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Why Study Math? – Functions and Rules

admin — Thu, 25 Mar 2010 13:39:51 +0000

One of the most important concepts in mathematics is that of a function. Although the topic of function can appear abstract, it is nothing more than a specific rule between two sets of mathematical objects. These sets are usually numbers, but they do not have to be restricted to such mundane entities. The sets might consist of more interesting objects, such as matrices or vectors. This notwithstanding, a function is nothing more than a rule that associates with each member of one set another member of the other set. Here we discuss this exciting concept in a little more detail so that the next time you see or hear about it, rather than avert your eyes or go cowering away in fear, you jump right in on the conversation.

Before we introduce the concept of function let us define what we mean by a set. A set is simply a well-defined collection of objects. Sets can be defined by listing their specific elements as in
S = {1, 2, 3} or by definition such as S = {2n| n is an integer} to define the infinite set of even integers. A function is simply a rule between two sets, such that this rule assigns to each element of the first set, call it set A, a unique element of the second set, call it set B. For example, let set A = {1, 2, 3} and B = {2, 4, 6}. We traditionally let the letter f stand for a function. We can define a function f from set A to B such that we associate 1 in A with 2 in B; 2 in A with 4 in B; and 3 in A with 6 in B.

If you have not realized it yet, we are doubling the elements of A. That is the function defined from set A to set B is that which multiplies each element or member in A (which is usually denoted by little “a”) by 2 to get each element “b” in set B. That is for each a in A we get b in B equal to 2a. Thus 2 = 2(1); 4 = 2(2); and 6 = 2(3). We write this function as f(a) = 2a. Remember that f(a) produces an element in the second set B.

In mathematics and particularly algebra, we often see the notation y = f(x). Here x in X is the first set, completely analogous with set A above; and y in Y is the second set, completely analogous with set B above. As you may realize now, functions between the letters x and y are common and x and y stand for the axes of the cartesian coordinate plane or x-y system. Depending on the complexity of the function or rule, the function y = f(x) can usually be graphed using graph paper and the pictorial relationship between the two sets X and Y can be seen and studied.

It is common to come across the terms “domain” and “range” when talking about functions. The first set in a functional relationship is called the domain and the second set is called the range. The notation f:X->Y is the symbolism which stands for the function from set X (the domain) to set Y (the range). The letters X and Y are often replaced by other characters, and it is not uncommon to use the letters g, H, and z for functions. As mentioned previously, X and Y are most common in functional notation because of convention in naming the axes in a coordinate plane by using these letters.

Functions occur throughout the realm of mathematics and indeed life. Specific functions model many real world phenomena and help solve many practical problems. For example, the function

s = -16t^2, between s (distance) and t (time) models the distance a body will free fall in time t.
The function P = A(1 + r)^t, between P (principal) and t (time) models the amount of principal accumulated after an initial deposit A at interest rate r.

Functions give us a unique opportunity to observe what we see around us—as in nature—and come up with some kind of rule which helps to explain what we experience. Without this utile concept, we would certainly not understand the world as well as we do. Functions hold the key. So the next time you see y = f(x) or b = f(a), remember you have entered the realm of functions. And oh what an interesting realm it is!

By: Joe Pagano

Why Study Math? Linear Equations and Slope-Intercept Form

admin — Thu, 18 Feb 2010 05:56:39 +0000

Linear equations or functions are some of the more basic ones studied in algebra and basic mathematics. The import of these functions is that they model many real world phenomena and a key component of them, the slope, is a springboard concept for the realm of the calculus. That’s right: the basic idea of rise over run, or slope, within these equations, leads to all kinds of interesting mathematics.

A linear equation, or function, is simply one of the form Ax + By = C. The x and y are variables and the a,b, and c represent numbers like 1, 2, or 3. Usually the early letters in the alphabet represent numbers, or fixed, quantities and the latter letters in the alphabet stand for variables, or changing quantities. We use the words equation or function interchangeably, although there is a slight difference in meaning. At any rate, the expression Ax + By = C is known as a linear equation in standard form. When we move these expressions around and solve for y, we can write this equation as y = -A/Bx + C. When we substitute m for -A/B and b for C, we obtain y = mx + b. This latter representation is known as slope-intercept form.

The simplicity and utility of this form makes it special in its own right. You see, when a linear equation is written in this form, not only do we have all the information about the line that we need, but also, we can quickly and accurately sketch the graph. Slope-intercept form, as the name implies, gives us the slope, or inclination, of the line, and the y-intercept, or point at which the graph crosses the y-axis.

For example, in the equation y = 2x + 5, we immediately see that the slope, m, is 2, and the y-intercept is 5. What this means graphically is that the line rises 2 units for every 1 unit that it runs; this information comes from the slope of 2, which can be written as 2/1. From the y-intercept of 5, we have a starting point on the graph. We locate the y-intercept at (0,5) on the Cartesian coordinate plane, or graph. Since two points determine a line, we go from (0,5) up 2 units and then to the right 1 unit. Thus we have our line. To make our line somewhat longer so that we can draw its picture more easily, we might want to continue from the second point and go 2 more units up and 1 unit over. We can do this as many times as necessary to produce the picture of our line.

