4. On the same axes, draw a suitable straight line which satisfies the equation.

5. Determine the solutions by reading off the x-coordinates of the point of

intersection of the two graphs.



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11.1: Linear Function

Complete the following table below



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Example 1:

Complete the following table for theequation xy =

x 0 2 3y 0 2 3

By using scientific calculator, substitute

each of the x value in equation above.

x xy =

0 0

2 2

3 3

Exercise 1

Complete the following table for theequation

i. xy = x 0 2 3y



x xy =

0

2

3

ii. xy =

x 0 2 3y

By using scientific calculator, substituteeach of the x value in equation above.

x xy =

0

2

3

Example 2

Complete the following table for theequation 32 += xy

x -1 0 3y 1 3 9



x 3)(2 += xy

-1 2(-1) + 3 = 1

0 2(0) + 3 = 33 2(3) + 3 = 9

Exercise 2

Complete the following table for theequation 13 += xy

x -1 0 3y



x 1)(3 += xy

-1

03


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11.2: Quadratic Function




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Example:

1. Complete the following table for

the equation y = 2x 2

x 1 2 3

y 2 8 18



x y

1 2(1) 2 = 2

2 2(2) 2 = 8

3 2(3) 2 = 18

Exercise:

Complete the following table for theequations:

1. y = - 2x 2

x -1 0 1y

By using scientific calculator, substitute each

of the x value in equation above.

x y=- 2(x) 2

-1

0

1

2. y =21 x 2

x -1 0 1

y

By using scientific calculator, substitute eachof the x value in equation above.

xy =

2

1(x) 2

-1

0

1

3. y = 3x 2

x -1 0 1

y



x y = 3(x)

2

-1

0

1

4. y = -2

5x 2



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x -1 0 1

y



xy = -

2

5(x) 2

-1

0

1

5. y = 5x 2

x -1 0 1

y



x y = 5(x) 2

-1

0

1

11.3: Cubic Function




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Example 1

Complete the following table for the

equation 33xy =

x 1 2 3y 3 24 81



x 3)(3 xy =

1 3)1(3 3 =

2 24)2(3 3 =

3 81)3(3 3 =

Exercise 1


equations:

i. 35xy =

x 1 2 3y



x 3)(5 xy =

1

2

3

ii. 32xy =

x 1 2 3y



x 3)(2 xy =

1

2

3



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Example 2


equation 52 3 += xy

x -2 0 2y -11 5 21


each of the x value in equation above.x 5)(2 3 += xy

-2 115)2(2 3 =+

0 55)0(2 3 =+

2 215)2(2 3 =+

Example 3


equation 43 += xy

x -1 -2 3y 5 12 -23


x 4)( 3 += xy

-1 54)1( 3 =+

-2 124)2( 3 =+

3 234)3( 3 =+

Exercise 2


equation 53 += xy x -2 0 2

y


each of the x value in equation above.x 5)(

3+= xy

-2

0

2

Exercise 3


equation

i. 33 += xy x -1 -2 3y


each of the x value in equation above.x 3)( 3 += xy

-1

-2

3

ii. 12 3 = xy x -1 -2 3y


each of the x value in equation above.x 1)(2 3 = xy

-1-2

3

Example 4


Exercise 4




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equation 943 = xxy

x -3 0 2y -24 -9 -9


x 9)(4)( 3 = xxy

-3 249)3(4)3( 3 =

0 99)0(4)0( 3 =

2 99)2(4)2( 3 =

equations:

i. 733 += xxy

x -3 0 2y


x 7)(3)(3

+= xxy

-3

0

2

ii. 3612 xxy =

x 1 0 2y



x 3)()(612 xxy =

1

0

2

11.4: Reciprocal Function




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Example 1:


equation y =x

1

x -2 -1 1 2

y -1/2 -1 1 1/2



x y

-2 -

-1 - 1/1 = -1

1 1/1 = 1

2

Exercise:


equations:

