Question

Can someone explain this Alg 2 question to me please?

Answer 1

Do you know what holes are?

Answer 2

no ideas

Answer 3

Look at the equation, what happens if x = -4?

Answer 4

it equals 0

Answer 5

Yes, the denominator equals 0 and what happens when you divide by 0?

Answer 6

Nothing happens it stays the same?

Answer 7

You cannot divide by 0. Try it on a calculator, it is not allowed. It is undefined and will produce an error, do you agree?

Answer 8

yes

Answer 9

This particular example has an exception since if you look at the bottom and look at the top of the fraction you'll notice that they both have (x+4) terms and they can cancel each other out, do you understand this?

Answer 10

so far yes

Answer 11

Here's the thing though, normally when you cancel out terms the value of the equation stays the same. The graph will look the same.

Answer 12

This is where, "holes" come in.

Answer 13

brb I have to reopen browser

Answer 14

ok back.

Answer 15

Now, do you agree that when you cancel out terms in a fraction that the value of the fraction will stay the same?

Answer 16

Yes

Answer 17

Exactly correct.... Most of the time!

Answer 18

If you cancelled out those terms in this equation, the shape of the equation will stay the same, but it will no longer be undefined when x = -4.

Answer 19

This is a very important concept to understand, this is what a "hole" in an equation is and where it comes from.

Answer 20

So the hole would be x=-4?

Answer 21

Exactly

Answer 22

Do you understand why though?

Answer 23

because it canceled out the term?

Answer 24

Yes. Because the function IS essentially equivalent to the function you will get when you cancel out those terms. The only difference is that the original function you have (before cancelling out the term) will be undefined when x = -4.

Answer 25

Following?

Answer 26

Yes surprisingly i am.

Answer 27

Do you understand asymptotes?

Answer 28

No /.\

Answer 29

It's not a surprise, you're a smart person. Everyone is smart, sometimes they just don't have the tools they need.

Answer 30

Do you know what an asymptote looks like on a graph?

Answer 31

Yes

Answer 32

Perfect, then I'll start from there. Consider the function you have given me. What happens when x = -1?

Answer 33

i see the same thing happens like last time the two terms cancel out

Answer 34

No. Stop.

Answer 35

This is a separate idea I'm talking about here I don't want you to relate them.

Answer 36

Remember the golden rule, "NEVER DIVIDE BY 0". When x = -1 here it will make the bottom of the fraction have a 0. So it will be undefined.

Answer 37

Still with me?

Answer 38

Ohhh so you just wanted me to plug in -1 all the way across? When i did that i got 3/0 but i also noted that the numerator and denomorator both had (-1+4) thats what i was talking about

Answer 39

Well I just want you to notice that when you plug -1 in the bottom you will get a 0 which is something to pay attention to. They are important in recognizing the characteristics of how a function behaves.

Answer 40

This term cannot be cancelled out. That's why it's different from the previous idea of a "hole".

Answer 41

Only the bottom has the term (x+1)

Answer 42

Ok im still following you.

Answer 43

:D That's what I like to hear.

Answer 44

Ok, now consider this. What happens when you divide by a number less than 1?

Answer 45

Something divided by a number between 0 and 1

Answer 46

This is the key to understanding asymptotes (well vertical asymptotes I should say)

Answer 47

The quotient is bigger than what you started with? Maybe thats worded weird idk >.<

Answer 48

No worries, I understand what you mean perfectly. That's exactly correct!

Answer 49

Now another thing to realize is the smaller the number you divide by, the larger the outcome

Answer 50

Try it out on a calculator. 17 / 0.5 and 17 / 0.00005

Answer 51

ok so 34 vs. 340000 :)

Answer 52

Now look at the function you gave me. You don't have to give me an exact answer, but try to guess what will happen when x = -0.9999999

Answer 53

It will get bigger?

Answer 54

It will get much bigger. In fact, the closer that x gets to being -1 the huger the value will be, agreed?

Answer 55

This is because as x approaches -1. x + 1 will be get a lot smaller.

Answer 56

And we know that dividing by a small number, makes our value bigger.

Answer 57

Do you understand this trend?

Answer 58

yes

Answer 59

That's why you see an asymptote on a graph shoot straight up or down. But do you know why it goes up on one side and down on the other?

Answer 60

Not really

Answer 61

Let's focus just on the term (x+1)

Answer 62

ok

Answer 63

What happens when x is a TINY bit smaller than -1? Will the value be positive or negative?

Answer 64

It will still be neg?

Answer 65

Nope, it will be positive. Think:
(x+1)
-0.9 + 1

Answer 66

= 0.1

Answer 67

Oh sorry i wasnt thinging right, but go on.

Answer 68

*thinking

Answer 69

Now what about when x is a TINY big "larger" than -1, I mean x = -1.000000001

Answer 70

What will the answer be? Positive or negative?

Answer 71

That would be negative.

Answer 72

It won't just be negative. It will be a very tiny negative number.

Answer 73

Now combine this with what we learned earlier about dividing by a small number, and you can figure out the rest. That's what an asymptote is. I want you to come up with the rest of the reasoning, I'm sure you're a smart person.

Answer 74

Please take a moment and try to explain it to me here and I will let you know.

Answer 75

I think i an do it from here

Answer 76

Explaining yourself is also a way to practice and improve your knowledge of something.

Answer 77

*can

Answer 78

Well, up to you. Good luck. I hope this helped.

Answer 79

It did so much thank you truly. :)