Cambridge International Mathematics

y

-2 x

12

2

O

@\=Qw_\(!\+\2)(!\-\1)(!\-2)

y

O 2 x

@\=\2!(!-2)X

² for a cubic in the form y=a(x¡®)^2 (x¡ ̄) the graphtouchesthex-axis at®andcutsit at ̄

² cubic functions have apoint of rotational symmetrycalled thepoint of inflection.

Example 1 Self Tutor

Use axes intercepts only to sketch the graphs of:

a f(x)=^12 (x+ 2)(x¡1)(x¡2) b f(x)=2x(x¡2)^2

a f(x)=^12 (x+ 2)(x¡1)(x¡2)

hasx-intercepts¡ 2 , 1 , and 2

f(0) =^12 (2)(¡1)(¡2) = 2

) they-intercept is 2

b f(x)=2x(x¡2)^2

cuts thex-axis when x=0and

touches thex-axis when x=2

f(0) = 2(0)(¡2)^2 =0

) they-intercept is 0

EXERCISE 23A.1

1 By expanding out the following, show that they are cubic functions.

a f(x)=(x+ 3)(x¡2)(x¡1) b f(x)=(x+ 4)(x¡1)(2x+3)

c f(x)=(x+2)^2 (2x¡5) d f(x)=(x+1)^3 +2

2 Use axes intercepts only to sketch the graphs of:

a y=(x+ 1)(x¡2)(x¡3) b y=¡2(x+ 1)(x¡2)(x¡^12 )

c y=^12 x(x¡4)(x+3) d y=2x^2 (x¡3)

e y=¡^14 (x¡2)^2 (x+1) f y=¡3(x+1)^2 (x¡^23 )

FINDING A CUBIC FUNCTION

If we are given the graph of a cubic with sufficient information, we can determine the form of the function.

We do this using the same techniques we used for quadratic functions.

Example 2 Self Tutor

Find the form of the cubic with graph:

aby

-1 x

-8

24

O

y

-3 x

6

O We_

Further functions (Chapter 23) 471

IGCSE01

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Y:\HAESE\IGCSE01\IG01_23\471IGCSE01_23.CDR Monday, 27 October 2008 2:18:31 PM PETER