1. y = -x

2

x -2 -1 1 2

y



xy = -

x

2

-2

-1

1

2

2. y =x

3

x -2 -1 1 2

y


xy =

x

3

-2

-1

1

2

3. y = -x2

1

x -2 -1 1 2



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y


x y = -x21

-2

-1

1

2

4. y =x

16

x -2 -1 1 2

y



xy =

x

16

-2

-1

1

2

5. y = -x

5

x -2 -1 1 2

y




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xy = -

x

5

-2

-11

2

11.21 : Quadratic Graph

Example :

a) Complete the following table for the equation 542 2 = xxy



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b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw the

graph of 542 2 = xxy for 52 x

c) From your graph, find

i) the value of y, when x = -1.5

ii) the values of x, when y= 0

d) Draw a suitable straight line on your graph to find the values of x which satisfy the

equation 03822

=+ xx for 52 x . State the values of x.

Answer For Example :

a) x = -1, y = 1

x = 2, y = -5

x = 4, y = 11

b) Refer graph on the next page

c) i) y = 5.5

ii) x = -0.9, 2.9

d)

Therefore, the appropriate graph that should be drawn is

x 0 1

y 8 4

From the graph, the solutions of the equation are

x = 0.4, 3.65

Another Method

We can also use the discrimination method to find the appropriate graph

to be drawn.


x -2 -1 0 1 2 3 4 5

y 11 -5 -7 1 25

0382 2 =+ xx

382 2 = xx

54385422

= xxxx

84 = xy

Rearrange the equation

so that one side of theequation is 2x2 4x5

84 = xy

03822

=+ xx


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)1.(..........542 2 = xxy

)2.(..........38202

+= xx

84:)2()1( = xy


Graph for Example Quadratic Graph

-2 -1 0 1 2 3 4 5

-5

--10

5

10

15

20

25

y

x

x

x

x

x

x

x

x

x

x

x

y = 4x - 8


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Exercise 1 :

a) Complete the following table for the equation 592 2 += xxy



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graph of 5922

+= xxy for 70 x

c) From your graph, findi) the value of y, when x = 1.7

ii) the value of x, when y = 15


equation 041022


Exercise 2 :

a) Complete the following table for the equation 752 2 = xxy


graph of 752 2 = xxy for 53 x



ii) the value of x, when y = 15


equation 04722


Exercise 3:

a) Complete the following table for the equation ( ) 952 = xxy


graph of ( ) 952 = xxy for 53 xc) Draw a suitable straight line on your graph to find the values of x which satisfy the

equation ( ) xxx 24952 = for 53 x . State the values of x.

Exercise 4 :

a) Complete the following table for the equation 523 2 ++= xxy


x 0 1 2 3 4 5 6 7

y 5 -2 -5 1 10 40

x -3 -2 -1 0 1 2 3 4 5y 26 0 -7 -10 -9 5 18

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

y 24 -2 -9 -12 -11 3 6


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b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 5 unit on the y-axis, draw thegraph of 523

2++= xxy for 43 x



ii) the value of x, that satisfy the equation of 5232

= xx


equation 025232


Exercise 5 :

a) Complete the following table for the equation ( )xxy = 8

2

1


graph of ( )xxy = 82

1for 70 x

c) Draw a suitable straight line on your graph to find the values of x which satisfy the

equation ( ) 108 =xx for 70 x . State the values of x.

Exercise 6 :

a) Complete the following table for the equation2238 xxy +=


graph of2238 xxy += for 43 x


i) the value of y, when x = 1.35

ii) the values of x, when y = -10

d) Draw a suitable straight line on your graph to find the values of x which satisfy theequation 026322


Exercise 7 :


x -3 -2 -1 0 1 2 3 4

y -11 0 4 -16 -35

x 0 0.5 1 2 3 4 5 6 7

y 0 1.88 6 7.5 7.5 3.5

x -3 -2 -1 0 1 2 3 4

y -19 3 8 9 6 -12


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a) Complete the following table for the equation2275 xxy +=







equation 011322

= xx for 53 x . State the values of x.

