Graphing Sine and Cosine Curves

As I usually include in most of my posts, please feel free to ask questions or leave comments on any material that I post, and I will try my best to address them. I recently received a request to go over how to graph sine and cosine curves.

Sine and cosine functions are PERIODIC FUNCTIONS, meaning that the function will evaluate to a repeated number after a defined length (period). You've probably seen these graphs several times before and not realized what they were. Sine and cosine graphs are virtually identical, EXCEPT that they are slightly shifted relative to one another... or they are "out of phase." This is an important distinction to make, as confusing which is which, of course, will get you the wrong answer.

The EASIEST way to remember, or to quickly plot out the graph, is to create a table of values. Let's work on the Sine curve first. The cosine one works exactly the same way. Creating this table of values will be a good review of how these trig functions work. On that note, we will evaluate for values for x that represent the angles of our special triangles, and then continue with more values at the same intervals. (I'm going to represent the square root function by "sq.rt.") If you want, you can continue to make the list longer if you wish, but I think this is sufficient to demonstrate the graph.

y = sin(x)

x |||||y
-----------
0 |||||0
30 |||||0.5
60 |||||(sq.rt.3)/2
90 ||||| 1
120||||| (sq.rt.3)/2
150||||| 0.5
180||||| 0
210||||| -0.5
240||||| -(sq.rt.3)/2
270||||| -1
300||||| -(sq.rt.3)/2
330||||| -0.5
360||||| 0

As you can probably see from looking at the list of values, the numbers seem to have a repeating pattern. If you continue to add values, they will begin to repeat themselves again. Essentially, we have gone in a full circle (360 degrees), and so that is why they repeat.

So, taking these values that we have and plotting them on a graph, we can see what a sine curve looks like:



The cosine curve will look very similar, and you can deduce it exactly the same way. As I said before, they look the same but are out of phase. The important difference to remember is that the sine curve passes through (0,0), and the cosine curve passes through (0,1), but the 'periodicity' is the same... that is, they repeat at the same regular intervals. You can extend the curve into the negative x direction, or further in the positive direction.

I hope that explains what the original poster was asking me to go over. These functions, like any other function, can also be modified by adding coefficients etc, in the same way as other functions that I've reviewed before. I will explain these concepts in a future post. Or for fun, add a coefficient to the front of the sine function (eg. A*sin(x)), or to the x value itself (sin(2x)) and create tables of values and compare the resulting graphs!

Pythagorean Identities

Starting with the fundamental trig identities that I've already covered, a little bit of derivation leads to several new identities that will always hold true, and are especially useful in simplifying more elaborate equations.  Expanding upon my last post, there are a few other identities that can be derived from the Pythagorean Identity, which also go by the same name to describe the group of them, due to how they are deduced by making use of the his famous Theorem.  (Click here to get a great explanation about these Pythagorean Identities.)

If we start with the first Pythagorean identity, which I already explained:

Sin²(ɵ) + Cos²(ɵ) = 1

we can divide each side by the sine term to give something new.

[Sin²(ɵ) + Cos²(ɵ)  ] / Sin²(ɵ)  = 1 / Sin²(ɵ)

The first term (sin / sin) reduces to 1:

1 + Cos²(ɵ)  / Sin²(ɵ)  = 1 / Sin²(ɵ)

The remaining terms can be now be simplified.  We can apply the laws of exponents to group the terms and express them in a slightly different way, which allows us to apply additional trig identity substitutions to express things in terms of inverse functions:

1 + [Cos(ɵ)  / Sin(ɵ)]²  = [1 / Sin(ɵ)]²

1 + Cot²(ɵ)  = Csc²(ɵ)

This is the second of the three Pythagorean identities.  Hopefully my explanation is clear enough to understand, and you can follow along with my derivation of it.  As I always say on this site, if you understand the derivation, then you don't have to worry about making mistakes in memorizing the actual formula.  This is a perfect example of that, because the starting point is very easy to remember, but even I sometimes will confuse myself if I try to write down these additional identities without thinking about how to get there.  Learning the derivations of identities, formulas, and equations will always help you!

As I said, there are three Pythagorean identities.  This third one is derived in a similar manner to what we have already done, using the same sort of logical steps but dividing instead by the cosine term (rather than the sine term).  I am not going to write down all of the steps here, but I highly recommend that you grab a pencil and paper and quickly work through it, following along with the process that I outlined above.  There are only a few steps to work through, and in the end, this is what you should end up with:

1 + Tan²(ɵ)  = Sec²(ɵ)

As it is with many identities and equations in mathematics, it doesn't take very many steps to make something look completely different, as I've shown here.  I hope that you've been able to follow along and that everything makes sense.  If the steps in the derivations confused you at all, then I suggest going back to my posts on the basic trig functions, and working with those until you are comfortable.  The steps I have applied in these transformations were very simple substitutions, and I think that the most confusing part knowing what to look for in the first place.  With practice, you will learn to see patterns and recognize where you can apply various rules to make things simpler.

There are still several other trigonometric identities which I will show you how to derive in coming posts, many of which require the same types of thinking.  Stay tuned for more tutorials that explain things like the sine law and cosine law!

Fundamental Trigonometric Identities

Trigonometric identities are specific equalities that express one trigonometric function in terms of other trig functions. These fundamental trigonometric identities (also sometimes called basic trig identities) are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of trig identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.

These fundamental trigonometric identities are traditionally visualized as a right angle triangle inscribed within a circle of radius r, with sides formed by length x, height y, and the radius as the hypotenuse, r:

The basic trig definitions and relationships can easily be observed.  I have covered these trigonometric functions in other blog posts, so I won't go into them any further here, other than to just remind you of the concept of SOHCAHTOA.  Refer to these specific posts for more information about sine, cosine, and tangent.

