Chapter 2 - Graphs of Functions II

7/30/2019 Chapter 2 - Graphs of Functions II

1/18

10

10

1

2

3

3

2

1

x2 4

4

5

y = 2x 3

3

y

1. Draw the graphs of the following linear functions.

1

2

0

4

6

x

2 4

8

y =x + 4

3

y

x

y = 3x 6

10

2

4

6

4

22 3

6

y

x

y = 5 2x

22 0

2

4

4

24

6

8

10

y

Draw the graph ofy = 2x 3 for 0x 4.

Step 1: Construct a table displaying the values ofx andy.

y = 2x 3

Step 2: Plot the points. Using a ruler, join the points to form a straight line.

Note:

For a straight line, only

2 points are needed.

(b) y =3x 6

x 0 3

y3 3

(a) y = x + 4 x 0 4

y 4 8

(c) y =5 2x x 2 4

y 9 3

x

y = 2x x2

10

1

3

2

12 3

y

Draw the graph of the following functions.

(a) y = 2x x2

(a) Step 1: Construct a table displaying the values

ofx andy.

Step 2: Plot the points. Join the points by a

smooth curve.

x 0 1 2 3y 0 1 0 3

(b) y = 3x x 3

(b) Step 1: Construct a table displaying the valuesofx andy.

Step 2: Plot the points. Join the points by a

smooth curve.

x 2 1 0 1 2y 2 2 0 2 2

x

y = 3x x3

112 0

1

2

2

12

y

x 0 3

y 6 3

Graphs of Functions II2CHAPTER

Understand and use the concept of graph of functions2.1

Example

Example

Chapter2.indd 10Chapter2.indd 10 7/23/08 7:30:23 AM7/23/08 7:30:23 AM


2/18

11

2. Draw the graphs of the following functions.

(a) y = 3x2 + 3x + 1 (b) y = x2 5x

(c) y =1

x3 2

(d) y =2x3 + 1

(e) y =2

x(f) y =

4

3x

x 1 01

1 2

2y 5 1 1.75 1 5

x 2 1 0 1 2y 4 0.5 0 0.5 4

x 1 2 3 4 5

y 2 1 0.67 0.5 0.4

x 1 0 1 11

22

y 3 1 1 5.75 15

x 0 1 2 21

3 4 5 6

2y 0 4 6 6.25 6 4 0 6

x

y = 3x2 + 3x + 1

11 0

1

3

2

5

6

4

12

y

x

y =x2 5x

210

2

4

6

6

4

24 5 63

y

x

112 0

4

1

2

3

4

1

2

3

2

y = x31

2y

x

21 0

2

4

6

8

2

10

12

14

16

1

y = 2x3 + 1

y

x

10

0.5

1

1.5

2

2 3 4 5

y =2

x

y

x

1

2

3

1234 0

y = 4

3x

y

x 4 3 2 1 1

2

y

0.33 0.4 4 0.67 1.33 2.67



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12

(b) (i) the value ofy whenx = 5.

(ii) the value ofx wheny = 2.6.

(c) (i) the value ofy whenx = 3.5.

(ii) the value ofx wheny = 4.

(d) (i) the value ofy whenx = 1.


3. From the following graphs, find

(a) (i) the value ofy whenx = 1.


Find the values of (a) y whenx = 2 and (b) x wheny = 2 for the following graphs.

From the graph,

(a) whenx = 2,y = 3.5.

(b) wheny = 2,x = 2.4.

From the graph,

(a) whenx = 2,y = 3.5.

(b) wheny = 2,x = 0.23 or 4.23.

x

y

12 0

4

2

2

4

6

8

6

10

12

2 31

y = x3 21

2

x

y

y= 2x + 5

112 0

4

2

2

4

8

6

10

2 3 4

x

y

14 0

4

2

2

4

8

6

10

12

14

16

42 13 2 3

y = x2 + 2x 11

2

x

y

10

2

1

4

5

6

3

4 5 62 3

y= + 22

x

x

4y =5x + 4

224 0

2

2

4

4

6

4

y

x212 145 3 0

4

2

4

2

8

10

6

12

14

43

y

y = (x + 5)(x 1)1

2

(a) (b)

(ii) x = 1.5 (i) y = 2.4 (ii) x = 3.4

(i) y = 12.1 (ii) x = 1.7 (i) y = 2.5 (ii) x = 2

(i) y = 7

Example



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13

x

y

0

5

5

y = x + 5

x

y

0

y = 3x 2

2

32 x

y

0

2y x = 31

21

3

4. Identify the type of function for the following graphs.

x x x

y y y

0

x

y

0

00

Identify the type of function for the following graphs.

