Chapter 19 I Graphs of Functions II ENHANCE mathematics spm exercise and notes

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 functionb) the type of function given a graphc) the graph given a function and vice versa.

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

Linear y = mx + c

m = gradientc = y-intercept

Quadratic y = ax2 + bx + c

Note : Paper 1 limits to y = ax2 + c

Cubicy = ax3 +bx2 +cx + d

Note : Paper 1 limits to y = ax3 + c

Reciprocal

y =

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 + bx2 + 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 = , where a is a constant.

19.1 To Identify The Type Of Function Given Its Equation

Graphs of Functions II 1

ax

O x

y

xx

y

O

y

O x

y

O x

Example 1No. Linear

y = mx + c

Quadratic

y = ax2 + bx + c

Cubic

y = ax3 + bx2 + cx + d

Reciprocal

y =

1. y = x y = x2 + 2x -3 y = x3 + 2x2 + 3x – 4 y =

2. y = 2x + 3 y = x2 + 2 y = x3 – 3 y =

3. y = 3x – 4 y = 4 – 3x2 y = 4 – 3x3 xy = 4

4. y = 5 y = 3x2 y = 2x3 y =

5. x+ y = 4 y = 3 – 4x + 5x2 y = 1 – 2x + 5x3 y =

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 = x2 + 2x – 4 4. y = -x3

5. y = 2x2 + 3 6. y = – x 7. y = x3 – 3 8. y = 1 + x2

9. y = 10. y = 6 – 3x2 11. y = 12. y = 20

13. y = 2 – x + 5x2 14. y = 15. y = 2x3 16. y =

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.

Graphs of Functions II 2

No. Functions Type No. Functions Type

1. y = 2x – 3 Linear 1. y = x2 – 5

2. y = Reciprocal 2. y =

3 y = –x3 Cubic 3 y = 10 – x

4. y = x2 + x Quadratic 4. y = x2 + 5

5. y = 5. y = 2 + 2x

6. y = (x – 1)2 6. y =

7. y = x ( x – 2) 7. x + 2y = 1

8. y = – 3 + x3 8. y = – 3

9. y = – x2 + 2 9. y = 4 – x2

10. y = 5 – 4 x2 10. y = 3 – 6 x

11. y = 2 11.

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

Example For You To Try...

Graphs of Functions II 3

y

Ox

- 3

3

O x

y

-2

2

y = x + 3

y = x

y = x – 2

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

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

Graphs of Functions II 4

y

O x- 2

2

O x O x

yy

y

O x1

1

O x

O x

y

y

O x

yy

Ox 3

3

-2

-2

y = – x + 3

y = –x

y = x – 2

O x

y

O x

y

O x

y

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

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

1. y = x2

(a) y = x2 + 2 (b) y = x2 – 3

2. y = x2 + 3

(a) y = x2 + 4 (b) y = x2 – 1

3 y = 2x2

(a) y = 2x2 – 1 (b) y = 2x2 + 3

4 y = 1 + 2x2

(a) y = 2 + 2x2 (b) y = – 3+2x2

More challenges for you ….

Graphs of Functions II 5

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

3

xO

y

xO

y

1

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 + x2

3.

y = x2 + 1 y = 2x2 + 1 y = 3 x2 + 1

4.

y = x2 – 3 y = 2x2 – 3 y = 3 x2 – 3

Case 2 : y = ax2 + c , a < 0

Graphs of Functions II 6

xO

y

xO

y

-4

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

1

xO

y

xO

y

xO

y

-3

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

1. y = – x2

(a) y = –x2 + 2 (b) y = – x2 – 3

2. y = –x2 + 2

(a) y = – x2 + 4 (b) y = –x2 – 1

3 y =

(a) y = – x2 – 1 (b) y =– x2 + 3

4 y = 1 – 2x2

(a) y = 2 – 2x2 (b) y = – 3 – 2x2

More challenges for you ….

Sketch the given quadratic functions :

Graphs of Functions II 7

xO

y

xO

y

xO

y

xO

y

2

xO

y

xO

y

1

xO

y

xO

y

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 – x2

3.

y = –x2 + 1 y = –2x2 + 1 y = – 3 x2 + 1

4.

y = –x2 – 3 y = – 2x2 – 3 y = – 3 x2 – 3

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

Graphs of Functions II 8

xO

y

xO

y

4

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

x

O

y

1

xO

y

xO

y

x

O

y

-3

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

1. y = x3

(a) y = x3 + 2 (b) y = x3 – 3

2. y = x3 + 3

(a) y = x3 + 4 (b) y = x3 – 2

3 y = – x3

(a) y = –x3 + 2 (b) y = –x3 – 3

4 y = 1 – x3

(a) y = 2 – x3 (b) y = – 3 – x3

19.5 To sketch graphs of reciprocal functions of the form .

No. Function Your turn to try !

