Question

sd

Answer 1

Alright, let's start with the first one. So, every x-intercept can be written as a factor of a polynomial. For example, when an intercept is at x= 3, you can write it as a factor (x-3). So the first thing we want to do is write every x-intercept as a factor like this.

Answer 2

ok

Answer 3

So did we get all the factors?

Answer 4

yes

Answer 5

-3 2 and 5

Answer 6

?

Answer 7

In terms of factors, though, do you know how we would write that?

Answer 8

(x+3)(x-2)(x-5)

Answer 9

Correct. Now, can we tell what degree the graph is?

Answer 10

no :(

Answer 11

is it odd or even?

Answer 12

Well, we can tell by end behavior. If the degree is even, it ends in the same direction as it starts. So that makes this an even degree function. Kinda see it?

Answer 13

ya

Answer 14

so by eventhe highest degree is an even right?

Answer 15

also when is it neither even nor odd

Answer 16

Well, when we're talking about degree of a function, never. Its either 1, 3, 5, 7, odd, etc, or even, 2, 4, 6, 8, etc.

Answer 17

okay

Answer 18

so this is even meaning the leading degree is even?

Answer 19

Now, any idea how to tell which even degree it is? 4, 6, 8, etc. And we need to be careful just saying even/odd. The degree is even, but it is not technically an even function. When I say even, I only mean the number of degree it is.

Answer 20

oh

Answer 21

and how do we tell what the degree is?

Answer 22

So, notice how at one of the intercepts, the graph only touches and turns around?

Answer 23

ya

Answer 24

Well, when the graph touches the axis and turns around, it means that the graph has 2 intercepts at the same point. So this could really be 4, 6, 8 etc, but those graphs would VERY VERY VERY likely have more turns and intercepts in it. A graph has a maximum amount of turns as 1 less than the degree. Since this graph has 3 turns, its logical to say the function is degree 4.

Answer 25

Well, I should say that the graph has an even amount of intercepts at the same point, to be more accurate.

Answer 26

ok

Answer 27

So we have a 4th degree polynomial with 4 intercepts. (x+3)(x-2)(x-2)(x-5) :P So thats where we can startoff. Now we need to multiply all of those and see what we get.

Answer 28

ok

Answer 29

but can't we just see that x+3 is squared because it looks like a parabola

Answer 30

i mean x-2

Answer 31

and x+3 looks like a straight line

Answer 32

Well, you cant just tell without looking. It has nothing to do with it being a parabola at all, only the fact that it does not cross the x-axis at the point.

Answer 33

ok

Answer 34

so what do we do?

Answer 35

Did you multiply out the 4 factors?

Answer 36

ya

Answer 37

Whatd ya get?

Answer 38

It does matter for me to know what ya got so we can get the rest, lol

Answer 39

x^4-6x^3-3x^2+52x-60

Answer 40

Well, now notice the y-intercept is at -2. That means when x = 0, y has to= -2.

Answer 41

isn't the y interecept -60?

Answer 42

Look at the graph, though.

Answer 43

its not on the graph

Answer 44

oh

Answer 45

well why does it become -60 when u do the math?

Answer 46

Because those are just the multiples of the factors. In order for it to be y-intercept of -2, we have to be missing a +58 somewhere.

Answer 47

ok

Answer 48

but just wondering, what was the purpose of expanding it?

Answer 49

we will have to eventually. And I skipped a step, so I apologize. Ill catch us up, though. In order to make the multiplication come out to have a negative 2 at the end, we have to start off with having an unknown constant before the multiplication. So what I should have had is A(x+3)(x-2)(x-2)(x-5). That being said, when x is 0, y has to be -2. So I am going to do this:
A(0+3)(0-2)(0-2)(0-5) = -2
A(3)(-2)(-2)(-5)= -2
-60A = -2
A = (1/30)
Now we would do our multiplication that we did, except with the A = (1/30) added in. After that, that is all we need really. So what you multiplied out is correct, just multiply it all by 1/30

Answer 50

okay

Answer 51

So now I just need to remember to do that for other problems if it happens, lol.

Answer 52

isn't that our answer 1/30 (x+3)(x-2)(x-2)(x-5)

Answer 53

It might be. I guess you wouldnt need to multiply it out if it didnt ask. If you think its fine like that then keep it xD

Answer 54

why did you multiply x-2 twice

Answer 55

Because remember, having the graph touch at an x-intercept without crossing means that there are an even amount of intercepts at that same point.

Answer 56

okay.

Answer 57

so lets do the next one

Answer 58

how do we do that?

Answer 59

Yeah, I was looking at the 2nd one and thinking the 3rd one is much easier, haha

Answer 60

Well, we can try the 2nd one and go in order if ya want.

Answer 61

sure

Answer 62

thanks for the help. I really appreciate it :)

Answer 63

Yeah, I dont mind. Its practice for me, too xD First off, do we know what a rational function is?

Answer 64

no

Answer 65

its basically a fraction graph. You have a rational function when you start having asymptotes and such. So we have 3 asymptotes. 2 vertical and 1 horizontal. Do you recognize them?

Answer 66

ya

Answer 67

2 and -1 vertical

Answer 68

i can't find the horizontal asymp tho

Answer 69

Alright, awesome. Those asymptotes are needed for us to get our graph. Do you know what causes vertical asymptotes and horizontal asymptotes? Andthe horizontal asymptote is y = 0. Horizontal asymptotes are special, they can actually be crossed, as long as the graph as it is ending in each direction does not cross it.

Answer 70

okay

Answer 71

im a little confused iwth the ending in each direction part

Answer 72

Alright, hang on.