(b) (c)(a) (d)

5. Sketch the graphs of the following linear functions.

x = 0,y = 0 + 5 = 5

y = 0,x = 5

x = 0,y = 3(0) 2 = 2

y = 0, 3x = 2

x =23

x

y

0

8

4

y = 2x 8

x

y

0

5

5

y = x + 5

Sketch the graphs of the following linear functions.

x

y

0

x

y

0

x

y

0

x

y

0

x

y

0

x

y

0x

y

0

y

0

(a) (b) (c) (d)

(e) (f) (g) (h)

Reciprocal function Cubic function Cubic functionQuadratic function

Linear function Quadratic function Reciprocal function Cubic function

(b) Linear function (c) Quadratic function(a) Reciprocal function (d) Cubic function

(c) 2y x = 3

x = 0, 2y = 3

y = 1

1

2

y = 0, x = 3

x = 3

(b) y = 3x 2

Example

Example

(a) y = 2x 8 m 0

The gradient is positive and they-intercept is 8.

x = 0,y = 2(0) 8

= 8

y = 0, 2x = 8

x = 4

(b) y = x + 5 m 0

The gradient is negative and they-intercept is 5.

x = 0,y = 0 + 5

= 5

y = 0,x = 5

(a) y = x + 5



5/18

14

x

y

6

2 30

y = (x + 2)(x 3)

y = x2 + 4

x

y

0 22

4

Sketch the graphs of the following quadratic functions.

x

y

27

30

x

y

x

y

9

300

4y = 9 x3

1

3

1

2

y = 32x3 + 4

y =x3 27

7. Sketch the graphs of the following cubic functions.

(a) y =x3 27 (b) y = 9 1 x3 3(c) y = 32x3 + 4

x = 0,y = 32(0) + 4 = 4

y = 0, 32x3 = 4

x3 =

18

x =

12

x

y

20

y = 2x2 4x

x

y

1

2

20

y = (x + 1)(x 2)

x

y

2

10

0

y = (5 x )(x + 2)

5

6. Sketch the graphs of the following quadratic functions.

x = 0,y = (1)(2) = 2

y = 0, (x + 1)(x 2) = 0x = 1, 2

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

x = 0,y = (5)(2) = 10

y = 0, (5 x)(x + 2) = 0 x = 5, 2

x = 0,y = 2(0) 4(0) = 0

y = 0, 0 = 2x(x 2) x = 0 orx = 2

(a) y = 2x2 4x (b) y = (x + 1)(x 2)

y = 3x3

x

y

0 x

y

2

4

0

y = x3 + 41

2

Sketch the graphs of the following cubic functions.

(b) y = 1

x 3 + 4 a 02

The function has a shape and the y-intercept is 4.

x = 0,y = 1

(0) + 42

= 4

y = 0,

1x3 = 4

2

x3 = 8

x = 2

(a) y = 3x3 a 0

The function has a shape and the

y-intercept is 0.

x = 0,y = 0

x = 0,y = 0 27 = 27

y = 0,x3 = 27

x = 3

13

x = 0,y = 9 (0) = 9

y = 0,1

x3 = 9

3x3 = 27

x = 3

Example

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

y =x2

x 6 a

0The function has a shape and the y-intercept

is 6.

x = 0,y = 0 0 6 = 6

y = 0, (x + 2)(x 3) = 0

x = 2 or 3

(b) y = x2 + 4 a 0

The function has a shape and the y-interceptis 4.

x = 0,y = 4

y = 0,x2 = 4

x = 2 or 2

Example



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15

x

y

0x

y

0x

y

0

y = 5

x

y =6

x

y = 4

x

x

y

0

y =2

x

x

y

0

y = 2

x

Sketch the graphs of the following reciprocal functions.

