Consider the leading term of the polynomial function. What is the end behavior of the graph? Describe the end behavior and provide the leading term.
-3x5 + 9x4 + 5x3 + 3

- anonymous

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- katieb

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- jtvatsim

So, have you been able to make sense of the question? Do you know what they are asking for? :)

- anonymous

no
:/

- jtvatsim

Alright, so I'm guessing you don't know what "end behavior" is talking about?

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- anonymous

lol, no.

- jtvatsim

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. :)

- anonymous

okay tell me what to do

- jtvatsim

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.

- anonymous

no, i havent studied any of this.

- jtvatsim

OK, very good. Thanks for the heads up. Let's start by reviewing what a "term" is. :)

- jtvatsim

In math, we have two basic operations: addition and multiplication (let's ignore subtraction and division for now, because those are basically just opposites).

- anonymous

ok

- jtvatsim

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

- anonymous

in my case -3x5 + 9x4 + 5x3 + 3

- jtvatsim

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

- anonymous

got you

- anonymous

So is that our leading term ?

- jtvatsim

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.

- jtvatsim

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

- anonymous

okay

- jtvatsim

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.

- anonymous

Okay awesome

- jtvatsim

OK, now for that "end behavior" conversation. Have you had to graph anything in class yet?

- anonymous

no, well not in a long time. I'm not to good at math

- jtvatsim

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.

- anonymous

okay :)

- jtvatsim

OK, so let's start with a very simple graph. The graph of 2x.

- jtvatsim

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.

- jtvatsim

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?

- anonymous

yes !

- jtvatsim

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.)

- jtvatsim

Now, I will record the information that I just found above in picture form|dw:1439358865094:dw|

- jtvatsim

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.

- jtvatsim

It's at little like the game Battleship, if you've ever played that. :)

- anonymous

hmm i dont quite understand that graph :/

- jtvatsim

No worries, let me redraw it. It's not an obvious process. :)

- anonymous

oh wait nvm, sorry my eyes are getting blurry lol

- jtvatsim

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. :)

- jtvatsim

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|

- jtvatsim

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.

- jtvatsim

OK, with that?

- anonymous

mhmm :)

- jtvatsim

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" :)

- anonymous

wait, is the end behavior up and down, because the leading trm is negative and an odd nu,ber ?

- jtvatsim

Aha! You seem to know the trick! :)

- anonymous

haha okay. So thats how I would explain my answer then ?

- jtvatsim

In fact, you will get a picture like this for -3x^5:|dw:1439359428939:dw|

- jtvatsim

Yes, the leading coefficient (the highest power) always controls the end behavior. Your explanation was perfect! Congrats!

- jtvatsim

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. :)

- jtvatsim

It's a great way to see the meaning behind all the symbols.

- anonymous

okay, well thankyou for taking the time to explain it to me. :D Have a good night !

- jtvatsim

Your welcome! Good night.

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