Chapter 19 I Graphs of Functions II ENHANCE

Graphs of Functions II 1

CHAPTER 19. GRAPHS OF FUNCTIONS II

Important Notes:

For Paper 1 (Objective Questions), the questions asked are usually related to the shapes and

positions of basic graphs of LINEAR, QUADRATIC, CUBIC and RECIPROCAL

FUNCTIONS.

Students are expected to be able to identify:

a) the shape of graph given a type of function

b) the type of function given a graph

c) the graph given a function and vice versa.

The different types of functions for Paper 1, SPM Mathematics:

Linear

y = mx + c

m = gradient

c = y-intercept

Quadratic

y = ax2 + bx + c

Note : Paper 1 limits to

y = ax2 + c

Cubic

y = ax3 +bx

2 +cx + d

Note : Paper 1 limits to

y = ax3 + c

Reciprocal

y = a

x

Note : This module concentrates on graph sketching, the skills needed for Paper 1. For details on how

to plot graphs please refer to the other related modules (eg. Chapter 12 – The Straight Line – on how to

draw / plot straight lines)

19.0 Back to Basic

I. Linear Graphs : can be represented by the equation y = mx + c.

m is the gradient

c the y-intercept, ie, the place where the straight line cuts the y-axis.

II. The general form of a Quadratic Function is f(x) = ax2 + bx + c; a, b, c are constants and a ≠ 0.

Characteristics of a quadratic function:

Involves one variable only,

The highest power of the variable is 2.

III. The general form of a Cubic Function is f(x) = ax3 + bx

2 + cx + d ; a, b, c and d are constants and

a ≠ 0.

Characteristics of a cubic function:

Involves one variable only,

The highest power of the variable is 3.

IV. The simple Reciprocal Function is of the form y = a

x, where a is a constant.

O x

y

x x

y

O

y

O x

y

O x


19.1 To Identify The Type Of Function Given Its Equation

Example 1

No. Linear

y = mx + c

Quadratic

y = ax2 + bx + c

Cubic

y = ax3 + bx

2 + cx + d

Reciprocal

y = a

x

1. y = x y = x2 + 2x -3 y = x

3 + 2x

2 + 3x – 4 y =

3

x

2. y = 2x + 3 y = x2 + 2 y = x

3 – 3 y =

1

x

3. y = 3x – 4 y = 4 – 3x2

y = 4 – 3x3

xy = 4

4. y = 5 y = 3x2 y = 2x

3 y =

1

2x

5. x+ y = 4 y = 3 – 4x + 5x2 y = 1 – 2x + 5x

3 y =

6

x

Exercise 1

Below is a list of various functions. Try to identify the type of function and state whether it is a linear,

quafratic , cubic or reciprocal function. Hence complete the table given by writing the correct

functions in the respective columns.

1. y = x + 1 2. y = x3 – 2 3. y = x

2 + 2x – 4 4. y = -x

3

5. y = 2x2 + 3 6. y = – x 7. y = x

3 – 3 8. y = 1 + x

2

9. y = 2

x 10. y = 6 – 3x

2 11. y = 1

x

12. y = 20

13. y = 2 – x + 5x2 14. y =

3

x 15. y = 2x

3 16. y =

1

2x

17. x – 3 y = 4 18. x y = 3 19. y = – 4 + x3 20. y = 3x

–1

Linear

Quadratic

Cubic

Reciprocal


Exercise 2 : State whether the following are linear, quadratic, cubic or reciprocal functions.

No. Functions Type No. Functions Type

1. y = 2x – 3 Linear 1. y = x2 – 5

2. y = 3

x

Reciprocal 2. y = 10

x

3 y = –x3 Cubic 3 y = 10 – x

4. y = x2 + x Quadratic 4. y = x

2 + 5

5. y = 1

x 5. y = 2 + 2x

6. y = (x – 1)2

6. y =

x

6

7. y = x ( x – 2)

7. x + 2y = 1

8. y = – 3 + x3

8. y = – 3

9. y = – x2 + 2

9. y = 4 – x

2

10. y = 5 – 4 x2

10. y = 3 – 6 x

11. y = 2 1

x

11. 31

2y x

12. y = (x+2) (x – 3)

12. xy = – 3

19.2. To sketch graphs of linear functions : y = mx + c Remember : m = gradient , c = y-intercept

EXAMPLE FOR YOU TO TRY...

1. y = x

y = x + 3

y = x - 2

(a) y = x + 5

(b) y = x – 1

(c) y = x + 2

y

O x

- 3

3

O x

y

-2

2

y = x + 3

y = x

y = x – 2


Example For You To Try...

1. y = – x

y = – x + 3

y = – x – 2

(a) y = – x + 2

(b) y = – x – 1

(c) y = – x + 4

Example For You To Try... For You To Try...

