The Government Holds the Whip

We all believe nowadays that slavery is wrong. People own themselves, and they have a right to their own life, liberty, and property. To forcefully take that away from someone is wrong.

But when slavery still existed in many countries like the U.S., England, and Brazil, many people argued against the abolition of slavery. Here are a few of their remarks:

Slavery has always existed because there cannot be a strong society without it. In this argument, they are saying that there are some things that have to be done that a normal person will not do without force. Without force, the slaves’ lives and the slave owners’ lives would fall apart.

Slaves are not capable of taking care of themselves and without a master, they would die off. This was actually a pretty popular argument. But now there are many blacks whose ancestors were slaves who are now quite well off with a good education and everything. People are smart, and this applies to all races. They will figure it out.

The abolition of slavery would inspire bloodshed. Slaveholders would resist. When the American Civil War took place, there was a lot of bloodshed. Many people believed that this confirmed this argument, but most countries abolished slavery peacefully.

Slaves are uncivilized, and freeing the slaves would allow them to run around causing mayhem. Again, we see that there many blacks who have jobs and places in society. A master was not necessary to control them.

Abolishing slavery is far-fetched, uncalled for, and altogether impractical. So what if they are treated unfairly here? The world is not fair, and it’s really a lesser evil to deal with. Perhaps these slaves will have a better fate in the afterlife.

See, it may seem strange and kind of an exaggeration to compare slavery to government, but if you think about it, many of the arguments against slavery abolition applies to the arguments against the abolition of government. To paraphrase, people say that yes, government can be oppressive sometimes, but that oppression is a lesser evil that we will have to deal with in order to have protection against invasions, terrorists, and even ourselves. But what if, like the slaves, we figure it all out? We don’t need slavery, and neither did the slaves. Perhaps we don’t need government as much as we think we do.

See http://mises.org/library/ten-reasons-not-abolish-slavery

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Easy Parabolas Lesson 6

Alright so back to this problem: y = x²+6x+6. We tried to make a square out of this, but it ended up like this:

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A bit incomplete. To be exact, we’re missing three ones. So, we’re going to cheat. }:] We are going to add three ones.

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Hero Zero! Now we’re going to cancel out those three ones!

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Okay I know it looks like we did nothing to it, and no, I didn’t change the value. But let’s turn this into an equation and see what we’ve got. So this is an x+3 by x+3 square now, since we’ve added those three ones, plus three negative ones. In other words, y = (x+3)² -3! Familiarity! We now have our equation expressed in vertex form. According to this equation, our vertex is at (-3,-3). Now we can make the graph!

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Using this approach of completing the square, we can convert any quadratic () equation into vertex form, and then easily graph it.

Let’s try one more. Let’s try y = x²+4x+6

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Too many! Okay well this is easy. We don’t need Hero Zero to see that this is (x+2)² plus an extra 2. So the equation is y = (x+2)² +2, and our vertex is at (-2,2). Our parabola looks like this:

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Ta-da! That’s all, folks! Now take a break from parabolas and play with blocks.

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Easy Parabolas Lesson 5

So, completing the square. Let me tell you something about this. Completing the square is literally completing a square! A real square! Here, I’ll show you.

But first I’ll have to introduce you to something. We’re going to play with blocks! Here are the blocks we’re going to be using:

Blocks from 1-10,

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and the 10×10 block, or the 100 block.

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Now see the ten block at the end of  the triangle of blocks right there? Yeah, this one?

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See, the nice thing about this ten block is that it doubles as an x when we turn it on its side. See how it’s blank, without any squares? That means that it can be any length.

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And the 100 block doubles as an x times x when we flip it over.

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Anyway, with that aside, let’s move on to parabolas! Imagine you have your normal, ordinary equation like y = (x+2)². Now (x+2)² of course equals x+2 times x+2. How would that look with the blocks? It’s simple:

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Now, if you’re really tiny and you walk up one side, you walk up a distance of x+2, and if you walk across, you also walk x+2. You can see that the components (or ingredients) of y = (x+2)² are four x’s, one , and four ones. Another way to say this is y = x²+4x+4.  So the equation y = x²+4x+4 represents a square, and each of the terms is a component in the picture of the square!

We can therefore graph the equation y = x²+4x+4 by rewriting it in the form y = (x+2)². The graph of course looks like this (with a vertex at (-2,0)):

graph

Now let’s do something interesting. So let’s say you have an equation like y = x²+6x+6. Again, this is listing ingredients for a square, a parabola. This means that we’re going to need one , six xs, and six. Then we arrange them into a square.

So here are the ingredients …

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And let’s arrange them into a square. Here we go.

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Okay if you were suspicious from the start, good for you. So, this doesn’t make a complete square. So, we’ve got a problem, do we? Well guess what? You’ve just gotta hang tight until next lesson. 😉

Easy Parabolas Lesson 4

Warning! This is a very sad post. Because now, we are going to flip our parabolas upside down!

Sad parabola.

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Sorry, that was my sister’s idea. You can blame your tears on her.

But how on Earth did this sad story come to pass? Well, it all began with a very happy parabola …

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who’s y coordinates all became negative! This makes the dots, which get connected by a parabola, go down on the y axis. But how could this be? If you look at our square numbers here …

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The ys are all positive! And if we make the xs negative …

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the ys are STILL positive! A negative times a negative equals a positive, right? Our parabola is still happy! But we need to make our parabolas sad, because I’m evil.