Exercise 8 :

a) Complete the following table for the equation 2273 xxy +=







equation 013322



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

y -34 -4 5 11 8 -10

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

y -36 -6 3 9 6 -12


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11.31 : Cubic Graph

Example 1:

a) Complete the following table for the equation 583 += xxy


graph of 583

+= xxy for 43 x



ii) the value of x, when y= 25


equation 01123



a) x = -2, y = 13

x = 1, y = -2

x = 3, y = 8


c) i) y = 13. 5

ii) x = 3.7

d)

Therefore, the appropriate graph that should be drawn is

x 0 1

y 6 10

From the graph, the solutions of the equation are

x = 0.1, 3.5


x -3 -2 -1 0 1 2 3 3.5 4

y 2 12 5 -3 19.9 37

01123 = xx

1123

+= xx

58112583

++=+ xxxx

64 += xy

Rearrange the equation

so that one side of theequation isx3 8x + 5

64 += xy

01123

= xx


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Graph for Example Cubic Graph

-3 -2 -1 0 1 2 3 4x

-5

5

10

15

20

25

30

35

40

y

x

x

x

x

x

x

x

x

x

x

x

=y 64 +x


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Exercise 1:



graph of 5103 += xxy for 5.35.3 x

c) From your graph, find the value of y, when x = -2.5


equation 6103

= xx for 5.35.3 x . State the values of x.

Exercise 2:



graph of 18103

+= xxy for 43 x



equation 010103

= xx for 4.3 x . State the values of x.

Exercise 3:

a) Complete the following table for the equation 7123

+= xxy


graph of 7123 += xxy for 44 x



ii) the values of negative x that satisfy the equation 7123

= xx


equation 02153



x -3.5 -3 -2 -1 0 1 2 3 3.5

y -2.9 8 14 5 -7 12.9

x -3 -2 -1 0 1 2 3 3.5 4

y 21 27 18 6 25.88 42

x -4 -3 -2 -1 0 1 2 3 4

y -9 16 18 7 -4 -9 23


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Exercise 4:


b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 10 unit on the y-axis, draw

the graph of 6123 += xxy for 45 x


i) the value of y, when x = -2.5ii) the values of positive x, when y = 0


equation 12123


Exercise 5:

a) Complete the following table for the equation 1253 = xxy


graph of 1253 = xxy for 5.33 x



ii) the value of x which satisfies the equation 1253

= xx


equation 493

= xx for 5.33 x . State the values of x.


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

y -59 -10 22 17 6 -5 -3 22

x -3 -2 -1.5 -1 0 1 2 3 3.5

y -10 -7.9 -8 -12 -16 0 13.4


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11.41 : Reciprocal Graph

Example 1:

a) Complete the following table for the equationx

y8

=


graph ofx

y8

= for 55 x



ii) the value of x, when y = 4.4

d) From the graph, state the values of x when x and y have a same value.


a) x = -5, y = -1.6

x = -2.5, y = -3.2

x = 1.6, y = 5


c) i) y = -2.6

ii) x = 1.8

d) x = -2.7, 2.73


x -5 -4 -2.5 -1.6 -1 1 1.6 2.5 4 5

y -2 -5 -8 8 3.2 2 1.6


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x

Graph for Example Reciprocal Graph

-2

-4

-6

-8

0

2

4

6

8

10

y

-1-2-3-4 1 2 3 4

x

x

x

x

x

x

x

x

x

x

y = x


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Exercise 1:


y1

=


graph ofx

y1

= for 44 x



equation ( )0,11

= xxx

for 44 x . State the values of x.

Exercise 2:


y6

=


graph ofx

y6

= for 44 x

c) From your graph,i) find the value of y, when x = 1.3

ii) find the value of x, when y = 3.5


equation xx

=+13

for 44 x . State the values of x.

Exercise 3:


y2

=


graph ofx

y2

= for 44 x

c) From your graph,

i) find the value of y, when x = 1.5

ii) find the value of x, when y = -1.8


x -4 -3 -2 -1 -0.5 0.5 1 2 3 4

y -0.25 -0.33 -1 -2 1 0.5 0.25

x -4 -2.5 -1 -0.6 0.6 1 2 3 4

y 1.5 6 10 -10 -6 -3 -1.5

x -4 -3 -2 -1 -0.5 0.5 1 2 3 4

y -0.5 -0.7 -1 -2 4 1 0.7 0.5


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equationx

x4

12 =+ for 44 x . State the values of x.

