We have dealt with linear functions earlier. All types of
equations containing two unknown (x and y) variables may be
inserted in a coordinate system. These types of equations are known
as functions. A straight line is known as a linear
function.
The function need not necessarily respond like a straight line
equation. For example: If we have $50 000 deposited in the bank,
and receive a 2 % interest annually, our investment shall increase
as follows:
|
Year
|
Capital
|
Interest
|
Sum
|
|
1
|
50 000
|
50 000 · 0.02 = 1 000
|
51 000
|
|
2
|
51 000
|
51 000 · 0.02 = 1 020
|
52 020
|
|
3
|
52 020
|
52 020 · 0.02 = 1040.40
|
53 060.40
|
Compare that with what we would have with a linear increase
(2%):
|
Year
|
Capital
|
Increase
|
Sum
|
|
1
|
50 000
|
50 000 · 0.02 = 1000
|
51 000
|
|
2
|
51 000
|
50 000 · 0.02 = 1000
|
52 000
|
|
3
|
52 000
|
50 000 · 0.02 = 1000
|
53 000
|
In this case we may note that the increase was constant each
year. The investment may be described as:
where x equals the number of years.
However in the first case, the structure proceeds as:
Here we have an x-variable in the exponent. The
interest and thus also the function are exponentials.
Now we shall examine the differences displayed with the
functions in our example above in a coordinate system.
The lower straight line represents the linear increase and the
upper bowed curve represents the exponential increase. In other
words it is more profitable to have a compounded interest than a
fixed return.
An exponential function is a nonlinear function that has the
form of
An exponential function with a > 0 and b > 1, like the one
above, represents an exponential growth and the graph of an
exponential growth function rises from left to right.
An exponential function where a > 0 and 0 < b < 1
represents an exponential decay and the graph of an exponential
decay function falls from left to right.
When a quantity increases or decreases exponentially it
increases or decreases by the same percent over equal time periods
in comparison to when a compound increases or decreases linearly
when a quantity increases or decreases with the same amount over
equal time periods.
Video lesson: Graph the functions
