Cambridge International Mathematics

Discovery Cubic functions#endboxedheading

To discover the shape of different cubics you can either use the graphing package or your

graphics calculator.

What to do:

1aUse technology to help sketch graphs of:

i y=x^3 , y=2x^3 , y=^12 x^3 , y=^13 x^3 ii y=x^3 and y=¡x^3

iii y=¡x^3 , y=¡ 2 x^3 , y=¡^12 x^3 , y=¡ 101 x^3

b Discuss the geometrical significance ofain y=ax^3.

You should comment on both the sign and the size ofa.

2aUse technology to help sketch graphs of:

i y=x^3 , y=x^3 +2, y=x^3 ¡ 3

ii y=x^3 , y=(x+2)^3 , y=(x¡3)^3

iii y=(x¡1)^3 +2, y=(x+2)^3 +1, y=(x¡2)^3 ¡ 2

b Discuss the geometrical significance of:

² kin the family of cubics of the form y=x^3 +k

² hin the family of cubics of the form y=(x¡h)^3

² handkin the family of cubics of the form y=(x¡h)^3 +k.

3aUse technology to help sketch graphs of:

y=(x¡1)(x+ 1)(x+3), y=2(x¡1)(x+ 1)(x+3),

y=^12 (x¡1)(x+ 1)(x+3), y=¡2(x¡1)(x+ 1)(x+3):

b For each graph inastate thex-intercepts and they-intercept.

c Discuss the geometrical significance ofain y=a(x¡®)(x¡ ̄)(x¡°):

4aUse technology to help sketch graphs of:

y=2x(x+ 1)(x¡2), y=2(x+ 3)(x¡1)(x¡2) and y=2x(x+ 2)(x¡1):

b For each graph inastate thex-intercepts andy-intercept.

c Discuss the geometrical significance of®, ̄and°for the cubic

y=a(x¡®)(x¡ ̄)(x¡°).

5aUse technology to help sketch graphs of:

y=(x¡2)^2 (x+1), y=(x+1)^2 (x¡3), y=2(x¡3)^2 (x¡1),

y=¡x(x¡2)^2 , y=¡2(x+ 1)(x¡2)^2 :

b For each graph ina, state thex-intercepts and they-intercept.

c Discuss the geometrical significance of®and ̄for the cubic y=a(x¡®)^2 (x¡ ̄):

You should have discovered that:

² if a> 0 , the graph’s shape is or , if a< 0 it is or

² y=(x¡h)^3 +k is the translation of y=x^3 through

μ

h

k

¶

² for a cubic in the form y=a(x¡®)(x¡ ̄)(x¡°) the graph hasx-intercepts®, ̄and°and the

graph crosses over orcutsthex-axis at these points

GRAPHING

PACKAGE

470 Further functions (Chapter 23)

IGCSE01

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\IGCSE01\IG01_23\470IGCSE01_23.CDR Monday, 27 October 2008 2:18:28 PM PETER