(b) y = 2 a 0

x(a) y =

2

a

0 x

8. Sketch the graphs of the following reciprocal functions.

(b) y = 6

x

(a) y = 5

x(c) y =

4

x

1. Find the points of intersection of the given graphs.

(b)(a)

x

21123 0

1

1

2

2

3

4

5

6

4 53

y

y = x2 +x + 41

2

y =2

x

x

211234 0

1

2

3

4

4

3

1

2

5

6

43

y

y = x3 4x1

2

y = x + 13

2

From the graph, the points of intersection are

(3.4, 6.1), ( 0.2, 0.8) and (3.2, 3.8).

From the graph, the points of intersection are

(3.8, 0.5), (0.4, 4.4) and (2.3, 0.8).

Find thex-coordinates of the points of intersection of these two graphs:y = 2x2 3x 5 andy = 2x + 1.

x

2112 0

2

2

4

6

4

6

8

10

4 53

y

y = 2x + 1

y = 2x2 3x 5

From the graph,x = 3.4 andx = 0.8.

Example

Example

Understand and use the concept of the solution of an equation bygraphical method

2.2



7/18

16

Solve the equationx2 +x + 1 = 1

x + 5 graphically.2

Step 1: Draw the graphs ofy =x2

+x + 1 and

y = 1

x + 5.2

From the graph,

x = 1.4 and 2.9.

Step 2: The solution will be the x-coordinates of thepoints of intersection.

x

21123 0

1

2

3

4

5

6

7

3

y

y = x + 5

y =x2+x +1

1

2

2. Solve the following equations by graphical method.

(c) 2x2 5x 3 = 2

x

(d) x3 6x 2 = 4 2x

x

2 3112345 0

1

2

3

4

5

3

1

2

y

y = x2 2x + 5

1

2y = x + 3

x

2 3 411234 0

1

2

3

4

4

3

1

2

y

y =x 2

4

xy =

x

2 3 4112 0

1

2

3

4

5

6

7

3

1

2

y

y = 2x2 5x 3

2

xy =

x

2 3112 0

1

2

3

4

5

4

5

6

7

8

3

1

2

y

y = 4 2x

y =x3 6x 2

From the graph,

x = 3.1 and 0.7.

From the graph,

x = 3.2 and 1.2.

From the graph,

x = 2.9, 0.4 and 0.8.

From the graph,

x = 2.5.

(a)1

x + 3 = x2 2x + 52

(b) 4

=x 2 x

Example



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17

x

211234 0

2

24

6

8

4

6

8

y

y = 4 + 2x

y = 2x2+ 5x 4

(c) x2 4x = 0.5x 1

x2 4x + 2 = 0.5x 1 + 2

x2 4x + 2 = 0.5x + 1

From the graph,x = 0.27 and 3.7

(c) x3 4x + 1 = 0

x3 + 4x 1 = 0

x3 + 3x = 1 x

From the graph,x = 1.8, 0.25 and 2.1.

x

y

2145 123 0

1

1

2

3

2

3

4

5

6

y = x+ 1

y = x2 4x + 2

1

2

x

y

y = 1 x

y = x3 + 3x

1 2123 0

2

2

4

6

8

(a) Complete the table below for the equationy = 2x2 + 5x 4.

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

y = 2x2 + 5x 4 for 3.5x 1.5.

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

2x2 + 3x = 8 for 3.5 x 1.5.

(a) (b)

(c) 2x2 + 3x = 8

2x2 + 3x + 2x = 8 + 2x

2x2 + 5x = 8 + 2x

2x2+ 5x 4 = 8 + 2x 42x2 + 5x 4 = 4 + 2x

From the graph,x = 1.4 and 2.9.

x 3.5 3 2 1.5 1 0 1 1.5

y 3 1 6 7 3 8

x 3.5 3 2 1.5 1 0 1 1.5

y 3 1 6 7 7 4 3 8

Tip:

Draw the graph

ofy = 4 + 2x.

