Exponential

1
Chapter 2 Exponents and Logarithms
2.2
Exponential Functions as Models of Growth or
Decay
2.2.1
MATHPOWERTM 12, WESTERN EDITION
2
Exponential Functions
An exponential function is a function of the form
f(x) Abx, where A is a constant and b gt 0.
Many real phenomena can be related to exponents
for example, population growth, the growth or
decay of substances, and the value of
investments earning compound interest.
Example A person sends a letter to each of 2
people. They, in turn, send a letter to 2 other
people, and so on ...
The number of letters grows exponentially
and can be represented by the equation y
2x.
1
2
4
8
2.2.2
3
Exponential Functions contd
Using the Graph of an Exponential Function
y 2x
(6, 64)
We can use the graph of y 2x to interpolate
information. For example, how many letters have
been written after the 6th round of writing
letters?
Number of Letters
Round of Letter Writing
2.2.3
4
General Observations of the Graph of y bx
y 2x
y 2x
y 4x
The y-intercept is 1.
y 3x
There is no x-intercept.
The domain is x x Î R.
The range is yy gt 0.
There is a horizontal asymptote at y 0.
2.2.4
5
General Observations of the Graph of y bx
contd
y 2x
The y-intercept is 1.
There is no x-intercept.
The domain is x x Î R.
The range is yy gt 0.
2.2.5
6
Using the Graph of y bx
You invest 400 in an account paying 8 per annum
compounded semi-annually.
Estimate when you would expect your investment to
double.
A(t) 400(1.04)2t
n 2t i 0.04
A(t) P(1 i)n
Investment ()
A(t) 400(1 0.04)2t
(17.69, 800.46)
A(t) 400(1.04)2t
From the graph, it would double in approximately
18 years.
2.2.6
Number of Years
7
The Exponential Function
For every metre that a diver descends below the
surface of the water, the light intensity is
reduced by 3.5.

• Write an exponential function relating the light
  intensity
• to the depth of the diver.
• About what percent of the original intensity
  remains at
• 10 m below the surface?
• Use a graph to estimate how far below the surface
  the
• diver would be when the light intensity is 40
  of the
• surface intensity.

Since the light is being reduced, there is 96.5
(100 - 3.5) of the light intensity remaining
for each metre descended.
Id I0(0.965)d
Id
intensity at a depth of d metres
original intensity
I0
2.2.7
8
The Exponential Function contd
B) When d 10 m
Id I0(0.965)d
Light Intensity ()
Id I0(0.965)10
Id 0.7003
At 10 m below the surface, about 70 of the light
remains.
(10, 0.7)
Id I0(0.965)d
C) Estimate the depth when the light
intensity is 40.
(25.69, 0.4)
From the graph, the diver would be at a depth of
about 26 m.
Depth (m)
2.2.8
9
Assignment
Suggested Questions Pages 81 and 82 1-23, 25, 26
abc, 29
2.2.9