Cambridge Additional Mathematics

Surds, indices, and exponentials (Chapter 4) 119

2 Explore the family of curves of the form y=2x+d wheredis a constant.

For example, consider y=2x, y=2x+1, and y=2x¡ 2.

a What effect does changingdhave on the position of the graph?

b What effect does changingdhave on the shape of the graph?

c What is the horizontal asymptote of each graph?

d What is the horizontal asymptote of y=2x+d?

e To graph y=2x+d from y=2x what transformation is used?

3 Explore the family of curves of the form y=2x¡c.

For example, consider y=2x, y=2x¡^1 , y=2x+2, and y=2x¡^3.

a What effect does changingchave on the position of the graph?

b What effect does changingchave on the shape of the graph?

c What is the horizontal asymptote of each graph?

d To graph y=2x¡c from y=2x what transformation is used?

4 Explore the relationship between y=bx and y=b¡x where b> 0.

For example, consider y=2x and y=2¡x.

a What is they-intercept of each graph?

b What is the horizontal asymptote of each graph?

c What transformation moves y=2x to y=2¡x?

5 Explore the family of curves of the form y=a£ 2 x whereais a constant.

a Consider functions where a> 0 , such as y=2x, y=3£ 2 x, and y=^12 £ 2 x.

Comment on the effect on the graph.

b Consider functions where a< 0 , such as y=¡ 2 x, y=¡ 3 £ 2 x, and y=¡^12 £ 2 x.

Comment on the effect on the graph.

c What is the horizontal asymptote of each graph? Explain your answer.

FromDiscovery 1you should have found that:

For the general exponential function y=a£bx¡c+d where b> 0 , b 6 =1, a 6 =0:

² bcontrols how steeply the graph increases or decreases

² ccontrols horizontal translation

² dcontrols vertical translation

² the equation of the horizontal asymptote is y=d

² if a> 0 , b> 1

the function is

increasing

² if a> 0 , 0 <b< 1

the function is

decreasing

² if a< 0 , b> 1

the function is

decreasing

² if a< 0 , 0 <b< 1

the function is

increasing.

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_04\119CamAdd_04.cdr Tuesday, 14 January 2014 2:29:01 PM BRIAN