Tip:

Change the left-hand

side of the equation to

2x2 + 5x 4.

3. (a) Complete the table below for the equation

y = x2 4x + 2.

(b) By using a scale of 1 cm to 1 unit on bothaxes, draw the graph ofy = x2 4x + 2 for

5x 1.

(c) Draw a suitable straight line on your graph

to solve the equation x2 4x = 0.5x 1.

(b)


y = x3 + 3x.

(b) 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 ofy = x3 + 3x for 2.5x 2.

(c) By drawing a suitable straight line, solve the

equationx3 4x + 1 = 0.

(b)

x 5 4 3 2 1 0 1

y 3 2 5 6 5 2 3

x 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2

y 8.1 2 1.1 2 1.4 0 1.4 2 1.1 2

Example



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18

x

y

y = 5

212 13467 5 0

2

3

1

2

1

3

54

6

7y = x2 + x1

2

5

2

5. (a) Show that the volume of the cuboid is (5x2 5x) cm3.

(b) Given thaty = 5x2 5x, complete the table below.

(c) 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 of

y = 5x2 5x for 0x 2.0.

(d) Given that the volume of the cuboid is 8 cm3.

Find the value ofx.

(a) Volume of cuboid = 5x (x 1)

= (5x2 5x) cm3

(d) If the volume of the cuboid is 8 cm3, theny = 8.

From the graph,x = 1.85.

x cm5 cm

(x 1) cm

(c)

x

y

y = 8

y = 5x2 5x

1 2 30

1

1

2

3

4

5

6

78

9

10

x 7 6 5 4 3 2 1 0 1 2

y 7 3 0 3 2 0 3 7

x 0 0.3 0.5 0.8 1.2 1.5 1.8 2.0

y 0 1.1 1.3 0.8 1.2 3.8 7.2 10

x 7 6 5 4 3 2 1 0 1 2y 7 3 0 2 3 3 2 0 3 7

Example

(a) Show that the area of the trapezium is ( 1 x2 + 5 x) cm2. 2 2(b) Given that y =

1x2

+

5x, complete the table below. 2 2

(c) By using a scale of 1 cm to 1 unit on both axes, draw the graph ofy =1

x2 +5

x for 7x 2. 2 2

(d) If the area of the trapezium is 5 cm2, find the values ofx.

(a) Area of trapezium =1

(x + 5)x (c)2

= ( 1 x2 + 5 x ) cm2 2 2

(b)

(d) If the area of the trapezium is 5 cm2, theny = 5.

From the graph,x = 1.5 and 6.5.

5 cm

x cm

x cm



10/18

19

1. Determine whether the following points satisfyy = 3x + 7,y 3x + 7 ory 3x + 7.

y = 2

3x + 7 = 3(1) + 7= 10

Since 2 10, therefore

(1, 2) satisfiesy 3x + 7.

2. Determine whether the following points satisfyy =

1x 5,y

1x 5 ory

1x 5.

2 2 2

Identify the inequality representing the following region.

x

y y = x + 6

6

6 0

Step 1: Choose a convenient point, say (0, 0), from the shaded region.

Step 2:y = 0

x + 6 = 0 + 6 = 6

Since 0 6, therefore (0, 0) satisfiesyx + 6.Step 3: Hence, the inequality representing the region isy x + 6.

x

y

y =3x

0

3. Identify the inequality representing the following regions.

(a) (b) (c)

y 3x y 2 5y 8x + 40

x

y

0

8

5y = 8x + 40

5

x

y

0

2

(a) (20, 5) (b) (1, 2) (c) (2, 1)

(a) (2, 0) (b) ( 6, 8) (c) (6, 5)

y = 5

1x 5 =

1(6) 5

2 2= 2

Since 5 2,

(6, 5) satisfiesy1x 5.

2

y = 5

3x + 7 = 3(20) + 7= 53



y = 8

1x 5 =

1( 6) 5

2 2= 8

Since 8 = 8,

( 6, 8) satisfiesy =1x 5.