Sin(θ) = y/r..... opposite  /hypotenuse
Cos(θ) = x/r..... adjacent / hypotenuse
Tan(θ) = y/x..... opposite / adjacent

For the first identity that I will show you how to derive, we need to make use of the Theorem of Pythagoras (I would also here like to refer you to my post about the Theorem of Pythagoras, which itself is derived from the cosine law, for more information on this trigonometric math concept.)  If we now apply the Theorem of Pythagoras to this right angle triangle we have inscribed within a circle, we can see:

r² = x² + y²

Dividing everything by r² gives:

1 = (x²)/(r²) + (y²)/(r²)
1 = (x/r)² + (y/r)²

And then, substituting in the basic trig identities listed above, we get:

1 = [Cos(θ)]² + [Sin(θ)]²

And that is the first of the 8 fundamental trigonometric identities. Nothing to it. In fact, this is an important mathematical expression.  As such, it has a special name or designation given to it to indicate that it is important.  It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it below in proper notation to clean it up a bit.  Notice how you can write the 2 (for the squares) in two different ways.  Either as above, like [Cos(θ)]², where you put the trig function in brackets to square it (note that you don't square the angle!), or more commonly as below, with the 2 placed next to the trig function to indicate that it is being squared.

Similarly, we can derive another basic relationship from a standard trigonometric identity.  For this one, it starts with:

Tan(θ) = y/x

Now to substitute in some more expressions.  This one is a little more complicated than how we arrived at the Pythagorean Identity above, but follow along and you should hopefully be able to see what I do.  The next step then, is to substitute in the basic Sine and Cosine definitions from the list at the top of this page (isolated for x and y, respectively) to give:

Tan(θ) = (r x Sin(θ)) / (r x Cos(θ))
Tan(θ) = Sin(θ) / Cos(θ)

And that's it again. Like the Pythagorean Trigonometric Identity I described above, this one also has a specific name.  This time, this one is called the Ratio Identity:

Those are now two of the simplest basic trig identities from which most of the others can be derived.  As I mentioned at the top, I always find that it is easier to memorize HOW to derive these fundamental trigonometric identities, rather than memorizing them explicitly and then possibly getting confused.  You WILL memorize them eventually (which always happens when you work with them so much!), but when you are first trying to understand them, I believe that it is always better to understand what is going on instead of just memorizing the final solutions.  This approach is a very wise strategy to apply throughout your mathematics training!


*Update 03-April-2012:  I have recently come back to this topic of basic trig identities, and have started a completely new post called the Trig Identities Cheat Sheet.  Please visit that site for a more thorough explanation of some of these derivations and fundamental identities of trigonometric functions.  Also, please click the +1 button below if you found this helpful!  Thanks.  

Stretching Graphs and Compressing Graphs

If you understand how to shift a curve horizontally or vertically, stretching graphs or compressing them isn't much different. Once again, it's only a small modification to the equation that causes the stretch or compression.  (Please click the Facebook Like button at the top of this post, or hit the Google +1 button at the end if you find this post helpful!)

Stretching and compressing graphs vertically is determined by the coefficient in front of the x (or more specifically, in front of the other direct modifications to x).
Let's look at a basic example.... f(x) = x², a standard parabola.

Now, to vertically compress this curve, you put a 'fraction coefficient' in front of the x component of the graph. i.e. f(x) = (1/2)*x². This squashes the graph down by a factor of 2. Or, another way to look at it, every y value in this curve is 1/2 of the value in the starting curve. Plot your own points to convince yourself of this. Note that the curve crosses (-2,4) and (2,4) in the original curve, whereas the new one crosses at (-2,2) and (2,2).

Now, naturally, if you put a whole number coefficient in front of the x term, you will be stretching graphs. For example: f(x) = 2x² looks like this:

You can see that this has caused the parabola to stretch upwards. Note that it now crosses (1,2), not (1,1). Or once again, to look at it from a different angle, every y value is now twice the value as in the original graph.  Compared to the original version, this is a vertical stretch graph.

The only other thing that you should keep in mind is that the coefficient to stretch or compress the graph vertically MUST be in front of any brackets that might be surrounding x, and the coefficient will act on any horizontal translation component and the exponent. Convince yourself of this by looking at graphs such as:

f(x) = (x-3)²....... and f(x)=2(x-3)²
f(x) = (x+1)³...... and f(x)=1/2(x+1)³
f(x) = x + 5...........and f(x) = 4x + 5
f(x) = (x-1)⁵ + 7...... and f(x) = 4(x-1)⁵ + 7

So far, all that I have talked about explains the concept of vertical compression and stretching.  But, let's now consider how we would go about showing a horizontal compression or stretch.  If we want to squash a graph together along the x-axis, what would we do?  Well, for any value of x, we would want the graph to be at some fractional value of that x.  So mathematically, this means that we have to modify the x value.  To perform a horizontal compression or stretch on a graph, instead of solving your equation for f(x), you solve it for f(c*x) for stretching or f(x/c) for compressing, where c is the stretch factor.  The simplest way to consider this is that for every x you want to put into your equation, you must modify x before actually doing the substitution.  Let me show you.

Consider again the parabola f(x) = 2x².  If we want to start plotting this graph, we could start by building a table of values and solving for f(1), f(2), f(3), etc.  Doing the quick math, you can see that f(1) = 2.  Now let's say we want to horizontally compress this graph by a factor of 10.  In this case, we do the modification on x before subbing in, so we apply the f(x/c) for compressing and see that f(1) becomes f(1/10).  You then substitute 1/10 in for x, and solve.  You then come up with the ordered pair of (1, 1/50).  This is a ten times compression along the x-axis.  If you draw some of these out, you will easily see that squishing or stretching along x is different from squishing or stretching along y.

Continuing with this same example, say we want to graph a horizontal stretch of a factor of 4.  We then solve f(c*x), giving us f(4*x).  For x = 1, f(4*1) becomes f(4), which works out to y = 32.  Again, it is easy to see how much we have stretched the graph by this simple modification.

As you can see, stretching graphs (or compressing them), both vertically and horizontally, really isn't that difficult.  There are obviously a few things that you need to remember, especially the distinction between horizontal and vertical changes.  But these modifications are just an extension of what you already know, building on your knowledge of horizontal or vertical shifts. Keep practicing, and you'll get it in no time. :)  (Remember to please hit Facebook Like button back at the beginning, or the Google +1 button if you liked my post!)

Manipulating graphs

In the next few days, I'm going to begin posting on graph manipulations. For any graphical expression, small changes to the expression can result in a very predictable change in the position of the graph. For instance, the graph can be shifted along the x-axis (horizontally), or along the y-axis (vertically). Similarly, the graph can be stretched or compressed along either axis as well. This may sound confusing, but the alterations to the equations are simple and easily recognizable. I will provide examples and go into detail in the coming days.

