anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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jtvatsim
  • jtvatsim
So, have you been able to make sense of the question? Do you know what they are asking for? :)
anonymous
  • anonymous
no :/
jtvatsim
  • jtvatsim
Alright, so I'm guessing you don't know what "end behavior" is talking about?

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anonymous
  • anonymous
lol, no.
jtvatsim
  • 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
  • anonymous
okay tell me what to do
jtvatsim
  • 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
  • anonymous
no, i havent studied any of this.
jtvatsim
  • jtvatsim
OK, very good. Thanks for the heads up. Let's start by reviewing what a "term" is. :)
jtvatsim
  • 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
  • anonymous
ok
jtvatsim
  • 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
  • anonymous
in my case -3x5 + 9x4 + 5x3 + 3
jtvatsim
  • 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
  • anonymous
got you
anonymous
  • anonymous
So is that our leading term ?
jtvatsim
  • 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
  • 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
  • anonymous
okay
jtvatsim
  • 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
  • anonymous
Okay awesome
jtvatsim
  • jtvatsim
OK, now for that "end behavior" conversation. Have you had to graph anything in class yet?
anonymous
  • anonymous
no, well not in a long time. I'm not to good at math
jtvatsim
  • 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
  • anonymous
okay :)
jtvatsim
  • jtvatsim
OK, so let's start with a very simple graph. The graph of 2x.
jtvatsim
  • 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
  • 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
  • anonymous
yes !
jtvatsim
  • 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
  • jtvatsim
Now, I will record the information that I just found above in picture form|dw:1439358865094:dw|
jtvatsim
  • 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
  • jtvatsim
It's at little like the game Battleship, if you've ever played that. :)
anonymous
  • anonymous
hmm i dont quite understand that graph :/
jtvatsim
  • jtvatsim
No worries, let me redraw it. It's not an obvious process. :)
anonymous
  • anonymous
oh wait nvm, sorry my eyes are getting blurry lol
jtvatsim
  • 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
  • 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
  • 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
  • jtvatsim
OK, with that?
anonymous
  • anonymous
mhmm :)
jtvatsim
  • 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
  • anonymous
wait, is the end behavior up and down, because the leading trm is negative and an odd nu,ber ?
jtvatsim
  • jtvatsim
Aha! You seem to know the trick! :)
anonymous
  • anonymous
haha okay. So thats how I would explain my answer then ?
jtvatsim
  • jtvatsim
In fact, you will get a picture like this for -3x^5:|dw:1439359428939:dw|
jtvatsim
  • jtvatsim
Yes, the leading coefficient (the highest power) always controls the end behavior. Your explanation was perfect! Congrats!
jtvatsim
  • 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
  • jtvatsim
It's a great way to see the meaning behind all the symbols.
anonymous
  • anonymous
okay, well thankyou for taking the time to explain it to me. :D Have a good night !
jtvatsim
  • jtvatsim
Your welcome! Good night.

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