1. y = x + 2

(a) y = x + 3 (b) y = x – 1

2. y = – x + 1

(a) y = – x + 3 (b) y = – x

(c) y = x – 2

(d) y = –x + 5 (e) x + y = 4

y

O x - 2

2

O x O x

y y

y

O x 1

1

O x

O x

y

y

O x

y y

O x

3

3

-2

-2

y = – x +

3

y = –x

y = x – 2

O x

y

O x

y

O x

y


19.3 To sketch graphs of quadratic functions of the form y = ax2 + c

Case 1 : y = ax2 + c , a > 0

No. Function (Sketched) Graph Your turn to try !

1. y = x2

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

2 – 3

2. y = x2 + 3

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

2 – 1

3 y = 2x2

(a) y = 2x2 – 1 (b) y = 2x

2 + 3

4 y = 1 + 2x2

(a) y = 2 + 2x2 (b) y = – 3+2x

2

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

3

x O

y

x O

y

1


More challenges for you ….

Sketch the given quadratic functions :

No. Quadratic Function Quadratic Function Quadratic Function

1.

y = x2 – 4

y = x2 – 2

y = x

2 + 2

2.

y = 3 + x2

y = – 3 + x2 y = 1 + x

2

3.

y = x2 + 1

y = 2x2 + 1 y = 3 x

2 + 1

4.

y = x2 – 3

y = 2x2

– 3 y = 3 x2 – 3

x O

y

x O

y

-4

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

1

x O

y

x O

y

x O

y

-3


Case 2 : y = ax

2 + c , a < 0


1. y = – x2

(a) y = –x2 + 2 (b) y = – x

2 – 3

2. y = –x2 + 2

(a) y = – x2 + 4 (b) y = –x

2 –

1

3 y = 21

2x

(a) y = – 2

1 x

2 – 1 (b) y =–

2

1x

2 +

3

4 y = 1 – 2x2

(a) y = 2 – 2x2 (b) y = – 3 – 2x

2

x O

y

x O

y

x O

y

x O

y

2

x O

y

x O

y

1

x O

y

x O

y



Sketch the given quadratic functions :

No. Quadratic Function Quadratic Function Quadratic Function

1.

y = – x2 + 4

y = – x2 – 2

y = – x2 + 2

2.

y = 3 – x2

y = – 3 – x2 y = 1 – x

2

3.

y = –x2 + 1

y = –2x2 + 1 y = – 3 x

2 + 1

4.

y = –x2 – 3

y = – 2x2

– 3 y = – 3 x2 – 3

x O

y

x O

y

4

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x

O

y

1

x O

y

x O

y

x

O

y

-3


19.4 To sketch graphs of cubic functions of the form y = ax3 + c


1. y = x3

(a) y = x3 + 2 (b) y = x

3 – 3

2. y = x3 + 3

(a) y = x3 + 4 (b) y = x

3 – 2

3 y = – x3

(a) y = –x3 + 2 (b) y = –x

3 – 3

4 y = 1 – x3

(a) y = 2 – x3 (b) y = – 3 – x

3

y

O x

y

O x

y

O x

y

O x

3

y

O x

y

O x

1

y

O x

y

O x


19.5 To sketch graphs of reciprocal functions of the form a

yx

.

No. Function Your turn to try !

1.

y = 1

x y =

2

x

2.

y = – 1

x y =

4

x

3

xy = 2 ( can be written as y = 2

x )

xy = 3

4

xy + 2 = 0 ( can be written as y = 2

x )

xy + 5 = 0

x

y

O

x

y

O

x

y

O

x

y

O

x

y

O x

y

O

x

y

O

x

y

O



EXERCISE A : Match each graph with the correct equation of function :

1.

2.

3.

O x

y

x

x

y

O

y

O

x

y

O

x

O x

y

x

y

O x

x

y

O

y

O x

y = x2 – 1 y = 2x

– 1 2y

x

y = x

3 – 1

y = x – 2 y = x

2 – 2 y = 2 – x

2 x + y = 2

3y

x

y = –2 + x

3 xy = –4 y = – x

3 +2

O x

y

x

y

O x

y

O x x

y

O


Exercise B : Sketch the given functions :

1.

y = x2 + 1

y = 2x + 1

y = x3 + 1

2.

y = x + 3

y = – 3 + x x + y = 4

3.

y = 3

x

xy = 2

xy = – 1

4.

y = x3 – 3

y = 3 + x3 y = – x

2 + 2

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y

x O

y


Questions Based On Examination Format

1. Which of the following graphs represents y = x2 – 2 ?

A. B.

C. D.

2. Which of the following equations represents the graph of a quadratic function?

A. y = 2 + x2 B. xy = 2 C. x – y = 2 D. y = 2x

–2

3. The diagram below shows a sketch of a function.

What is the equation of the function?

