Exponential Functions

India is the second most populous country in the world, with a population in 2008 of about 1.14 billion people. The population is growing by about 1.34% each year. We might ask if we can find a formula to model the population, P, as a function of time, t, in years after 2008, if the population continues to grow at this rate.

In linear growth, we had a constant rate of change – a constant number that the output increased for each increase in input. For example, in the equation , the slope tells us the output increases by three each time the input increases by one. This population scenario is different – we have a percent rate of change rather than a constant number of people as our rate of change. To see the significance of this difference consider these two companies:

Company A has 100 stores, and expands by opening 50 new stores a year.

Company B has 100 stores, and expands by increasing the number of stores by 50% of their total each year.

Looking at a few years of growth for these companies:

Notice that with the percent growth rate, each year the company grows by 50% of the current year’s total, so as the company grows larger, the number of stores added in a year grows as well.

To try to simplify the calculations, notice that after 1 year the number of stores for company B was:

100 + 0.50(100)      or equivalently by factoring

100(1 + 0.50) = 150

We can think of this as “the new number of stores is the original 100% plus another 50%”.

After 2 years, the number of stores was:

150 + 0.50(150)      or equivalently by factoring

150(1 + 0.50)          now recall the 150 came from 100(1 +  0.50). Substituting that,

100(1 + 0.50)(1 + 0.50) = 100(1 + 0.50)² = 225.

After 3 years, the number of stores was:

225 + 0.50(225)      or equivalently by factoring

225(1 + 0.50)          now recall the 225 came from 100(1 + 0.50)² . Substituting that,

100(1 + 0.50)²(1 + 0.50) = 100(1 + 0.50)³ = 337.5.

From this, we can generalize, noticing that to show a 50% increase, each year we multiply by a factor of (1 + 0.50), so after n years, our equation would be

B(n) = 100(1 + 0.50)ⁿ

In this equation, the 100 represented the initial quantity, and the 0.50 was the percent growth rate. Generalizing further, we arrive at the general form of exponential functions.

To see more clearly the difference between exponential and linear growth, compare the two tables and graphs below, which illustrate the growth of company A and B described above over a longer time frame if the growth patterns were to continue.

2	200	225
4	300	506
6	400	1139
8	500	2563
10	600	5767

In all of the preceding examples, we saw exponential growth. Exponential functions can also be used to model quantities that are decreasing at a constant percent rate. An example of this is radioactive decay, a process in which radioactive isotopes of certain atoms transform to an atom of a different type, causing a percentage decrease of the original material over time.

Finding Equations of Exponential Functions

In the previous examples, we were able to write equations for exponential functions since we knew the initial quantity and the growth rate. If we do not know the growth rate, but instead know only some input and output pairs of values, we can still construct an exponential function.

In this example, you could also have used to evaluate the 6^th root if your calculator doesn’t have an n^th root button. We chose to use the f(x) = ab^x form of the exponential function rather than the f(x) = a(1 + r)^x form, but this choice was entirely arbitrary – either form would be fine to use.

When finding equations, the value for b or r will usually have to be rounded to be written easily. To preserve accuracy, it is important to not over-round these values. Typically, you want to be sure to preserve at least 3 significant digits in the growth rate. For example, if your value for r was 0.00317643, you would want to round this no further than to 0.00318.

Graphs of Exponentials

To get a sense for the behaviour of exponentials, let us begin by looking more closely at the function f(x) = 2^x . Listing a table of values for this function:

x	-3	-2	-1	0	1	2	3
f(x)				1	2	4	8

Notice that:

1. This function is positive for all values of x.
2. As x increases, the function grows faster and faster (the rate of change increases).
3. As x decreases, the function values grow smaller, approaching zero.
4. This is an example of exponential growth.