Trigonometry - Secant, Cosecant, Cotangent

In addition to the three basic trig functions we've already looked at (Sine, Cosine, Tangent), there are three other related functions. These are Secant, Cosecant, and Cotangent. These functions have similar meanings as the first three, in that they represent the ratios of various side lengths of a right angle triangle, and can be used to find angles or unknown side lengths. I will not go into extensive detail on these functions, as they are less commonly required, but I will show you what they mean.  Please remember to click on the Like button if this is helpful, and at the end, please hit the +1 button as well.

So far, with the help of SOHCAHTOA, we have seen that:

Sine = opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent

These new functions are related to the originals because they represent the inverse ratios.

Cosecant = hypotenuse / opposite... (compare to Sine)
Secant = hypotenuse / adjacent....... (compare to Cosine)
Cotangent = adjacent / opposite...... (compare to Tangent)

Also, these functions can be abbreviated: Cosecant = Csc, Secant = Sec, Cotangent = Cot.

At the middle school and high school math level, you will rarely have a need to use these functions, but it is good for you to know what they are. However, most trig problems at this stage can easily be solved with the original three functions.  Just in case, though, it's always good to know all the trig functions: sine, cosine, tangent, secant, cosecant, cotangent.

Please remember to Like and +1 this post if you found it helpful!  Thanks!

Area of a Triangle

With all this talk about triangles and their side lengths and angles, we shouldn't forget to discuss how to find the AREA of a triangle.

You are probably familiar with one formula for finding the area of a triangle:

Area = 1/2 (base)(height)

Compare this to finding the area of a rectangle:

The area of the rectangle is equal to the product of (base) x (height)..... (or length x width). However, by drawing a diagonal within the rectangle which joins two opposite corners, you can see that each newly-formed triangle is equal to half of the area of the original rectangle. Therefore, the area of a triangle is one-half the area of the rectangle, as shown by this triangle area formula. Even if you are looking at a triangle that doesn't immediately look like it is half of a rectangle, this formula still applies.

To prove it, you can draw a line in to represent the height, as I have shown here, thus creating two smaller triangles, and you can rearrange them to see that they indeed are equal to the area of half a rectangle:

That is one way to find the area of a triangle. However, if instead of base and height measurements, you are given lengths of sides or angles, this method won't work for you. In this case, you need to use a trig equation to solve for the area of a triangle.

Let's start with the first equation we had above, and modify it. By the standard trig identities, we can show that:

height = (a)(SinC)

So substituting that into our formula:

Area = 1/2(base)(height)
Area = 1/2(b)(a)(SinC)

And this is the trig formula for solving the area of a triangle!

Area = (1/2)abSinC

You can use this to find the area of a triangle where you know any two sides and the angle between them! It's that easy!

Special Angles in Trigonometry

Some angles in trigonometry are so common, they are known as special angles.  The values of trig functions of these specific angles can be represented by known ratios, and are good to commit to memory to help you work through problems faster. These angles can be remembered by examining two different special angle triangles. Specifically, the trig functions for angles of 0, 30, 45, 60, 90 degrees are the special ones.  (Please hit the Like button or +1 button at the end if you think this post helps you with special angles!)

Let's look at the first of the special angle triangles, and this will hopefully become clear. Take a right angle triangle with two 45 degree angles, and with sides of 1 unit length. By the Theorem of Pythagoras, the hypotenuse of this triangle is of length √2. This is what this triangle looks like:

So then, from these values and SOHCAHTOA, you can obtain the trig values for this special angle of 45 degrees. You can see that:

Sin(45) = 1/√2
Cos(45) = 1/√2
Tan(45) = 1

Don't worry if you can't remember these exact ratios... the simplest thing to remember is how to construct the special angle triangle... which is as easy as remembering a right angle triangle with a 45 degree angle and 2 sides of length 1... you can easily fill in the rest, and then work out the trig ratios yourself!

The second of the special angle triangles, which represents the rest of the special angles to remember, is slightly more complex, but still straightforward. Take a right angle triangle with angles of 30, 60, and 90 degrees. The simplest lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse). This 30 60 90 triangle looks like this:

So then, using the these values, you can obtain the trig values for these special angles as well:

Sin(30) = 1/2
Cos(30) = √3/2
Tan(30) = 1/√3

Sin(60) = √3/2
Cos(60) = 1/2
Tan(60) = √3/1 = √3

Again, just remember how to construct the triangle, and the ratios are easy to come up with!

For 0 and 90 degrees, there isn't a simple triangle scheme to remember the values (although please feel free to correct me if I am wrong!). However, these aren't scary square root numbers or weird fractions:

Sin(0) = 0
Cos(0) = 1
Tan(0) = 0

Sin(90) = 1
Cos(90) = 0
Tan(90) = undefined

If you can familiarize yourself with all of these special angles, or at least understand how to derive them from the special angle triangles, then you will have a much easier time working on trigonometry questions.  Again, please remember to Like and/or +1 this post if it helped.  Thanks.

Trigonometry - Cosine Law, Theorem of Pythagoras

The Cosine Law works similarly to the Sine Law that I have already discussed. The Cosine Law is the general form of the Pythagorean Theorem, which itself applies strictly to right angle triangles. Therefore, this law allows us to work with any triangle. It's a bit more of an equation to remember than the Sine Law unfortunately, but it is extremely useful. Here is the equation:

c(squared) = a(squared) + b(squared) -2abCosC

(I don't see a way to type superscript text on blogger, unfortunately.)

So, as you can see, the Cosine Law is useful for finding the third side of a triangle when any 2 sides and the angle between them is known. Let's try an example:

So by the Cosine Law:
c^2 = 6^2 + 8^2 - 2(6)(8)Cos(60)
c^2 = 36 + 64 - 96Cos(60)
c^2 = 36 + 64 - 48
c^2 = 52
c = 7.2

Now that you have done that, you have obtained a complete ratio to use with the Sine Law! And now, you can find the rest of the angles to fully describe the triangle!

Sin(60)/7.2 = SinB/8
and
Sin(60)/7.2 = SinA/6

I'll leave those for you to solve. But that's it! Using a combination of the Cosine Law and Sine Law, you can completely solve any triangle that you are given. They are extremely powerful and useful equations!