So, we’re going to cheat. Because, like I said, I’m evil. We are going to make our y negative!

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Muahahahaha!

Okay, that was way easy. But how do we do that in our equation? Simple: y = (x²) Now, y equals negative ! Oh dear, our parabola doesn’t look too happy about that.

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You know what? I’m getting rid of the eyes! This is just too depressing!

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There we go. Now how about we play around with this parabola and move it around a bit? Let’s try y = (x²+7)+5.

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And y = (x-6)²+1

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See? It’s the same parabola, just … well … sad.

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Sorry, couldn’t help myself!

Easy Parabolas Lesson 3

So, during the last lesson we learned about what happens when we add or subtract before we square the x, but what if we add or subtract after we square it? For example, y = +3. Don’t ask me why I like the number 3.

Well, since Area = Side², and x corresponds to the Side, represents the Area, which corresponds to y. So that means that when you add 3 to , you are adding 3 to the y axis. So this means that now our vertex would be here:

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*triumphant fanfare plays*

That was easy. Now we can play around a bit. Let’s subtract 3 from our !  So now, our equation looks like this: y = -3

Now we are subtracting 3 from the y axis, so our vertex should be here:

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Btw, the ends of the parabola continue infinitely, but I’m too lazy to make the lines go out of the  picture, so I’m not going to.

Alright, that was kind of short. So now, just for fun, let’s use the stuff we learned from the last lesson and the stuff in this lesson to move the parabola around anywhere we want! For example, this equation: y = (x-3)²+5.

Lets do the blue bit first. So what we gotta do is see what x has to be for the Side to be zero. Hero Zero! We’re going to need x to be +3 in order to cancel out the -3 and make Hero Zero. Thus, our graph looks like this:

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But we’re not done! There’s still a +5 right outside of our blue stuff!

Well like I said, we are adding 5 to the y axis, because it’s red, so we move our parabola up 5!

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yay! Now let’s go crazy and play with it some more! Let’s try y = (x-8)²-13!

Again, what does x have to be for -8 to be 0?

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And now what are doing to y?

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Ta-da! Go impress your friends!

Btw, if my graph looks a little off, sorry about that. I made this graph, and I loose track of my place very easily.  -_-

Easy Parabolas Lesson 2

Okay so lesson 2! We’re going to learn how to move the parabola from side to side!

For this lesson, the first thing you aught to know is that the parabola really ALWAYS looks like this:

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This is the TRUE FORM of the parabola, and when you change the equation, you are really just moving this parabola around. No new parabolas, no different kinds of parabolas, or anything like that.

See the cross right there at the base of the parabola? That’s the vertex. If you remember from the previous lesson, the Area of the square is orange, and the Side is green. Also, like I said in the previous lesson, the Area corresponds with y, so they are represented with analogous colors. Also, the Side corresponds with x, so those two are analogous. And of course, when you move the parabola around, you move the vertex around with it.

Now, our base equation, which makes your standard parabola in it’s standard position, is this: y = . With this plain and boring equation, our Side matches up with x, and our Area matches up with y. But what if we were to, say subtract 3 from x before we square it? So that means that our equation will now look like y = (x-3)².

Now we just have to do a bit of simple algebra to ask ourself: what does x have to be in order for the Side to be 0? And that, my friend, is how you move the the parabola horizontally.

Let’s put it this way:

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Here it’s easy to see where the vertex should be. When the Side, or x-3, is 0, x is 3, meaning that we should put the put the vertex at 3 on the x axis, like this:

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Yay! Now let’s try more examples, just to make sure you get it. What if we added 3 to x before squaring it?

Well, our equation now is y = (x+3)², so again, to find the vertex, we ask what x has to be in order for the Side to equal zero. Okay to tell you the truth I don’t want to make another one of those if/then graph things, so I’m just going to show you show you another way of doing it. Hero Zero!

See, we need to cancel out the 3 to make 0, like the Side should be, so we use the x. We make the x -3, so it cancels out the +3 and makes 0. Thus we see that x has to be -3 for the Side to be 0, so the vertex is now here:

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Hooray! Now you know how to move parabolas side-to-side! Hope you found this helpful

Easy Parabolas Lesson 1

I know, boring title for this post. But I promise you, this will be a BLAST. I hope.

Okay so the first thing you have to know when doing parabolas is that Area = Side², like a square. The side of a square times itself, or squared, is the area. Another thing you need to know is that y = . Also, the Area corresponds with y, and the Side corresponds with the x. This is why the color for the Area is analogous to the color for y, and the color for the Side is analogous to x. Sorry if that doesn’t help you. I think in colors, so this helps me. Anyway, the thing you have to know is that parabolas are squares.

See, here’s how it works. If x is 0, , aka y, is 0. If x is 1, y is 1. If x is 2, y is 4. If x is 3, y is 9. And so on. So, it looks something like this:

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Now how do we graph it? Well, here we have our ordinary graph …

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And now we see that y is the vertical axis, and the x is the horizontal axis, since they’re the same colors. So, if we put dots where the Side and Area are, we get this:

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now we draw a line through them!

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Ta-da! Half a parabola!

And, since a negative times a negative equals a positive, if all the x numbers were negative, the y numbers would stay the same. Now lets graph that.

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A parabola! See, that wasn’t hard! Now you know that y = is a parabola, and that parabolas are really squares!