2

y = 0

1x 5 =

1(2) 5

2 2= 6

Since 0 6,

(2, 0) satisfiesy1x 5.

2

Understand and use the concept of the region representing inequalities intwo variables

2.3

Example

Determine whether the following points satisfyy = 2x + 3, y 2x + 3 ory 2x + 3.

(a) (1, 5) (b) (2, 3) (c) (1, 6)

(a) y = 5

2x + 3 = 2(1) + 3 = 5

Since 5 = 5, therefore

(1, 5) satisfiesy = 2x + 3.

(b) y = 3

2x + 3 = 2(2) + 3 = 7



(c) y = 6

2x + 3 = 2(1) + 3 = 1



Example

y = 1

3x + 7 = 3(2) + 7= 1

Since 1 = 1, therefore

(2, 1) satisfiesy = 3x + 7.



11/18

20

x

yy = 2x 5

2

5

0 1

2

y = 2x 5

2

5

12

x

y

y = x + 3

3

3

0

x

y

y = x + 33

3

0

4. Shade the region satisfying the following inequalities.

Shade the region satisfyingy 3x 3.

Step 1: Choose a convenient point, say (0, 0).

Step 2: y = 03x 3 = 3(0) 3 = 3

(0, 0) satisfiesy 3x 3.

Step 3: (0, 0) lies above the line. So any point above

the line satisfiesy 3x 3.

x

y y =3x 3

3

0 1Note:

Ify 3x 3, then the line

will be a solid line.

x

y y =x + 3

3

3

0

x

y

y = 5x 5

1

5

0

x

y y = 2x + 1

1

01

2

x

y

y = 5 3x

5

0

12

3

5. Shade the region representing the following inequalities.

(b) y 6x 1,y 1

x + 2 (c) y1 + 4x,y x + 52

x

y

43

1

2

0

1

4y = x 1

2

3y = x + 2

x

y

y = 6x 1

41

61

2

0

y = x + 212

x

y y = 1 + 4x

5

5

1

0

y = x + 5

1

4

(d) y 1

x + 3,y 3x 4 (e) y4x 2,y 2x + 2 (f) y 3x 7,y 5 x

2

Step 4:y 3x 3 is the region below the line.

Shade the region representingy 2x 5 andyx + 3.

Step 1: Identify the regions y 2x 5 and y x + 3

separately.

Step 2: Combine the two graphs and obtain

the common region between them.

x

yy = 3x 4

6

1

24

3

0

y = x + 3

1

31

x

yy = 4x 2

1

2

2

0

y = 2x + 2

1

2

x

y

y = 5 x

5

7

5

0

y = 3x 7

21

3

(a) y x + 3 (b) y5x 5 (c) y 2x + 1 (d) y 5 3x

(a) y 2

x + 2,y 1

x 1

3 4

Example

Example



12/18

21

Shade the region which satisfies the following inequalities: y 2,y 2x + 1,y 3x 3.

6. Shade the region which satisfies the following inequalities.

(a) x0,y 2,y 2x + 4 (b) x 2,y 0,y x 5

y = 3x 3

32

1

2

x

y

y = 2x + 1

y = 2

1

1

0

x

yy = 2x + 4

y = 2

4

2

02

x

y

y =x 5

2

x = 2

50

5

x

y

x = 2y = 2x 3

y = 3x + 4

2

4

3

0 11

21

1

3x

y

y = 3x + 3

y = 33

3

61 0

y = x + 31

2

x

y

y =xy = x

8 y = 8

0

x

y y = 2x

y = 6

x = 1

6

01

(c) x 2,y 3x + 4,y 2x 3 (d) y 3,y 1

x + 3,y 3x + 32

(e) y8,y x,y x (f) x 1,y 6,y2x

Example



13/18

22

x

y

0

2

x

y

0

x

y

0

55

2

x

y

0

x

y

0

x

y

0

x

y

0

x

y

0

x

y

0

33

3

3 x

y

0

x

y

0

x

y

0

1. Which of the following represents the graph of

y = 2x3 5?

A C

B D


y =x2 + 3?