Exercise 4:


y4

=

b) By using a scale of 2 cm to 1 unit on the x-axis and 2cm to 2unit on the y-axis, draw the

graph ofx

y4

= for 55 x

c) From your graph,

i) find the value of y, when x = -1.4ii) find the value of x, when y = 4.4


equation 02

3= x

xfor 45.0 x . State the values of x.

Exercise 5:

a) Complete the following table for the equation 12

3+=

xy

b) By using a scale of 2 cm to 0.5 unit on the x-axis and 2cm to 0.5unit on the y-axis, draw

the graph of 12

3+=

xy for 45.0 x

c) From your graph,i) find the value of y, when x = 1.3

ii) find the value of x, when y = 2.8


equation 02

3= x

xfor 45.0 x . State the values of x.


x -5 -4 -2 -1 -0.5 0.5 0.8 1.5 2.5 5

y -1 -4 -8 8 5 1.6

x 0.5 1 1.5 2 2.5 3 3.5 4

y 4 2.5 1.75 1.6 1.43 1.38


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PAST YEAR EXAMINATION QUESTIONS

1. Nov 2003, Q12

(a) Complete the following table for the equationx

y4

= .

(b)

x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4

y 1 1.6 8 -8 -4 -1.25 -1

[2 marks]

For this part of the question, use the graph paper. You may use a flexible curve.

By using a scale of 2 cm to 1 unit on thex-axis and 2 cm to 2 units on they-axis,

draw the graph ofx

y4

= for -4 x 4.

[5 marks](c) From your graph, find

(i) the value ofy whenx = 1.8,

(ii) the value ofx wheny = 3.4.[2 marks]

(d) Draw a suitable straight line on your graph to find all the values ofx which

satisfy the equation 324

+= xx

for -4 x 4.

State these values ofx.[3 marks]

2. July 2004, Q12

(a) Table 2 shows values ofx andy which satisfy the equation 1053

+= xxy .

x -3.4 -3 -2 -1 0 1 2 3 3.4

y -12.3 k 12 14 10 6 m 22 32.3

TABLE 2

Calculate the value ofkand ofm. [2 marks]

(b) For this part of the question, use the graph paper on page 33.You may use a flexible curve rule.By using a scale of 2 cm to 1 unit on thex-axis and 2 cm to 5 units on they-axis,

draw the graph of 1053 += xxy for -3.4 x 3.4.




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(i) the value ofy whenx = 0.7 ,

(ii) the value ofx when 01053 =+ xx .[2 marks]



= xx for -3.4 x 3.4.

State these values ofx. [4 marks]

Answer:

(a) k = ..

m = .

(b) Refer graph on page 33

(c) (i) y = ..

(ii) x = .

(d) x = ..

3. Nov 2004, Q12

(a) Table 1 shows values ofx andy which satisfy the equation 342 2 = xxy .

x -2 -1 0 1 2 3 4 4.5 5

y k 3 -3 -5 -3 m 13 19.5 27

Table 1

Calculate the value ofkand ofm. [2 marks]

(b) For this part of the question, use the graph paper. You may use a flexible curve.By using a scale of 2 cm to 1 unit on thex-axis and 2 cm to 5 units on they-axis,

draw the graph of 3422

= xxy for -2 x 5.

[5 marks]

(c) From your graph, findi. the value ofy whenx = -1.5,

ii. the value ofx wheny = 0.[2 marks]

(d) Draw a suitable straight line on your graph to find a value ofx which satisfies the

equation 02322

=+ xx for -2 x 5.

State this value ofx.[3 marks]



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4. Nov 2005, Q12

(a) Complete Table 2 for equation 32 2 = xxy .

x -2 -1 0.5 1 2 3 4 4.5 5

y 7 -2 -2 3 12 33 42

Table 2

[2 marks]

(b) For this part of the question, use the graph paper. You may use a flexible curve.By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-

axis, draw the graph of 32 2 = xxy for -2 x 5.