Also, on a side note... as I mentioned at the start of this post, the Cosine Law is a generalization of the Pythagorean Theorem, which specifically applies to right angle triangles. You can see that if you are working with a right angle triangle and substitute in 90 degrees to the Cosine Law, it reduces down to the Pythagorean theorem:

c^2 = a^2 + b^2 -2abCosC
but
Cos(90) = 0... so the 2abCos(90) term reduces to 0
and so
c^2 = a^2 + b^2... The Pythagorean Theroem! :)

Trigonometry - Sine Law (Law of Sines)

The trig functions that I've discussed so far (Sine, Cosine, and Tangent) will be incredibly useful to you when working specifically with right angle triangles. However, of course, not all triangles have a 90 degree angle in them. So can you still use these functions? Well, yes, but in a different way. One way is through application of the Law of Sines.

Let's consider a triangle that has three different angles, none of which are right angles. The standard naming scheme still applies, although now there is no hypotenuse (remember, the hypotenuse is opposite to the 90 degree angle).

But what is sine law?  The trigonometry sine law says that the ratio between the sine of an angle and the side opposite to it will be equal for all three angles. In other words:

a/SinA = b/SinB = c/SinC

Since you only really work with 2 ratios at a time, with a little rearranging, you can see that:

aSinB = bSinA

So let's try some sine law examples. Say we have a triangle with 1 known angle of 40 degrees, sides 4 and 6 units long. Find angle B:

The Sine Law says that the ratios of angle to opposite sides will be equal. With this, we know how to find an angle using sine law.  So we have:

6 x (Sin 40) = 4 x (SinB)
SinB = 0.964
B = 74.62 degrees

Now, if the question had asked to find all the angles in this triangle, you can simply say that since the 3 angles in the triangle will add up to 180, you can just subtract your known angles from 180 to get the third angle:

180 = 40 + 74.62 + C
C = 65.38

And now that you have angle C, you can use the Law of Sines with that angle to solve for the final unknown side:

cSinA = aSinC
c x (Sin 40) = 4 x (Sin 65.38)
c = 5.66

Whenever you are given a triangle that does not include a right angle, but you are provided with 1) an angle, 2) the length of the side opposite to that angle, and 3) any one of either the other sides or angles, then you can use the Sine Law.  Now you know how to find an angle using sine law!

Trigonometry - Tangent, SOHCAHTOA

So far, I've explained the concepts of Sine and Cosine... the third basic trigonometry function is TANGENT. If you've been following along and understanding those lessons, tangent isn't anything new for you. The tangent function of a right triangle relates an angle of the triangle to the ratio of its opposite side and its adjacent side. It does not refer to the hypotenuse at all. Referring to the same triangle we've been looking at, you can therefore see that:

TanB = b/a

In general terms, TanB = opposite/adjacent. We write "Tan" as the shorthand form for tangent.

Naturally, SOH CAH TOA helps us remember tangent as well. As you can probably guess by now, the "TOA" component stands for "Tangent is Opposite over Adjacent."

Again, try it out for yourself to see how all these functions work. See that in this triangle:

TanB = 3/4, so...
B = 36.87 degrees

These are the basic definitions for the standard trig functions, and they all work the same way to provide you with the same measurements for the angles or sides... the only things that are different are the sides that are being included in the ratios! So, pay attention to what sides you are looking at!

Trigonometry - Cosine, SOHCAHTOA

In continuing with my trig homework help series, this post will explain another of the basic trig functions, the cosine function.

Solving trigonometry problems that use the cosine function is extremely similar to the sine function explained in the last post. The cosine function relates an angle of a right angle triangle to the ratio of its adjacent side and the hypotenuse. The ADJACENT side to the angle is one of the sides that makes up the angle, but is not the HYPOTENUSE (the longest side of the triangle). Referring back to the same trig triangle we've been working with, it can be seen that side "a" is adjacent to angle B, and so:

CosB = a/c

In more general terms, CosB = adjacent/hypotenuse. We write "Cos" as the shorthand form of Cosine.

SOH CAH TOA helps us to remember the cosine function as well. The "CAH" term is short for "Cosine means Adjacent over Hypotenuse."

Take a look at the previous labeled trig triangle and see for yourself how the cosine function works. If you can do the sine function without problems, cosine shouldn't give you any difficulties either in your trig homework. Once again, it is absolutely important to understand the relative namings of the sides.

Trigonometry - The Sine Function and SOHCAHTOA Explained

Solving trigonometry problems can be easy, but it first requires you to have a solid understanding of the basic trig functions.  There are three of these functions, and the first one I will discuss is SINE.  I will refer to the triangle pictured in the previous post.  In subsequent posts, I will highlight the other two functions: cosine and tangent.

The sine function relates an angle of a triangle to the ratio of its opposite side and the hypotenuse. So, if we look at angle B (and keeping in mind the notation for sides), we can see that:

SinB = b/c

In more general terms, SinB = opposite/hypotenuse. We write "Sin" as the shorthand form of Sine.

Similarly, SinA = a/c (again, it is opposite/hypotenuse... remember how I said it was important to understand the RELATIVE notation scheme!).

SOHCAHTOA is the trig acronym that describes the three basic functions and their ratios. It may be easier to see if you look at it with spaces: SOH CAH TOA. The one that we are interested in this post is the SOH term, which is short for "Sine is Opposite over Hypotenuse."  There are many other mnemonics that can be used to remember the order of these relationships, as you can see by the multiple comments below.  Thanks to everyone who submitted something!  As you can see, there is no one right way to have soh cah toa explained.  Whatever works for you to help you remember is all that is important.

Working with the Sine function is fairly straightforward, and usually just a matter of plugging the appropriate numbers into the ratio.

For example, you can imagine a triangle with known side lengths, and be asked to find the angles. This is a very common and simple question that you will get.  In this case, you would say Sin(B) = opposite/hypotenuse (where you substitute in the known values for the sides.) This will give you Sin(B) = "some value." And now, just as in working with addition or multiplication, when you want to solve for a specific variable, you have to isolate it... and to isolate it, you must do the same thing to both sides. Therefore, to get rid of the Sine, you must do 'inverse sine' to each side (which is usually the same calculator button, but pushing SHIFT to access it). Then you'll get B = inverse sine of (some value), which is your answer.  Technically, you are taking the arcsine of the ratio, though I will cover than in a future post in more detail.