A C

B D


y = 3 ?__

x

A C

B D

4. The equationx3 5x =x2 3 can be solved

by drawing 2 graphs. Which of the following

equations represent the 2 suitable graphs?

A y =x3 5x andy = 3 x2

B y =x3 5x andy =x2 3

C y = 5xx3 andy =x2 3

D y =x3 5x andy =x2 + 3

5. Which of the following statements is true?

A A linear function has an equation in the form

ofy = ax2 + b.

B They-intercept of a linear equation is 5 if the

equation isy = 8x 5.

C A linear function is a straight line graph.

D A linear function has a maximum point.

6. Table 1 shows the values ofx andy for the

equation 2x =y + 6.

TABLE 1

Find the value ofa.

A 4 C 0

B 2 D 2

7. Diagram 1 shows a graph.

DIAGRAM 1

Which region satisfies the inequalitiesx 10

andy 10?

A I C III

B II D IV

8. Diagram 2 shows the graph ofy = 8 x3.

DIAGRAM 2

State the value ofk.

A 8 C 2

B 2 D 8

SPM

2005

SPM

2004

x

y

I IV

II

10 0

10

III

x

y

(0, k)y = 8 x3

0

SPM

2002

SPM

2006

x 3 4 5 6

y a 2 4 6

Paper 1

SPM Practice2



14/18

23

x

y

0 1

3

x

y

0 1

3

x

y

0

3

x

y

01

3

1

y

0

9

y

0 9

y

0 9

9

y

0

x

y = 2 xy =x + 4y

IIII

IV

II

4

4

0 2

2

x

y

0x

y

0

x

y

0x

y

0

9. Which of the following graphs represents the

inequalityy 3x 3?

A

B

C

D

10. Which of the following graphs represents

y = 9 x2

?A

B

C

D

11. Which of the following statements is false?

A A quadratic function is of the form

y = ax2 + bx + c.

B A quadratic function has a parabolic shape.

C A quadratic function has 2 lines of symmetry.

D A quadratic function has a maximum or

minimum point.

12. Diagram 3 shows a graph.

DIAGRAM 3

Which region satisfies the inequalitiesyx + 4

andy 2 x?

A I C III

B II D IV

13. Which of the following graphs represents

y = ax n + b, when n = 2?

A C

B D

14. Diagram 4 shows the graph ofy = (x + 3)(x 1).

DIAGRAM 4

Find the coordinates ofP.

A (3 , 0)

B (1 , 0)

C (0 , 3)

D (0 , 1)

x

y

P0

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15. Which of the following graphs satisfies the

inequalitiesy 4 andyx 4?

A

B

C

D

16. Diagram 5 shows the graph ofy =xn

4.

DIAGRAM 5

Find the values ofn and k.

A n = 3 , k= 4 C n = 1 , k= 4

B n = 2 , k= 4 D n = 3 , k= 4

Questions 17 and 18 are based on Diagram 6.

Diagram 6 shows the graph ofy =x3

+ 4.__2

DIAGRAM 6

17. Find the coordinates ofP.

A (4 , 0)

B (2 , 3)

C (2 , 0)

D (0 , 4)

18. Find the coordinates ofQ.

A (2 , 0)

B (0 , 8)

C (2 , 8)

D ( 4 , 0)

19. Diagram 7 shows the graph ofy = 5x 2.

DIAGRAM 7

From the graph, find the value ofy when

x = 0.6.

A 0.5 C 3 B 1 D 6

20. Diagram 8 shows the graph ofy =x2 4.

DIAGRAM 8

From the graph, find the values ofx wheny = 1.

