Friday, March 1, 2013

Top 5 Most Popular Posts of February 2013

Another month has come and gone, so in trying to maintain my new year's resolution, here is my monthly summary of my top 5 most popular posts of February 2013.  This is tabulated simply from the number of visitors each page on my blog receives.  I have recently started a Facebook page for my site as well, so eventually I would like to have these "best of" posts reflect the most shared or most liked content.  For now though, I don't have enough traffic to compile meaningful stats.

This is a great opportunity for anyone to catch up on some of my more popular pages, if they haven't been following along from the very beginning... which I'm sure is most people.  ;)  So, here we go.  My top 5 most popular posts of the month!

  1. Converting Point-Slope Form to Standard Form.  This is easily my most visited post in the history of my site, and this month it was back on top again.  Choosing the proper way to express your equation of a line is important, so learning how to convert between point-slope form and standard form is crucial!

  2. Stretching and Compressing Graphs.  When you first learn how to graph functions, you are amazed that numbers actually mean something.  Then, with this concept, you add on top of that the ability to modify your image by stretching it or squashing it.  This post discusses the all you need to know to perform these modifications on your graphs.

  3. Trigonometry - Secant, Cosecant, Cotangent.  These trigonometric functions are the less famous variants of the ubiquitous sine, cosine, and tangent.  There isn't anything special about them, you just have to understand their connection to the main three functions.  In this popular post, I explain them and show you what their symbols are so that you won't be confused anymore!

  4. Which Measure of Central Tendency to Use? Mode, Mean, or Median?  This is one of the first things that is discussed in statistics courses: how to measure the center of a data set.  Of course, there are different ways of representing the center.  So, this post gives some pointers on which measure is appropriate for various types of data.

  5. Special Angles in Trigonometry.  There are a couple of special angles that are very important to trigonometry.  They are easy to memorize, and will make other trig work seem much easier.  They are all based on triangles, and if you follow the tips in this post, you'll have all you need to know to tackle trigonometry questions.

To find more great explanations and discussions of math concepts on my site, browse the Math Concepts Explained table of contents.  Alternately, you can enter your topic of interest in the search bar at the top of every page.

If you enjoy Math Concepts Explained, I invite you to join the many other students, teachers, and math enthusiasts who follow my site:
Thanks to all of my visitors for your support!

Monday, February 25, 2013

Understanding Point Slope Form

When you are graphing straight lines, one of the most common formats for describing the equation of the line is called "point slope form."  In this representation, the equation identifies one ordered pair that is on the line, and the slope.  If you were given only those pieces of information, you would have all that you would need to construct the line.  Continue reading to learn more about this line graphing concept, and make sure that you Like my post if it is helpful to you!

By using this formula...
- if you know one point that is on the line (x1, y1),
- and you know the slope of the line (m),
... with a little bit of mathematics and algebraic rearrangement, you can determine any other point (x, y) on the line.

Here's one type of problem that you will likely encounter:  Express in point slope form the line that passes through (4, 2) and has a slope of 8.  

To correctly solve this problem, all that you need to do is substitute the given values into the equation shown above.  Therefore, the correct expression is y - 2 = 8(x - 4).  It's as easy as that!

A more complicated problem would be something like this: What is the y-coordinate when x = 3 on the line that passes through (1, 1) and has a slope of 5?

To successfully work through this problem, you first approach it as you did the previous one.  That is, find the equation of the line.  In this case it is y - 1 = 5(x - 1).  Now, to find y when x = 3 is as simple as subbing in x = 3 into this equation, doing a bit of rearranging, and simplifying to isolate y.  Try it out for yourself, and you will see that y = 11 when x = 3.  In other words, the point (3, 11) is on the line that is described in the question.

Now that you've seen a few questions that can be asked about point slope form, perhaps it might help you to better understand the concept if you see what it actually means.  First off, consider all of the variables that are included in the expression:

We have an ordered pair (x1, y1), an unknown point (x, y) that can be any point on the line at all, and the slope (m).  Now, my question to you is: where have you seen these variables together in one place before?

If you answered that this is a rearrangement of the slope formula, then you get a gold star!  As we've seen before, the slope of a line is equal to rise over run.  In other words, the slope corresponds to the ratio of the change in vertical height to the change in horizontal distance of the line.  Mathematically, here is what you this means:

So, really, all we're really dealing with for any of this is the definition of the slope!  The point slope formula is just a different way of looking at it.

Now, it should be said that this may not be the most intuitive way of representing an equation of a line. Many people find expressing their equation in the form of y = mx + b to be more familiar and descriptive, since by definition it denotes the slope and y-intercept.  The intercept is a very easy "starting point" from which to extend your line, and with the known slope, is it simple to count spaces to plot another point.  Whichever way you express the equation of your line, assuming that the math is correct, they are just different ways of describing the same thing and so they aren't wrong... of course, unless your teacher specifically asks for one form or the other.

If you think about it, you should be able to see the connection between "point slope" and "slope intercept" forms.  Consider that in y = mx + b, the b term is the y-intercept (or y1), which means that x1 = 0.  So, you can say that:
y = mx + y1
y - y1 = mx
Looks familiar, except it is missing the x1 term... but we already designated that it is 0 anyways!

Another way to show that these two formulas are the same thing is to equate them to the same variable, which therefore means the equations are equal.  This is like solving a system of equations.

If we solve for m in the point slope equation, we have: m = (y - y1) / (x - x1).
If we solve for m in the slope intercept equation, we have: m = (y - b) / x.
Since we're talking about the same line, it obviously is the same slope in each version, so it is fair to equate them. So:
(y - y1) / (x - x1) = (y - b) / x
Written like this, it is easy to see how the two equations correspond to each other.

A third way of representing a line's equation is to express it in standard form.  What this does is put everything on to one side of the expression, simplifies it, and sets it equal to 0.  Your expression will then have the form Ax + By + C = 0, where A is the "x" coefficient, B is the "y" coefficient, and C is the constant not associated with either x or y.

The whole point of all of this is to make a very simple point: all you need to fully describe the equation of a straight line is its slope and a point on it.  If you only have two points given, it is easy to calculate the slope from the slope formula, and then it is only a matter of plugging numbers into whichever expression you like.  If you only have the slope and no points, then you have the shape of a line but no point to which you can anchor it.  Get the slope, get a point, pick a way of expressing the equation of your line, and that's all there is to it.  If you made it all the way to the end, please remember to click the Like or +1 buttons (or both) below if you enjoyed this post!  Thanks!

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