Similarly, if you know a mixtures of some of the sides and some angles, you may be asked to find the unknowns. You can then say Sin(known angle) = opposite/hypotenuse (where one of these sides is known and the other unknown), and then just simply solve for the unknown side length.

Here's a quick triangle example:

So from this triangle, we can tell that:
SinA = (4/5)

SinA = 0.8 (now push inverse sin on your calculator...)
A = 53.13 degrees

Also:
SinB = (3/5)
SinB = 0.6
B = 36.87 degrees

A useful trick for quickly solving triangles is to understand that if you sum up the three angles, they will always total 180 degrees. So for this triangle, after solving angle A, you could subtract it and 90 from 180 to find B. Of course, this defeats the purpose of practicing Sine in this example... ;)

On the other hand, if we already knew that angle B = 36.87 degrees, and we wanted the length of the unknown hypotenuse, then we do:
Sin(36.87 degrees) = 3/hypotenuse
0.6 = 3/hypotenuse
hypotenuse = 3/0.6 = 5

As you can see, there really isn't anything complicated about performing these steps. For the most part, it's simple arithmetic with a few things about triangles thrown in for good measure.  I hope this explains the sine function so that it is understandable, and I also hope that now that you have had sohcahtoa explained, that makes more sense too. As always, please don't hesitate to comment if you're unsure or if you would like additional help. I'll discuss cosine and tangent in the next few posts.

What is Trigonometry?

Try asking any number of young math students the question "what is trigonometry?"  Inevitably, you will learn that it is one of the most feared subjects in math that students have to learn.  However, solving these questions and equations is much easier than they often give it credit for.  (For some interesting background history and applications of this discipline in mathematics, wikipedia has a good article.)  On this page I'm going to go over some of the basics to get you started.  Some of my following posts will go into more details on the specific functions, so be sure to check those out as well.  And please remember to hit the Like button and/or +1 button if this helps you at all.

Trigonometry Basics

Trigonometry, which literally means "triangle measurement," deals with the relationships between the angles and sides of triangles.  For the purposes of explaining the trigonometry basics, this is going to specifically deal with triangles that have one 90 degree angle (right-angle triangles).  These particular kinds of triangles have important distinctions to point out regarding their nomenclature.  The longest side, which is always the side that is directly opposite to the right angle, is called the HYPOTENUSE.  There is a common triangle notation of naming the sides and angles, as explained below, for which I will refer to this diagram:

The first concept that is essential for you to really understand is how to name a triangle.  The standard convention of naming triangles is to name them by the letters of their 3 corners. So in this example, this is triangle ABC. Similarly, the sides are named by the corners at either end of the side, so here we have sides AB, BC, and AC. The angles may be named with a single letter, or designated by 3 letters representing the points that make up the angle, in order, with the corner point in the middle (ie. angle B = angle ABC, angle A = angle CAB, angle C = angle ACB).

An alternate way to look at triangles is to name the ANGLES with capital letters. Then, the side that is directly opposite the angle is given the same letter, but in lower case. Therefore, from the above example, side AC = side b, AB = c, and BC = a. Somewhat confusing, but not really... it makes sense. The keyword in this method of notation is OPPOSITE, which tells you that the angle and the side opposite to it are related.  If this doesn't quite make sense to you automatically, you can also consider that the side c is the only one that doesn't help make angle C.  Check it out and see for yourself that this concept applies to all of the angles in any triangle.  If you can understand this basic naming scheme used in triangle problems, solving trigonometry questions will be that much easier.

In working with triangle sides and angles, you will also need to be able to recognize their RELATIVE positions.  So, in addition to the hypotenuse, which can easily be identified, the other two sides can be labeled relative to the angles.  So, this means that for angle B, you could be interested to label its OPPOSITE side and ADJACENT side. The opposite side is easy to identify: it is the side that does not touch the corner you are looking at. The adjacent side equally easy, though care must be taken to not mistake it for the hypotenuse: it touches the corner, but is not the hypotenuse.  Also recognize that in a right angle triangle, the two acute angles will always be created by an adjacent side and the hypotenuse.

A solid understanding of these trigonometry basics, including triangle notations and the naming of angles and sides (in particular, the relative naming designations) is important before you can move on to confidently do calculations involving these measurements. These calculations in themselves, however, are very straightforward, and as will be explained in separate posts, may be figured out very easily by remembering just one word - SOHCAHTOA.  This is one of the most important key points there is to memorize on this topic!  Make sure you check out some of my other posts (specifically, my lessons about sine, cosine, and tangent, where you will find sohcahtoa explained simply).

I really hope that I'm doing a decent job at explaining trigonometry easily.  Please hit the Facebook Like or +1 buttons to recommend my post if you found any value in it, and leave me some feedback in the comments below - especially if I didn't answer "what is trigonometry" well enough for you.

Why does "m" represent slope?

Why is the symbol for slope "m"?

Few people know how to answer this question properly, which doesn't help when dealing with inquisitive students!  There actually isn't a definitive answer to this question, and scholars are still looking for its first use!  Unfortunately, that likely won't cure any lingering curiosity.  And, frustratingly for teacher, this bit of mathematics history won't actually help students solve their graphing problems any easier, though it probably won't stop them from asking about it!

Some records indicate that it first appeared in print in the mid-1800's, in a geometry paper by a British mathematician named Matthew O'Brien, though other records suggest it dates back even further to Italy in the mid-1700's by Sandro Caparrini.  There is also a suggestion that it originated in the US (interesting email transcript here).  One theory suggests that because "slope" used to be called "the modulus of slope," they shortened this to just "m", though it is difficult to find evidence to back this up.  Similarly, the m may come from the words for "mountain"... in Latin it is mons and in French it is montagne.  There also is no evidence to support the myth that the French word for "to climb," monter, provides the "m." To dispute these suggestions, however, is the notion that the famous French philosopher Rene Descartes never recorded slope as m.  One would think that if the origin was from a French word, a noteworthy French scholar would have used it!  An alternate theory, which seems rational to me, is that the mathematician M. Risi suggested that the early letters a, b, c, etc are used to represent constants, the later letters z, y, x, etc are used to represent unknowns, and the middle letters represent parameters.  Therefore, since slope is considered a parameter, it may have arbitrarily been assigned this mid-alphabet designation.  For all we know, it was a practical joke that got taken too far and is now an accepted concept of the mathematics of graphing!

