At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
So, have you been able to make sense of the question? Do you know what they are asking for? :)
Alright, so I'm guessing you don't know what "end behavior" is talking about?
OK, great! It's always helpful to know where we are starting from. We can always build up our knowledge, but it's best to start on common ground. :)
okay tell me what to do
Before we dive into "end behavior," let me finish the "interrogation" :) and ask if you could identify the leading term? Not trying to embarrass you or anything, just want to make sure I'm using language we both understand.
no, i havent studied any of this.
OK, very good. Thanks for the heads up. Let's start by reviewing what a "term" is. :)
In math, we have two basic operations: addition and multiplication (let's ignore subtraction and division for now, because those are basically just opposites).
Usually, in algebra class we like (or not) to study polynomials: They usually look something like this: 2x^2 + 4x + 5 or this 5x^7 + 32x^4 + 3x + 1
in my case -3x5 + 9x4 + 5x3 + 3
That's right! Notice that our friends addition and multiplication are the "starring actors" here. We have multiplying numbers like the 2 in 2x^2 or the 4 in 4x. Or in our case the -3 in -3x^5 but we also have addition, like the 2x^2 + 4x or the 4x + 5. Or again, the -3x^5 + 9x^4
So is that our leading term ?
Almost there. Wait for the punch line. :) When we multiply numbers, they "feel" closer. Even visually the -3 seems closer to the x^5 than the + sign. -3x^5 + 9x^4.
A "term" is just the piece that is separated by a + sign. So, in our example, we have 4 terms: -3x^5, 9x^4, 5x^3, 3
We like to arrange our terms from "highest power of x" to lowest power. So in our example, -3x^5 is our leading term. That's the first part of your question.
OK, now for that "end behavior" conversation. Have you had to graph anything in class yet?
no, well not in a long time. I'm not to good at math
That's fine, it's a rare person who is good at math automatically. It takes a lot of practice (let me tell you). :P Let's have a quick review of graphing, because that is where this "end behavior" stuff really comes to life.
OK, so let's start with a very simple graph. The graph of 2x.
If I wanted to make this polynomial into a picture, what could I do? It's not very obvious... but one way I could start to understand what 2x means is to plug in some numbers for x and see what the equation does to it.
For example, when x = 0, 2x = 2(0) = 0 x = 1, 2x = 2(1) = 2 x = 2, 2x = 2(2) = 4 and so on. Does that make sense so far?
Great, now for the picture part. We set up two number lines, one horizontal that we put our x on, and one vertical that we put our 2x on (most people call this the y-axis, but that's because they will say that y = 2x, it's an extra step, but we can skip over that for now.)
Now, I will record the information that I just found above in picture form|dw:1439358865094:dw|
See how the dots match our information? The first dot is at x = 0, 2x = 0. The second dot is at x = 1, 2x = 2. And the third dot is at x = 2, 2x = 4.
It's at little like the game Battleship, if you've ever played that. :)
hmm i dont quite understand that graph :/
No worries, let me redraw it. It's not an obvious process. :)
oh wait nvm, sorry my eyes are getting blurry lol
You got it? A good way to read it is to first look horizontally. For example find x = 1. Then with your eyes look up vertically until you see a dot. Then look to the left and you will see the value for 2x, in this case 2. :)
OK, so if you can imagine we kept doing this, plugging in numbers and so forth, we would eventually get something like this |dw:1439359121271:dw|
All of the dots we plot would sit on that line. It turns out that 2x does in fact make the picture of a line.
OK, with that?
Alright, then! What I am going to do now is to skip over a lot of actual graphing and just be "the expert" and tell you what many different graphs look like. This will help us understand what something like x^5 would look like if we graphed it. Prepare for an "art gallery" :)
wait, is the end behavior up and down, because the leading trm is negative and an odd nu,ber ?
Aha! You seem to know the trick! :)
haha okay. So thats how I would explain my answer then ?
In fact, you will get a picture like this for -3x^5:|dw:1439359428939:dw|
Yes, the leading coefficient (the highest power) always controls the end behavior. Your explanation was perfect! Congrats!
For future problems, I'd recommend checking your answer with this graphing tool: https://www.desmos.com/calculator just type in an equation and you can see the computer plot it for you. :)
It's a great way to see the meaning behind all the symbols.
okay, well thankyou for taking the time to explain it to me. :D Have a good night !
Your welcome! Good night.