Linear functions model many real world phenomena. A simple example would be the following: Suppose you are a waitress at the local diner. You earn a fixed $20 per 8-hour shift and the rest of your income comes in the form of tips. After working at this job for six months, you have figured that your average tip income is $10 per hour. Your income can be modeled by the linear equation y = 10x + 20, where x represents hours and y represents income. Thus for the 8-hour day, you can expect to earn y = 10(8) + 20 or $100. You can also graph this equation on a coordinate grid using the slope of 10 and y-intercept of 20. You can then observe at any point in your day where your income stands.

Simple models like these show us how mathematics is used in the world around us. Having read and digested the contents of this article, try to come up with your own example of a linear equation or model. Who knows? You just might start liking math more than ever before.

By: Joe Pagano

Why Study Math? – The Hyperbola

admin — Fri, 08 Jan 2010 03:30:40 +0000

As we continue the “Why Study Math” series of articles, here we look at the conic section called the hyperbola. The hyperbola is obtained by intersecting the double-napped cone (see the other articles in this series on this point) with a plane so that both parts of the cone are cut. Those familiar with the parabola might note that this curve almost looks like two parabolas pasted back to back with a space in between them. Mathematically, the hyperbola is not a parabola, although these two conic sections have a similar outward appearance.

The hyperbola is the least known of the four conic sections. It is also the most difficult curve to derive algebraically. Probably for this reason, students who study the conic sections, like the hyperbola the least. However, when students see the reason we study this curve, their attitude changes significantly. For this reason, we will now examine some of those applications connected to the hyperbola.

Everyone at one time or another has thrown a pebble into a still pond. Picture throwing not one but two pebbles into this pond at the same time. The outward concentric circles that form intersect each other at points which trace out the curve known as the hyperbola. This application is used in radar tracking stations. LORAN, the terrestrial navigation system, uses low frequency radio transmitters to locate objects. Objects are located by sending out sound signals from two sources to a receiving station, such as one found on a boat or plane. The constant time difference between the signals from the two stations is represented by a hyperbola.

As we discussed with the applications of the ellipse, most celestial bodies follow elliptical orbits. In the case of comets, however, a hyperbolic path is followed as they shoot through space. The hyperbola is also the shadow cast on a wall by a lamp with a cylindrical shade. And for something a little more earthy, the shape of that horse saddle you get on to ride forms an interesting solid curve called a hyperbolic paraboloid. So you see, the conic sections—even the hyperbola—might be closer than you think.

See more at Math Ebooks

By: Joe Pagano

Why Study Math? Trigonometry and SOHCAHTOA

admin — Wed, 21 Oct 2009 06:27:32 +0000

Beginning with “S,” oh that’s the Sine,

Opposite next, to keep you in line,

Hypotenuse follows, and then you will know,

Opposite over Hypotenuse equals Sine there you go.

Cosine will follow the “C” gives the clue,

Next is Adjacent, followed by Hypotenuse too,

Now you are rolling and having some fun,

Adjacent over Hypotenuse, the Cosine is won.

Then onto the “T” the Tangent we face,

Opposite and Adjacent finish the race,

It’s clear now, you have the key to the throne,

Opposite over Adjacent, the Tangent we own.

Let no man deceive you, let none blind your path,

To this mystery Chief who teaches you math,

Many have hearkened his voice to hear clear,

And master their trig, year after year.

-From the poem “Chief SOHCAHTOA” from the collection Poems for the Mathematically Insecure

Trigonometry is that curious branch of mathematics that deals with the measurements and relationships of the various triangles and their sides and angles. It would appear hard to imagine that so seemingly unglamorous a discipline as this would find itself intertwined in so many physical applications of the world around us and in many branches of physics and upper mathematics. Yet this is indeed the case.

Trigonometry specifically deals with the measurement of the sides of triangles using special ratios. Right triangle trigonometry, as the name implies, deals with the measurement of the sides of right triangles using the ratios of sine, cosine, and tangent. These three mathematical entities are nothing more than simple ratios as expressed by the sides of a right triangle. If you recall from your basic geometry days, a right triangle has one angle whose measure is 90 degrees. The side opposite the right angle is called the hypotenuse and the other two sides, legs.

The manner in which we calculate the sine, cosine, and tangent of any right triangle is to use ratios based on the sides of the right triangle. Mathematicians have come up with a curious mnemonic to remember these formulas, and if you have already read Poems for the Mathematically Insecure, then you know from Chief SOHCAHTOA the way to do it. That is, the sine is equal to the length of the opposite side to the length of the hypotenuse; the cosine, the adjacent to the hypotenuse; and the tangent, the opposite to the adjacent.

Once we know these formulas, we can put these ratios into all sorts of uses. For example, the sine and cosine can give us the resultant force when doing leg presses in our local gym. The tangent can tell us the height of a building when we know how far we are standing from it. These ratios can also tell us how to ease the exertion required by a person belaying a rock wall climber.

Not too bad for three simple ratios! What is even more fascinating is that higher up in mathematics we see these curious creatures popping up all over the place to help aid us in solving a myriad of mathematical problems. So go meet Chief SOHCAHTOA and these three trigonometric ratios. You too will then be able to use them in some everyday ways.

See more at Cool Math Ebooks and come meet Chief SOHCAHTOA here Cool Math Poems Ebook

By: Joe Pagano

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