A. y = x2 – 5

B. y = – x2 – 5

C. y = x3 – 5

D. y = – x3 – 5

y

O

y

x O

y

x O

x

– 2 – 2

– 2

y

O – 2

x

y

x

O

– 5


4. Which of the following graphs represents 4

yx

?

A. B.

C. D.

5. Which of the following graphs represents ?5 3xy

A B

C D

x

y

O

y

x O

y

x O

x

y

O

y

x O

5

y

x O 5

y

x O

5

y

x

O -5


6

Diagram 1

Diagram 1 shows the graph of 16 nxy . State the value of n and k.

A. n = –1, k =16 B. n = 3, k =17 C. n = –3, k =16 D. n = 3, k =15

7

Diagram 2

Diagram 2 shows the graph of y = x3 + k. Find the value of h.

A. 2 B. 3 C. – 2 D. – 3

8. Which of the following equations represents a linear graph ?

A. y = 1 + x2 B. xy = 4 C. x – y = 2 D. y = 3x

–1

9. Which of the following equations represents the graph of a reciprocal function?

A. y = 1 + 2x B. y – x = 4 C. x y + 2 = 0 D. y = 3x3

10. Which of the following represents the graph of y = x3 – 1 ?

A

B C D

(–1, k)

x

y

0

16 nxy

.

O x

y

x

x

y

O

y

O

x

y

O

x

( h, 0)

x

y

0 .

8


11. Which of the following represents the graph of 10

yx

?

A

B C D

12.

The graph labelled G most likely represents

A. y = –2x2 B. y = 2x

2 – 2 C. x y + 2 = 0 D. y = 2x

3

13.


A. y = x – 2 B. y = – x – 2 C. y = – x + 2 D. y = - 2x

O x

y

x

x

y

O x

y

O

y

O

x

O

y

x

y = 2x2

G

O

y

x

y = x + 2

G


14.


A. y = 2

1 x B. y = – x C. y = x + 2 D. y = 2x

15. Which of the following represents the graph of 31y x ?

A

B C D

16. Which of the following represents the graph of 22y x ?

A

B C D

17.

Diagram 3

Diagram 3 shows the graph of y = - 4xn. Find the value of n

A. 2 B. 3 C. – 1 D. – 2

O

y

x

y = x

G

y

O

x

y

O

x x

y

O

y

O

x

x O

y

O x

y

O x

y

O x

y

x

y

O


18.

The above graph represents

A. y = - x + 2 C. y = 2 – x 2

B. x

y6

D.

xy

6

19.

The above graph represents

A. y = – x2 + 16 C. y = (x + 4)

2

B. y = x2 + 16 D. y = 16 – (x+4)

2

20. Which of the following does NOT represent a linear graph ?

A. y = x B. x = 2y C. xy = 2 D. y = – 2

x

y

O

x

y

O

16

4 -4


Past-Year Questions on Graphs of Functions II

1 Nov 2003, Q12

Which of the following graphs represents y = x2 – 3 ?

A. B.

C D.

2 Nov 2004, Q29

Which of the following graphs represents 2

yx

?

A. B.

C. D.

x

y

O

y

x O

y

x O

y

x O


3 July 2005, Q28

Diagram 12 shows a sketch of a function.

What is the equation of the function?

A. y = x2 – 3

B. y = – x2 – 3

C. y = x3 – 3

D. y = – x3 – 3

4 Nov 2005, Q28

Which of the following graphs represents y = x2 – 4 ?

A. B.

C. D.

y

x

O

– 3

y

O

y

x O

y

x O

x

– 4 – 4

– 4

y

O – 4 x


5 July 2006, Q27

Which of the following graphs represents y = 4

1

A. B.

C. D.

6 Nov 2006, Q28

Which of the following graphs represents y = 10 – 2x3?

A. B.

C. D.


7 Jun 2007, Q 30

Which of the following graphs represents y = – 5x3

?

A. B

C D

8 Nov 2007, Q 28

Which of the following graphs represents 52 xy ?

A B

C D

x

y

0 x

y

0

x

y

0

x

y

0

– 5 – 5

y

x 0 5

y

x 0

5

y

x 0 5

y

x 0

5


9 Jun 2008, Q 29

Which graph represents ?32

1 xy

A B

C D

10 Nov 2008, Q 27

Which graph represents 22 xxy ?

A B

C D

y

x 0 –3

–6

y

x 0

–3

–6

y

x 0 –3

6

y

x 0

–3

6

y

x 0 –2 1

y

x 0 2 –1

y

x 0 –2 1

y

x 0 –1 2


ANSWERS

Questions Base On Examination Format

1. A 2. A 3. B 4. D 5. C 6. D 7. C 8. C 9. C 10. D

11. C 12. B 13. C 14. A 15. B 16. D 17. C 18. B 19. A 20. C

Past-Year Questions

1 B

2 D

3 D

4 A

5 C

6 B

7 B

8 B

9 B

10 C

Chapter 19 I Graphs of Functions II ENHANCE

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