A 3, 3 C 1.7, 1.7

B 2.3, 2.3 D 1.5, 1.5

x

y

y = xn 4

0

(0, k)

x

y

P

Q

0 2

y = + 4x3

2

x

2 31123 0

2

4

2

4

6

y

y =x2 4

x

2112 0

2

3

1

2

1

3

4

y

y = 5x 2

x

y

y = 4

y =x 4

0 4

4

4

x

y

y = 4

y =x 4

0 4

4

4

x

y

y = 4

y =x 4

0 4

4

4

x

y

y = 4

y =x 4

0 4

4

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Paper 2

4. Shade the region defined by the following

inequalities:y 3

x + 3 andy3

x + 3. 4 2

(a)

(b)

y = (x 1)(x + 4)

x

y

4

4

10

y = x3+ 1

x

y

0

1

1

y = 2x + 12

y = 12

x = 6

x

y

6

12

0

y = x + 3

x

y

4

3

02

3

4

y = x + 33

2

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2. Sketch the graph of(a) y = (x 1)(x + 4)

(b) y = x3 + 1

3. On the graph in the answer space, shade the

region which satisfies the three inequalitiesy 2x + 12,x 6 andy 12.

1.

On the graph, shade the region which satisfies

the three inequalities yx + 4,y 1

x 1

2

andx 1.

y =x + 4

x

y

224 4

2

4

0

2y = x 11

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5. Shade the region defined by the following

inequalities:x 0,y 0,y 1

x + 4 and2

y

1

x 3.__2

6. Diagram 1 shows the graph ofy =x3 4x.

DIAGRAM 1

Find the coordinates ofA andB.

x3

4x = 0 x(x2 4) = 0

x(x + 2)(x 2) = 0

x = 0, 2, 2

Coordinates ofA = (2 , 0)

Coordinates ofB = (2 , 0)


y =x3 + 5 by writing down the values ofy

whenx = 1 andx = 2.

(b) By using the 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 y = x3 + 5 for

3x 2.5.

(c) From your graph, find

(i) the valueofy whenx = 1.5,


(d) Draw a suitable straight line on your graph

to find the values ofx which satisfy the

equationx3

8x 4 = 0 for 3

x

2.5.State these values ofx.

x

y

86

4

0

3

y = x + 41

2

y = x 31

2

x

y

y =x3 4x

BA 0

(a)

(b)

x 3 2.5 2 1 0 1 2 2.5

y 22 10.625 3 5 6 20.625

x = 1,y = 4;x = 2,y = 13

(c) (i) 1.5

(ii) 2.45

(d) 0.5, 2.6

x

y

y = 8x + 9

y =x3 + 5

1123 2 30

5

10

15

20

25

5

15

10

20

25

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x 2 1 0 1 2 3 4 5

y 31 14 3 2 1 6 19 38

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(c) (i) y = 22(ii) x = 4.9

(d) 3x2 8x + 3 = x + 1

Draw the graph ofy = x + 1.

From the graph,x = 0.4 and 2.

x

y

y = x + 1

y = 3x2 8x + 3

2112 43 5 60

10

10

20

30

40

(b)

x

y

y =x + 6

2112345 43 50

5

10

15

5

10

15

y =16

x

(b)

9. (a) Complete Table 2 for the equation y = 16__

x

by writing down the values ofy when

x = 3 andx = 1.5.

TABLE 2

(b) By using a scale of 2 cm to 2 units on the

x-axis and 2 cm to 5 units on the y-axis,

draw the graph ofy = 16for 4x 4.__

x

(c) From your graph, find

(i) the value ofy whenx = 2.5,


(d) Draw a suitable straight line on your graph

to find a value ofx which satisfies theequationx2 + 6x = 16 for 4x 4.

x 4 3 2 1 1.5 2 3 4

y 4 5.3 8 16 10.7 8 5.3 4

(c) (i) y = 6.4(ii) x = 2

(d) x2 + 6x = 16

x(x + 6) = 16

x + 6 =

16

x

Draw the graph ofy =x + 6.

From the graph,x = 2.

8. (a) Complete Table 1 for the equation

y = 3x2 8x + 3.

TABLE 1

(b) By using a scale of 2 cm to 2 units on the

x-axis and 2 cm to 10 units on they-axis,

draw the graph ofy = 3x2 8x + 3 for

2x 5.

(c) From the graph, find

(i) the value ofy whenx = 4.2,


(d) Draw a suitable line on your graph to

find all the values of x which satisfy

the equation 3x2 8x + 3 = x + 1 for

2x 5.


Chapter 2 - Graphs of Functions II

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Transcript of Chapter 2 - Graphs of Functions II