A *linear combination* is just the sum of constant multiples of simpler functions. For example, ax^{2}+bx+c is a linear combination of x^{2}, x, and 1.

Each of these terms (ax

Notice that monomials have a variable base, whereas exponentials have a variable exponent. For example, x^{3} is a monomial whereas 3^{x} is an exponential.

An exponential function's rate of growth (or decay, if the base is less than 1) varies directly with it's current value. For example, a city of 2 million people should have twice as many babies born this year as a city of 1 million people if thier populations grow exponentially over time.

We will learn how to analyze exponential functions shortly, but first we will discuss some of their applications.

After 1 year (after adding interest) it would be worth P+0.12P = P(1+0.12).

After 2 years, it would be worth

P(1+0.12)+0.12(P(1+0.12)) = (1+0.12)(P(1+0.12)) = P(1+0.12)

After 3 years P(1+0.12)

It appears that compound interest can be modeled as follows: P_{N} = P(1+I)^{N}, where a principle (P) is invested at an interest rate (I) for a number of compunding periods (N) yeilding a future value (P_{N}).

Quite often though, one doesn't invest a lump sum at the start of an investment period, rather one invests the same amount of money per month or per year. Think about paying off a car loan, or investing in a retirement or education savings plan. This type of investment is called an annuity and can be modeled as a geometric series. A geometric series has the form:
ar+ar^{2}+ar^{3+}...+ar^{n}. Later, We will be discussing how to calculate the sum of a geometric series and how it applies to annuities.

Exponential functions can be use to model population growth. For example, predicting the world's population in 2020 or determining the concentration of bacteria in a sample after 60 minutes are both examples of exponential growth. They growth pattern is exponential because the rate of growth depends on the current population.

Another use for exponential functions is for modeling *exponential decay*. A hot frying pan will cool quickly in the surrounding air, but as it cools, its rate of cooling declines as its temerature approaches the room temperature. The temperature difference is an example of exponential decay. In carbon-14 dating, the concentation of carbon-14 isotopes are compared to that of the more common carbon-12. By comparing this ratio to the expected half-life of carbon-14, scientists can determine a fossil's age.

Sometimes, a population appears to grow exponentially, but reason would dictate that this rate cannot be maintained. For example, bacteria can grow in a sample exponentially until its food supply begins to run out, then its rate of growth will slow. This type of growth is called *logistical growth*. We will discuss *expontential growth and decay* later, but we will hold off on logisitical growth until the section on grade 12 calculus as it involves differential equations.

Exponential functions are used to model many probability distributions in the field of Statistics. One of the most important is called the Normal Distribution (also refered to as the Gaussian Distribution or the bell shaped curve). Many quanitities in the real world are normally distributed. Examples include human height, weight, and IQ scores. It can also be shown that given a sample from ANY population (regardless of its own probability distribution) that the means (or averages) of the samples become more normally distributed as the size of the samples increase.

The standard normal distribution (where the mean is 0.0 and the standard deviation 1.0) can be modeled as:

We will derive this formula in the grade 12 section of this website.A catenary is also the ideal shape for a freestanding arch or dome of constant thickness. Unlike a dome with a semi-circular cross section, a catenary dome's sides will not bulge out, nor will the top cave in. In other words, it is a shape that will support itself.

A soap film bubble that forms between 2 rings of equal radius will form the shape of a catenoid (a 3D shape with a catenary in cross section). A catenoid is the shape that will minimize the surface area. Other minimal surfaces include a sphere and a plane. These correspond to freely floating bubbles and the film that is initially inside the ring of a bubble wand.

Finally a word on the relationship between quadratic functions and catenaries. If you roll a parabola on a straight line, the path traced out by the parabola's focus will form a catenary. Also noteworth is the fact that a formula for a circle also involves an x^{2} term (x^{2}+y^{2}=r^{2}). If there is some relationship between quadratics and circles and also between quadratics and exponentials, then one may suppose that there is some relationship between exponentials and circles. There is in fact a relationship:

This formula was first published in 1748 by the great mathemetician Leonhard Euler and is one of the most important formulas ever discovered. This formula will make a lot more sense when we cover Imaginary and Complex numbers. The irrational constant