Cambridge Additional Mathematics

DYNAMIC

GRAPHING

PACKAGE

“ ” means

“approaches”.

!

x

y

1

1

-3 -2 -1 2 3

2

4

6

8

y=2x

O

118 Surds, indices, and exponentials (Chapter 4)

We have already seen how to evaluate bn when n 2 Q, or in other words whennis a rational number.

But what about bn when n 2 R,sonis real but not necessarily rational?

To answer this question, we can look at graphs of exponential functions.

The most simple generalexponential functionhas the form y=bx where b> 0 , b 6 =1.

For example, y=2x is an exponential function.

We construct a table of values from which we graph the function:

x ¡ 3 ¡ 2 ¡ 1 0 1 2 3

y^1814121248

When x=¡ 10 , y=2¡^10 ¼ 0 : 001.

When x=¡ 50 , y=2¡^50 ¼ 8 : 88 £ 10 ¡^16.

As x becomes large and negative, the graph of y =2x

approaches thex-axis from above but never touches it, since

2 x becomes very small but never zero.

Discovery 1 Graphs of exponential functions

#endboxedheading

In this Discovery we examine the graphs of various families of exponential functions.

Click on the icon to run thedynamic graphing package, or else you could use your

graphics calculator.

What to do:

1 Explore the family of curves of the form y=bx where b> 0.

For example, consider y=2x, y=3x, y=10x, and y=(1:3)x.

a What effect does changingbhave on the shape of the graph?

b What is they-intercept of each graph?

c What is the horizontal asymptote of each graph?

G Exponential functions

So, as x!¡1, y! 0 from above.

We say that y=2x is ‘asymptoticto thex-axis’ or ‘y=0

is ahorizontal asymptote’.

We now have a well-defined meaning for bn where b,n 2 R

because simple exponential functions have smooth increasing or

decreasing graphs.

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_04\118CamAdd_04.cdr Thursday, 30 January 2014 2:27:43 PM BRIAN