[4 marks]

(c) From your graph, find

i. the value ofy whenx = 3.6,ii. the value ofx wheny = 37.[2 marks]

(d) Draw a suitable straight line on your graph to find all value ofx which satisfy the

equation 10322

= xx for -2 x 5.

State these values ofx.

5. July 2005, Q12

(a) Complete Table 1 in the answer space for the equation 223 xxy = .

[2 mark]

Answer:

(a)

x -3 -2 -1 0 1 2 3 3.5 4

y -12 -3 3 0 -7 -25 -33

TABLE 1

(b) For this part of the question, use the graph paper provided on page 33.

You may use a flexible curve rule.


draw the graph of223 xxy = for -3 x 4.

[4 marks]Answer:

(b) Refer graph on page 33 .



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(i) the value ofy when x = 1.7 ,

(ii) the value ofx when y = 26 .

Answer:

(c) (i) y = .

(ii) x =



118

2=+ xx for -3 x 4.

State these values ofx.[4 marks]

Answer:

(d) x = ,

6. July 2006, Q12

(a) Complete Table 2 in the answer space for the equation 383 += xxy

by writing down the values ofy when x = -2 and 2 .

[2 marks ]

(b) For this part of the question, use the graph paper provided on page 35.You may use a flexible curve rule.


draw the graph of 383

+= xxy for -3.5 x 3.5 .



(ii) the value ofx when y = 9 .[2 marks ]



= xx for -3.5 x 3.5 .

State these values ofx.

[4 marks]Answer:

(a)

x -3 -2 -1 0 1 2 3 3.5 4

y -12 -3 3 0 -7 -25 -33

TABLE 1



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(b) Refer graph on page 33 .

(c) (i) y = .

(ii) x =

(d) x = ,

7. Nov 2006, Q13

(a) Complete Table 1 in the answer space for the equation y =x

24by writing down

the value of y when x = -3 and x = 1.5. [2 marks ]

(b) For this part of the question, use the graph paper provided on page 35. You mayuse a flexible curve rule.

By using a scale of 2cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis,

draw the graph of y =x

24for 44 x . [5 marks ]


i. the value of y when x = 2.9

ii. the value of x when y = -13.

[ 2 marks ]

(d) Draw a suitable line on your graph to find a value of x which satisfies the

equation 2x2

+ 5x = 24 for44

x .State this value of x.

[ 3 marks ]

Answer :

(a)

x -4 -3 -2 -1 1 1.5 2 3 4

y -6 -12 -24 24 12 8 6

TABLE 1

(b) Refer graph on page 35(c) (i) y = ..

(ii) x = ..

(d) x = ..



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8 Jun 2007 Q 16

(a) Complete Table 3 in the answer space for the equation 132 2 += xxy

by writing down the values ofy when x = 3 and x = 1 .


By using a scale of 2 cm to 1 unit on thex-axis and 2 cm to 5 units

on the y-axis, draw the graph of 132 2 += xxy for 5.34 x .




(d) Draw a suitable straight line on your graph to find the values ofxwhich satisfiy the equation 072 2 =+ xx for 5.34 x .

State these values of x .

Answer:

(a)

x 4 3 2 1 0 1 2 2.6 3.5

y 19 1 2 1 13 20.3 34

(b) Refer graph on page 37.

(c) (i) y = .

(ii) x = .

(d) x = . , .

9 November 2007 Q 12


[2 marks]

[4 marks]

[2 marks]

[4 marks]


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(a) Complete Table 1 in the answer space for the equation 36 xy =


(b) For this part of the question, use the graph paper provided on page 23.



on the y-axis, draw the graph of 36 xy = for 5.23 x .




(d) Draw a suitable straight line on your graph to find the values ofx

which satisfiy the equation

3

6 xy=

for5.23 x

.State these values of x .