Looking at the function

x	-3	-2	-1	0	1	2	3
g(x)	8	4	2	1

Note this function is also positive for all values of x, but in this case grows as x decreases, and decreases towards zero as x increases. This is an example of exponential decay. You may notice from the table that this function appears to be the horizontal reflection of the f(x) = 2^x table. This is in fact the case:

Looking at the graphs also confirms this relationship:

Consider a function for the form f(x) = ab^x . Since a, which we called the initial value in the last section, is the function value at an input of zero, a will give us the vertical intercept of the graph. From the graphs above, we can see that an exponential graph will have a horizontal asymptote on one side of the graph, and can either increase or decrease, depending upon the growth factor. A horizontal asymptote is a horizontal line y = b where the graph approaches the line as the inputs get large. The horizontal asymptote will help us determine the long run behaviour and is easy to determine from the graph.

The graph will grow when the growth rate is positive, which will make the growth factor b larger than one. When it’s negative, the growth factor will be less than one.

When sketching the graph of an exponential function, it can be helpful to remember that the graph will pass through the points (0, a) and (1, ab).

The value b will determine the function’s long run behavior:

If b > 1, as x→∞ , f(x)→∞ and as x→-∞ , f(x)→0.

If 0 < b < 1, as x→∞ , f(x)→0 and as x→-∞ , f(x)→∞ .

Example 4.1.6

Sketch a graph of

This graph will have a vertical intercept at (0,4), and pass through the point (1, ) . Since b < 1, the graph will be decreasing towards zero. Since a > 0, the graph will be concave up.

We can also see from the graph the long run behavior: as x→∞ , f(x)→0 and as x→-∞ , f(x)→∞ .

To get a better feeling for the effect of a and b on the graph, examine the sets of graphs below. The first set shows various graphs, where a remains the same and we only change the value for b.

Notice that the closer the value of b is to 1, the less steep the graph will be.

In the next set of graphs, a is altered and our value for b remains the same.

Notice that changing the value for a changes the vertical intercept. Since a is multiplying the b^x term, a acts as a vertical stretch factor, not as a shift. Notice also that the long run behavior for all of these functions is the same because the growth factor did not change and none of these a values introduced a vertical flip.

Example 4.1.7

Match each equation with its graph.

The graph of k(x) is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of h(x) can be identified as the only growing exponential function with a vertical intercept at (0, 4). The graphs of f(x) and g(x) both have a vertical intercept at (0, 2), but since g(x) has a larger growth factor, we can identify it as the graph increasing faster.

Compound Interest

In the bank guaranteed investment certificate (GIC) example earlier in the section, we encountered compound interest. Typically bank accounts and other savings instruments in which earnings are reinvested, such as mutual funds and retirement accounts, utilize compound interest. The term compounding comes from the behaviour that interest is earned not on the original value, but on the accumulated value of the account.

In the example from earlier, the interest was compounded monthly, so we took the annual interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by 12, the number of compounds in a year, to find the monthly interest. The exponent was then measured in months.

Generalizing this, we can form a general formula for compound interest. If the APR is written in decimal form as r, and there are k compounding periods per year, then the interest per compounding period will be r/k. Likewise, if we are interested in the value after t years, then there will be kt compounding periods in that time.

A Limit to Compounding

Because of compounding throughout the year, with compound interest the actual increase in a year is more than the annual percentage rate. If $1,000 were invested at 10%, the table below shows the value after 1 year at different compounding frequencies:

The amount we earn increases as we increase the compounding frequency. Notice, though, that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to believe that although increasing the frequency of compounding will increase our result, there is an upper limit to this process.

To see this, let us examine the value of $1 invested at 100% interest for 1 year.

These values do indeed appear to be approaching an upper limit. This value ends up being so important that it gets represented by its own letter, much like how π represents a number.

Because e is often used as the base of an exponential, most scientific calculators have a button that can calculate powers of e, usually labeled e^x. Some computer software instead defines a function exp(x), where exp(x) = e^x. Because e arises when the time between compounds becomes very small, e allows us to define continuous growth and allows us to define a new toolkit function, f(x) = e^x .

This type of equation is commonly used when describing quantities that change more or less continuously, like chemical reactions, growth of large populations, and radioactive decay.

Practice questions