Question

67 x 53 x 63 x 6

Answer 1

vertical asymptotes occur generally in function where we attempt to divide by 0. Given that, when does your function do so, aka at what value of x?

Answer 2

5x^4?

Answer 3

hmmm well what value of x would make that term divide by 0?

Answer 4

5?

Answer 5

I mean 4

Answer 6

well if we make x = 4 in that term, we aren't dividing by 0 right?
\[5x^{4}\]\[5(4)^{4}\]\[5(256)\]\[1280\] rather we should look at your second term in your function aka -7/x

Answer 7

what value of x would make -7/x divide by 0?

Answer 8

4?

Answer 9

ummm no. Do you understand what I mean by dividing by 0?

Answer 10

I don't think so

Answer 11

ah ok! So basically you want to find a value of x where it will make the denominator equal to 0. So if we have a function such as:
\[y=\frac{4}{x-3}\] in order to make the denominator 0 we would simply make x = 3. Right? Since that would make the function look like this:
\[y=\frac{4}{3-3}\]and 3-3 is 0 right so the function would become \[y=\frac{4}{0}\]and thus we are dividing by 0 and hence at x = 3, there is a vertical asymptote. Does this all make sense?

Answer 12

yes

Answer 13

ok so let's go back to your function of: \[y=5x^{4}-\frac{7}{x}+3\] but lets just look at the second term of your function. At what value of x would 7/x be dividing by 0?

Answer 14

3?

Answer 15

in other words how would we make:
\[\frac{7}{x}\] into \[\frac{7}{0}\]

Answer 16

well putting 3 in as x (aka x=3) would make \[\frac{7}{x}\] into \[\frac{7}{3}\]

Answer 17

so that can't be right.

Answer 18

do you understand what I mean by "what value of x"?

Answer 19

wait what about 0?

Answer 20

or 1?

Answer 21

so if we make x = 0; \[\frac{7}{x}\] becomes \[\frac{7}{0}\]
if we make x = 1; \[\frac{7}{x}\] becomes \[\frac{7}{1}\]
so x =0 works! And so a vertical asymptote of f(x) =5x^4 - 7/x + 3 is x =0! But I have good feeling that you don't actually understand what you are doing. Am I wrong?

Answer 22

I think I just need to practice more

Answer 23

hmm ok but do you understand what I mean by me asking "what value of x"?

Answer 24

like exponent?

Answer 25

when I say what value of x, it simply means to put that number as x. Aka to substitute it as such. So everywhere there is an x you should instead put that value. So if the function is: \[f(x) = x^{2} +1\] and my value of x is 2 the function becomes like this \[f(2)=2^{2}+1\]Do you see how I replaced x with 2 everywhere in the function? does this make sense?

Answer 26

ohhhhh ok yes

Answer 27

could you help me with another problem?

Answer 28

so when I was asking for the value of x so that 7/x is dividing by 0, it makes full sense as to why x has to be 0 for that to happen right?

Answer 29

sure.

Answer 30

so do you know what it means by the left side of the graph?

Answer 31

not really

Answer 32

the left side of the graph is basically when x gets really really small and by "really really small" I mean really negative. So your question is essentially asking where does this function: \[f(x) = -x^{4}-x^{3}+4x-2\] go when x is really negative. Does this make sense?

Answer 33

yea

Answer 34

ok cool so let me say this before asking a question which seems to have little relevance: a function behavior is mainly dictated by the highest ordered term. By "ordered term" I mean the term with the highest x exponent. In this case the term would be: -x^4 since 4 is the highest exponent of x in the function. So my question is do you know what happens when you square a negative number?

Answer 35

no

Answer 36

ah ok so whenever you raise a negative number by a even exponent it becomes positive, so "squaring" or raising by 2 a negative number would make it positive...Does this make sense?

Answer 37

yes

Answer 38

ok good! So let's bring it back your function, so I said that we need to basically look at the highest ordered term of your function. What is the highest ordered term of your function(f(x) = - x^4 - x^3 + 4x - 2)?

Answer 39

4?

Answer 40

that's the order number, the actual term is -x^4. Terms are separated by + and -. Does this make sense?

Answer 41

ohokyea

Answer 42

cool! So now since we are looking for what happens when x is really negative right? But we are raising that really negative number by 4. So what happens when we raise a really negative number by 4?

Answer 43

It becomes...?

Answer 44

positive?

Answer 45

smaller?

Answer 46

It does become positive! But it becomes bigger! But since it has a negative sign in front of the term it actually becomes negative... So it become a really negative number. Does this make sense?

Answer 47

yes

Answer 48

you still there? lol

Answer 49

sorry I went to take a shower...ok good so we basically have our answer don't we? if the function tends to go to a really negative number. The direction is down right?

Answer 50

does that all make sense?

Answer 51

right because positive would go up?

Answer 52

I have one more question then I'll leave you alone lol

Answer 53

exactly because of how graph are drawn right where positive is up and negative is down and since at the left side, the function becomes really negative because x is really negative...the function's direction is down!

Answer 54

Lol. Don't worry about "leaving me alone" I enjoy helping people which is why I do this...this is the internet you know? if I really want to be left alone, I would simply shut off my computer lol.

Answer 55

Aw thanks:) lol I REALLY appreciate this.You really have been helping me understand this more.

Answer 56

so do you know what it means by relative extrema?

Answer 57

nope

Answer 58

so relative extrema aka local maximums are the peaks of the function so here a drawing: in this drawing there are three relative extremas. Do you see the three peaks?