Answer:

(a)

x 3 2.5 2 1 0 1 2 2.5

y 33 21.63

14 6 5


(c) (i) y = .

(ii) x = .

(d) x = . , .

10 Jun 2008 Q 12


[2 marks]

[4 marks]

[2 marks]

[4 marks]


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(a) Complete Table 12 in the answer space for the quation 343

+= xxy


(b) For this part of the question, use the graph paper provided on page 25



on the y-axis, draw the graph of 343 += xxy for 45.3 x .

(c) From your graph in 12(b), find


(ii) the value ofx when y = 25 ,

(d) Draw a suitable straight line on the graph in 12(b) to find the values ofx

which satisfiy the equation 01716

3=

xx for.45.3 x


Answer:

(a)

x 3.5 3 2 1 0 1 2 3 4

y 25.9 12 3 3 0 18 51


(c) (i) y = .

(ii) x = .

(d) x = . , .

11 Nov 2008 Q 12


[2 marks]

[4 marks]

[2 marks]

[4 marks]


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(a) Complete Table 12 in the answer space for the quationx

y36

=



Using a scale of 1 cm to 1 unit on thex-axis and 1 cm to 1 unit

on the y-axis, draw the graph ofx

y36

= for 142 x .

(c) From your graph in 12(b), find


(ii) the value ofx when y = 5 ,

(d) Draw a suitable straight line on the graph in 12(b) to find the values ofx

which satisfiy the equation 01436

=+ xx

for .142 x


Answer:

(a)

x 2 2.4 3 4 6 8 10 12 14

y 18 15 12 9 4.5 3 2.6


(c) (i) y = .

(ii) x = .

(d) x = . , .


[2 marks]

[4 marks]

[2 marks]

[4 marks]


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ANSWERS

Chapter 19 : Graphs of Fumctions Paper 2



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1 SPM 2003(N)

(a) 4 , 2

(b) Graph

(c) (i) y = 2.2 (ii) x = 1.2

(d) 32 = xy , x = 2.3 , 0.85

2 SPM 2004(J)

a) k = 2 , m = 8

(b) Graph

(c) (i) y = 6.8 (ii) x = 2.9

(d) 122 += xy , x = 2.25 , 0.32 , 2.6

3 SPM 2004(N)

(a) k = 13 , m = 3

(b) Graph

(c) (i) y = 7 (ii) x = 2.6 , 0.5

(d) 205 += xy , 3.1 x 3.2

4 SPM 2005(J)

(a) 2 , 18

(b) Graph

(c) (i) y = 4.7 (ii) x = 3.55

(d) 72

3=

xy , x = 2.85 , 3.1

5 SPM 2005(N)

(a) 0 , 25



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(b) Graph

(c) (i) y = 19 (ii) x = 4.7

(d) 72 += xy , x = 1.6 , 3.1

6 SPM 2006(J)

(a) y = 11 , y = 5

(b) Graph

(c) (i) y = 8 (ii) x = 3.4

(d) 55 = xy , x = 0.64 , x = 3.24 ,

7 SPM 2006(N)

(a) y = 8 , y = 16

(b) Graph

(c) (i) y = 8 (ii) x = 1.85

(d) 52 += xy , x = 2.45

8 SPM 2007(J)

(a) y = 8 , y = 4

(b) Graph

(c) (i) y = 14 (ii) x = 2.85

(d) 62 += xy , x = 2.14 , x = 1.64 ,

9 SPM 2007(N)

(a) y = 7 , y = 2

(b) Graph



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(c) (i) y = 2.5 (ii) x = 1.65

(d) xy 8= , x = 0.8 , x = 2.4 ,

10 SPM 2008(J)

(a) y = 7 , y = 2

(b) Graph

(c) (i) y = 2.5 (ii) x = 1.65

(d) xy 8= , x = 0.8 , x = 2.4 ,

11 SPM 2008(N)

(a) y = 6 , y = 3

(b) Graph

(c) (i) y = 5.5 (ii) x = 3.75

(d) 2012 += xy , x = 3.3

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Chapter 11 II Graphs of Functions ENHANCE