Graphs of Functions II 9

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

1.

y = y =

2.

y = – y =

3

xy = 2 ( can be written as y = ) xy = 3

4

xy + 2 = 0 ( can be written as y = ) xy + 5 = 0

More challenges for you ….

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

Graphs of Functions II 10

x

y

O

x

y

O

x

y

O

x

y

O

x

y

Ox

y

O

x

y

Ox

y

O

1.

2.

3.

Exercise B : Sketch the given functions :

Graphs of Functions II 11

O x

y

xx

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 y = x3 – 1

y = x – 2 y = x2 – 2 y = 2 – x2x + y = 2

3y

x y = –2 + x3 xy = –4y = – x3 +2

O x

y

x

y

O x

y

O xx

y

O

1.

y = x2 + 1 y = 2x + 1 y = x3 + 1

2.

y = x + 3 y = – 3 + x x + y = 4

3.

y = xy = 2 xy = – 1

4.

y = x3 – 3 y = 3 + x3 y = – x2 + 2

Questions Based On Examination Format

Graphs of Functions II 12

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

xO

y

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 – 5B. y = – x2 – 5 C. y = x3 – 5D. y = – x3 – 5

4. Which of the following graphs represents ?

Graphs of Functions II 13

y

O

y

xO

y

xO

x

– 2 – 2

– 2

y

O – 2 x

y

xO

– 5

A. B.

C. D.

5. Which of the following graphs represents

A B

C D

6

Graphs of Functions II 14

x

y

O

y

xO

y

xO

x

y

O

y

xO

5

y

xO 5

y

xO

5

y

xO-5

(–1, k)

x

y

0

.

Diagram 1

Diagram 1 shows the graph of . 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

Graphs of Functions II 15

O x

y

xx

y

O

y

O

x

y

O

x

( h, 0)x

y

0.

8

11. Which of the following represents the graph of ?

A B C D

12.

The graph labelled G most likely represents

A. y = –2x2 B. y = 2x2 – 2 C. x y + 2 = 0 D. y = 2x3

13.

The graph labelled G most likely represents

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

Graphs of Functions II 16

O x

y

xx

y

Ox

y

O

y

O

x

O

y

x

y = 2x2

G

O

y

x

y = x + 2

G

14.

The graph labelled G most likely represents

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

15. Which of the following represents the graph of ?

A B C D

16. Which of the following represents the graph of ?

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

Graphs of Functions II 17

O

y

x

y = x

G

y

O

x

y

O

xx

y

O

y

O

x

xO

y

O x

y

O x

y

Ox

y

x

y

O

18.

The above graph represents

A. y = - x + 2 C. y = 2 – x 2

B. D.

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

Graphs of Functions II 18

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 ?

A. B.

C. D.

Graphs of Functions II 19

x

y

O

y

xO

y

xO

y

xO

3 July 2005, Q28 Diagram 12 shows a sketch of a function.

What is the equation of the function?A. y = x2 – 3B. 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.

Graphs of Functions II 20

y

xO

– 3

y

O

y

xO

y

xO

x

– 4 – 4

– 4

y

O – 4 x

5 July 2006, Q27

Which of the following graphs represents y =

A. B.

C. D.

6 Nov 2006, Q28

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

A. B.

C. D.

Graphs of Functions II 21

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 ?

A B

C D

Graphs of Functions II 22

x

y

0 x

y

0

x

y

0

x

y

0

– 5 – 5

y

x0 5

y

x0

5

y

x0 5

y

x0

5

9 Jun 2008, Q 29

Which graph represents

A B

C D

10 Nov 2008, Q 27 Which graph represents ?

A B

C D

Graphs of Functions II 23

y

x0–3

–6

y

x0

–3

–6

y

x0–3

6

y

x0

–3

6

y

x0–2 1

y

x0 2 –1

y

x0–2 1

y

x0–1 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

Graphs of Functions II 24

1 B2 D3 D4 A5 C6 B7 B8 B9 B10 C