Whatever the true origin of this seemingly strange symbol, it won't provide any insight into easier ways to work with slopes, lines, and graphing!  So, the next time someone asks this question, you don't have to feel bad to say "I don't know and I don't think anyone else does either.  Now go do your homework."  :)  If you enjoyed reading this short post, please do me a favor and click the +1 button below to share this wealth of information!

Functions - Complex, Piecewise

The last post introduced you to functions and how they work, but there are few other things worth discussing. The first is more about function notations.

When I introduced functions, I referred to the standard notation as f(x). However, you will quickly come across problems that use different letters, but mean the same things. You will probably see things like g(x) or h(x), and they are just used to differentiate between different functions. An example will probably clear this up:

f(x) = 2x + 3
g(x) = x - 8
h(x) = 4x + 1

That shows that function f (or g, or h) is equal to the given expression. So f(x), g(x), and h(x) all represent different functions. It's the equivalent of saying 3x = 12, or 5a = 10... the math is the same, just with different letters.

So, with multiple expressions, we can do slightly more complicated things. If we say solve f(1), we substitute 1 in for x, and solve it to get 5:

f(x) = 2x + 3
f(1) = 2(1) + 3
f(1) = 5

However, we can now ask for something like f(g(x))... read as "f of 'g of x'." Looks complicated, but keeping in mind what the function notation is saying, we can easily solve this by substituting the expression for g(x) in for x (instead of just subbing in a single number like before):

f(x) = 2x + 3, g(x) = x - 8
f(g(x)) = 2(x - 8) + 3
f(g(x)) = 2x - 16 + 3
f(g(x)) = 2x - 13

Furthermore, if we wanted to solve f(g(5)), we can do one of two things. We can evaluate the expression as we just did, then solve when x= 5... or we can solve g(5) first, then substitute that into f(g(x)):

Method 1:
f(g(x)) = 2x - 13
f(g(5)) = 2(5) - 13
f(g(5)) = (-3)

Method 2:
g(x) = x - 8
g(5) = 5 - 8
g(5) = (-3)...
then...
f(x) = 2x + 3
f(g(5)) = f(-3) = 2(-3) + 3
f(g(5)) = (-3)

Both methods get the same answer, because they are doing the same things. It's just a matter of what you are more comfortable with. Function notation can get very confusing like this, but for problems like this, you just have to basically work from the inside out, just as in any other math expression you've seen! (eg. (2x + 7(3x -4) + 2).... solve the inside bracket first, then the outside). They may seem tricky, but you will be surprised at how easy they are when you see through the notation and know what they are asking!

Another point to discuss is functions that are not described by a single equation, but remain continuous. Remember back to what we said a function is... a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. Also remember the vertical line test. But we never said anything about it being entirely expressed by one equation! So then, we can say that the following is also a function:

Obviously, it is composed of two straight lines, each having their own equation. This is called a PIECEWISE FUNCTION. Since it is just one line with two distinct 'pieces,' it is a single function, not two, and it is written as:

That says that f(x) is described by two separate equations, over the domains of x stated after each equation. Given this graph or equation, if you were then asked to solve f(2), you must look to see within which domain x = 2 falls, and evaluate that expression. For this example, f(2) is in the top equation, so f(2) = 3. You can check by looking at the graph to see that when x = 2, the function evaluates to (y equals) 3. Alternately, f(4) is within the bottom expression, so f(4) =3/2.

Try to remember that as difficult and tricky as some functions look, they aren't that difficult if you relate to them what you know already about solving math problems and you pay attention to what is being asked! Remember, if you need something clarified further, don't hesitate to post a comment!

Functions - Domain, Range, Vertical Line Test

Functions may seem like a totally foreign and incomprehensible concept at first, but they're not really the monsters that many students think they are. The notation may seem a little weird from what we're used to, making them look difficult, but when you understand the concepts of what is being presented, you will see that they're not really doing anything that doesn't make sense.

To give a very simple definition, a function is basically just a way of relating two variables... although their notation makes it look otherwise. But we've already seen something like this before! If we take a regular X-Y graph and draw a straight line on it, the equation of the line is the way of relating the x variable with the y variable. However, the standard way of writing a function is to write f(x) instead of y. We say it like f of x. So for example, taking a line, we can do the following:

y = 3x + 5
f(x) = 3x + 5

The function notation says that "the whole expression with x on the right" evaluates to some number (the equivalent of y). The math is still the same, no tricks with what you can do with the expression. It's just a different way of describing things. But we also now have a way of saying "tell me what that thing evaluates to when x=10" and we can write that out like this:

f(10) = 3(10) + 5 = 30 + 5 = 35
f(10) = 35

Again, it's essentially the same thing as saying 'solve for y' from our graph equations, but you can't say to solve y = 3x + 5 at a certain point for x, without actually writing that instruction out in words. With function notation, you can say to solve the expression when x=10, 50, or 25,000, just by writing f(10), f(50), or f(25,000).

There are a few other terms you will likely see when dealing with functions. Domain refers to all the values of x that are valid for the expression. Range refers to all the values of y in the expression. Domains and ranges could be infinite, as in the case of a straight line that goes forever in either direction, or they could have defined numbers as in line segments. The notation for domain and range sets is like [x₁, x₂] or [y₁, y₂], where those numbers represent the two extremes of the domain (furthest left and right) or range (highest and lowest), and the square brackets indicate that those numbers are included in the range. (If you had an 'open' point on a line, this indicates that that exact point is excluded from the line, and if this occurred as an endpoint, you would write (x1, x2) to denote that the number is not included in the set. Also, you could have on the line a closed and an open point, in which case you'd have something like (x1, x2] or vice versa.)

The domain (the x-values) for line segment A is [-8, 1] and its range (y-values) is [5, 8]. In this case the domain and range extremes are the actual end points of the line, but pay attention to the particular question as this is not always the case. Remember, this notation is defining the domain and range (all the values that x or y could take), not the end points of the line.

Similarly for line segment B, the domain is [2, 7] and the range is [-3, 6].

Line C demonstrates a very important concept about functions. Line C is not a function. We should expand our definition of a function to say that it is a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. That means, that for every x, there can only be one y. To test this on a graph, you can use the vertical line test to prove a line is a function or not. If you draw a vertical line through your function at any value for x, the vertical line will pass through the function line ONLY ONCE. If it passes through it more than once, then you are not looking at a graph of a function. Therefore, looking back to line c, you can tell that a vertical line would pass through the line twice at some positions, and so this proves it is not a function of x. Similarly, the circle in D does not pass the vertical line test, and so is not a function either.

