Question

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

Answer 1

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

Answer 2

no
:/

Answer 3

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

Answer 4

lol, no.

Answer 5

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. :)

Answer 6

okay tell me what to do

Answer 7

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.

Answer 8

no, i havent studied any of this.

Answer 9

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

Answer 10

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

Answer 11

ok

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

got you

Answer 16

So is that our leading term ?

Answer 17

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.

Answer 18

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

Answer 19

okay

Answer 20

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.

Answer 21

Okay awesome

Answer 22

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

Answer 23

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

Answer 24

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.

Answer 25

okay :)

Answer 26

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

Answer 27

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.

Answer 28

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?

Answer 29

yes !

Answer 30

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.)

Answer 31

Now, I will record the information that I just found above in picture form

Answer 32

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.

Answer 33

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

Answer 34

hmm i dont quite understand that graph :/

Answer 35

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

Answer 36

oh wait nvm, sorry my eyes are getting blurry lol

Answer 37

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. :)

Answer 38

OK, so if you can imagine we kept doing this, plugging in numbers and so forth, we would eventually get something like this

Answer 39

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.

Answer 40

OK, with that?

Answer 41

mhmm :)

Answer 42

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" :)

Answer 43

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

Answer 44

Aha! You seem to know the trick! :)

Answer 45

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

Answer 46

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