These concepts may seem different, but you will become accustomed to them. Just remember our definition of what a function actually is, and what the notation is saying, and you will be fine. When you begin to grasp these concepts and advance to higher levels, you will get to see some really cool stuff with functions. (sneak peek.... f(x) = x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.... it's still a function!)

If any of that isn't clear and you'd like something more explained or clarified, please don't hesitate to leave a comment for me! I'm always glad to help!

Graphing - Standard Form of the Equation

Just a short explanation for what is meant by "standard form" of the equation of the line. We have been looking at line equations in the form of y=mx+b. However, you may be asked to express this in standard form, or as a standard form equation.  Graphing standard form equations will give you the exact same line as graphing something expressed as y=mx+b... standard form is just a different way of displaying the equation.  (Please hit the Like button and/or the +1 button if this post is helpful for you!)

The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we've been looking at but in a different position from what we recognize.

To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let's look at some examples:

Given that y = 3 x+ 5, standard form of this is 3x - y = (-5).

Given y = (1/2)x -15, standard form of this is (1/2)x - y = 15... also, if you don't want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x - 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000... looks different, but it's not! Reduce it down and see for yourself!)

For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it's still quite straight forward.

Given 5x - 15y = 10, you just have to rearrange things to get y by itself on one side:
(-15y) = (-5x) + 10
y = (1/3)x - (2/3)...
and then you can see it is a line with slope 1/3 and y-intercept (-2/3).

Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably.  Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations.  They just look different because the numbers are rearranged.  This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything.  You do not have a new line.

Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren't any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent... exponents make the graph do cool things later), it is automatically a straight line.

(check this post for some additional pointers)

If you liked my post, you can help me spread the word by tweeting about it!  Also, please remember to click the Like button or +1 if it helped!

Graphing - Parallel and Perpendicular Lines

How can you determine if two lines are perpendicular?  How can you determine if two lines are parallel?  If you have two lines on a graph, and you have determined their equations or slopes, you may be asked if the two lines are parallel or perpendicular to each other.  These are two favorite questions of teachers and you will undoubtedly have to answer them!  Keep reading to find out how to easily answer them, and please remember to click the +1 button at the end.

Parallel lines are at the same angle and will never cross... like two railroad tracks. It doesn't matter what direction the lines travel. As long as they are going the same way, they are parallel. In mathematical terms, two lines are said to be parallel if they have the exact same slope.

Remember our equation for a line, y = mx + b.  Two parallel lines, each defined by their own equation, will have the same value for m, the slope.  So, y = 3x + 5 and y = 3x + 200 are parallel lines (they differ in their y-intercepts, but they have the same slope m).  You can plot this out for yourself quickly to see that this is indeed the case.

The opposite of parallel lines are perpendicular lines.  But how can you determine if two lines are perpendicular?  Perpendicular lines have a bit of a twist to them. Two lines are perpendicular if they cross (remember, any two straight lines that are NOT parallel will cross at only a single point.  They cannot ever intersect again unless they curve back on themselves, in which case they are not straight!) and they form a 90 degree angle, or rather, a T-shape. For example: The x-axis and y-axis are perpendicular to each other. Mathematically, if line 1 has a slope of m₁, then a perpendicular line 2 will have a slope m₂=(-1/m₁)... that is, it's slope will be the negative inverse of the first.

Try it out... y = 2x + 1 and y = (-1/2)x + 5... m₁ = 2 and m₂ = (-1/2). Check it out on the graph to see that they indeed form a 90 degree angle where they intersect.  If you don't believe that this is correct, go get a protractor and measure it for yourself!  You can then convince yourself that the relationship holds true.

Perpendicular graph

Also of interest to you: the symbol for "perpendicular" is ┴, an upside-down capital T, whereas the symbol for "parallel" is two vertical lines next to each other, like ||.  You would write AB┴CD to say that AB is perpendicular to CD, and similarly AB||CD to say that they are parallel.

So, I've shown you that to in order to determine if lines are parallel or perpendicular, all you need to know is their slopes!  If you know the equations of the lines, then this is only a matter of simply reading the m value, or at the worst, performing some basic mathematical rearrangements to find this info.  And once you have the numbers, you can easily see if either relationship applies.  This is much faster than manually creating a table of values and plotting out the graphs by hand!  Now, you should have no problem answering questions that ask you to quickly identify true parallel or perpendicular line relationships!  If this post helped solve your questions, please remember to do me a favor and click the +1 button below to help me get the word out!  Thanks!

Graphing - Equation of the Line

If you are given a line on a graph (or enough information to construct a line), you will likely also be asked to find the equation of the line. The equation of the line is unique to each line; that is, every line has a different equation. With it, you can readily tell the slope of the line, and you can calculate what the x-value is for any y-value on the line (and vice versa). It is very handy!

The most common and basic form of the equation of the line is:

y = mx + b

where m is the slope and b is the y-intercept (where the line crosses through the y-axis).

Another way to write it, which is more general is called the POINT-SLOPE FORMULA:

(y-y1) = m(x-x1)

where m is the slope, and x1 and y1 are the coordinates of a known point on the graph. (If you pay close attention, you can see that this way of writing it is really the same as the slope formula, but rearranged!) The version basically says 'give any point at all, and a slope, and you have enough information to draw the line."

Let's use the graph from the slope lesson to practice, using points (2,1) and (7,7). To get our final answer, we are going to have to do a couple of steps first. Let's use the y=mx+b equation. The steps we are going to do are:
1) Find the slope
2) Find the y-intercept
3) Write the equation of the line

Finding the slope is easy now, if you read the posting on slopes (I always leave numbers as fractions, instead of changing to decimals):

m = (y2-y1)/(x2-x1)
= (7-1)/(7-2)
= 6/5

The y-intercept is easy now... just plug numbers into y=mx+b, and solve for b. So, b=y-mx. We have our slope now, and for y and x, we just substitute in the coordinates of a single point (x and y MUST be from the same point!)

b=y-mx
= 1-(6/5) x 2
= 1-12/5
= (-7/5) (you have to do some fraction math!)

So now we can write the equation of our line!

y = mx+b
y = (6/5)x - (7/5).... (if we leave it like this, we can read the values for slope and y-int right from the equation!)
y = (6x - 7)/5

If we work through using the other formula... (y-y1) = m(x-x1)... we get the same answer, and we don't have to explicitly solve for b! First, solve for slope. Second, substitute in values for slope and x1,y1, and then rearrange!

slope = 6/5 (same thing we did above)

so (y-y1) = m(x-x1)..... (sub in slope, and point (2,1) for (x1,y1))
(y-1) = (6/5)(x-2)...
y=(6/5)(x-2) + 1...
y = (6/5)x - (6/5)2 + 1...
y=(6/5)x -12/5 + 1...
y=(6/5)x -(7/5)... (same slope and y-int)
y=(6x-7)/5

Same answer! That is the equation of the line. Now if you put in any value for x, you can say exactly what the value for y would be... if you wanted to know what y is when x is 1000, you can do that now! You will see that it will always be on the same straight line.

If we work through and do the same thing for the red line on the graph, with points (-6,2) and (-4,-5), we can get the equation for that line too! Let's try it this time using just the one formula.

(y-y1) = m(x-x1)
((-5)-2) = m((-4)-(-6))
(-7) = m(2)
m=(-7/2)

Now put that back into the same formula along with a point, and rearrange:

(y-y1) = m(x-x1)
(y-2) = (-7/2)(x-(-6))
y-2 = (-7/2)(x+6)
y= (-7/2)x + (-7/2)(6) + 2
y = (-7/2)x -21 + 2
y = (-7/2)x -19

(see slope (-7/2) and y-int (-19)... look at the graph, and you will see that makes sense! Negative slope, very low y-int!)

That's all there is to it! Just remember that there are a few steps to follow, depending on the formula you are using... basically, remember to always find the slope first, then the y-intercept, and plug directly into y=mx+b... or use the point-slope formula to find the slope using 2 points, and then resubstitute it back in with a single point and rearrange. It's not that complicated once you practice and understand what you are doing! Either method is going to give you the same answer, so pick your favorite and stick with it!

Graphing - Slopes

When you have a line on a graph, you will probably need to know its SLOPE. A slope on a graph means essentially the same thing as if you were talking about the slope of a driveway or the slope of a ski hill... it is a measure of how steep the driveway/hill/line is.

The minimum amount of information you need to find the slope of a line is the location of two points on the line. These could be endpoints for a line segment, or just points on a line that goes on forever. Since it's a single, straight line, ANY two different points that are on the line can be used and will give you the exact same slope as any other two points on the same line. Makes sense right? The slope is a property of the line, and all the points on the line make up the line.

The slope formula is easy to remember. The complicated way of saying it is 'the slope is the difference in height of two points on a line, divided by the difference in width of the same two points.' The much easier way is 'slope equals RISE OVER RUN.'

slope = rise / run

The rise (think rising = height) is the difference of the y-coordinates of two points on a line. The run (think about running on the street = horizontal) is the difference of the x-coordinates of the same two points.

Usually, slope is represented by the letter 'm.' So then, the slope equation can be written like:

m = (y2-y1) / (x2 - x1)

This appears to be a little more complex than the first one, but it means the same thing. The 1 and 2 are just names for the y's and x's. (They could be anything... A and B, or whatever.) The (y2-y1) means 'the difference between two y-coordinates, and the (x2-x1) means 'the difference between two x-coordinates.' IMPORTANT: Make sure that the point you use for y2 is the same point you use for x2, and likewise for y1 and x1. (Otherwise, you'll get the wrong slope.)

Another IMPORTANT thing to recognize: if the graph rises to the right, it is said to have a POSITIVE SLOPE. If it is falling towards the right, it has a NEGATIVE SLOPE. That means your slope value will have a positive or negative sign with its number. You can check that when you're done. (It's easy to make sign errors. It happens to everyone.)

This figure should hopefully clear things up.

Let's look at the black line first. Let's call the top point 'point 1' and the bottom point 'point 2'. So:

rise = (y2-y1) = (7-1) = 6
run = (x2-x1) = (7-2) = 5 **Notice that I didn't use (2-7)!

slope = rise/run = 6/5 or 1.2

That's it! Let's look at the red one now. It's a little trickier because of the negative signs, but you do the exact same thing. Point 2 on the left, point 1 on the right:

rise = (y2-y1) = (2-(-5)) = (2+5) = 7
run = (x2-x1) = ((-6)-(-4)) = ((-6)+4) = (-2)

slope = rise/run = 7/(-2) = (-7/2) or (-3.5)

Notice how the first graph had slope of (positive) 1.2 and was going up to the right, and the second one had slope (-3.5) and was falling to the right.

As long as you keep y2 and x2 coming from the same set of coordinates (and y1, x1), and you keep track of your signs, you'll get the right answer for your slope! And with the slope, you can do more complex things, like find an EQUATION that describes the line.

Graphing - Points, Ordered Pairs

The first graphing concept to understand is x-y coordinates. A POINT has both an x-coordinate AND a y-coordinate. The x-coordinate is how far left/right the point lies, and the y-coordinate is how far up/down. A single point is determined by both how far left/right and up/down it is. Together, the x- and y-coordinates of a point are called an ORDERED PAIR. Every point is represented by a different ordered pair. (If they were the same, they'd be in the same place!) The standard way of writing an ordered pair is (x,y). Also, the point itself is often named with a letter and placed before the ordered pair... like M(x,y) for point M that has x- and y-coordinates.

Refer to this graph for the following examples

So, for point S with x=3 and y=5 , it will be written as S(3,5) and will be 3 units to the right and 5 units up.

For point T with x=(-2) and y=1, it will be written as T(-2,1) and will be 2 units to the left (pay attention to the negative sign) and 1 unit up.

Point U(-4,-4) is 4 units left and 4 units down (two negative signs this time).

V(2,-5) is 2 units right and 5 units down.

Try for yourself... where would you draw points (1,1) and (-5,7)? What about the two points on the graph that don't have any labels... can you figure out what they should be? Let's call the point on the right W, and the point on the left N. Easy, right? They are W(8,1) and N(-8,7)! All you have to do is count in the proper direction, and you know where your point goes and what its coordinates are! Once you are familiar with how points are labeled and how their positions are calculated, you will be able to work with these numbers to do all sorts of things!