precalculus

[grim sand]

So my teacher teaching me wrong

Never trusted him anyways

[viscid thistle]

We never said that

[grim sand]

I know

[uncut mulch]

you may be misrepresenting them

[grim sand]

Probably 😔

[viscid thistle]

Yeah, x is equal to 1 is far different from saying that 1x is the same as x

[grim sand]

Oh

So x is the same as 1x

[viscid thistle]

If you insist, yes

[grim sand]

Oki

So back to the 4727/x+3

[uncut mulch]

()

[grim sand]

How’d u turn x into 0

[uncut mulch]

you don't

[viscid thistle]

Please remember to read all my messages, i will repeat myself.

We use x+3=0 to solve for the values of x that make the denominator 0

[grim sand]

Oh

So x=-3

[viscid thistle]

Yes.

[grim sand]

So for example if it was 100/x+100

[viscid thistle]

You can see clearly that when x=-3, the function does not exist

[grim sand]

I can still do

[uncut mulch]

and that is the value you should exclude from the set of reals

[grim sand]

X+100=0

And make it x=-100

[viscid thistle]

He doesn't know what reals is, so i refrain from using that word yet lol

[uncut mulch]

parentheses though

[viscid thistle]

^

[grim sand]

U add parenthesis in X=-3 too?

I thought it was just for (x,Y)

[uncut mulch]

\verb|100/x+100| reads as $\frac{100}{x} + 100$

[obsidian monolith]

ramonov:

[grim sand]

Oh

[viscid thistle]

By putting parens we mean 100/(x+100)

And not 100/x+100

[grim sand]

Ohh

Ok

I never knew that

I just kept doing it like that back in sophomore year

[viscid thistle]

And make it x=-100
@grim sand but this logic was correct tho

[grim sand]

Oki

[viscid thistle]

x=-100 is not on the domain

[grim sand]

Now that I got the =0 part

What next

[viscid thistle]

Good

[uncut mulch]

using appropriate grouping symbols in plain text trains understanding of order of operations which is why we're so strict on it

[viscid thistle]

Now one last example before anything: Domain of $y=\sqrt{4x+2}$

[grim sand]

using appropriate grouping symbols in plain text trains understanding of order of operations which is why we're so strict on it
@uncut mulch what

[viscid thistle]

Try this one

Remember what a sqrt can't take

[grim sand]

So that would be

[viscid thistle]

It's different from the fraction

[grim sand]

How do u know what symbol it is

or <

How do u know if it’s > or <

[uncut mulch]

properties of the square root

[viscid thistle]

Okay, answer this question: what an sqrt can't take to receive a real solution

[grim sand]

Negative numbers

[viscid thistle]

Good

But you CAN take 0

[grim sand]

That equation has no negatives tho

[viscid thistle]

Remember $\sqrt{0}=0$ has no problems with it

[obsidian monolith]

Al𝟛dium:

[viscid thistle]

Ok?

[grim sand]

Ok

[undone pawn]

That equation has no negatives tho
the equation has an x in it though.. and for specific values of x, the thing inside the sqrt may become negative

[grim sand]

Oh

[uncut mulch]

$\sqrt{\text{you want this to be greater or equal to 0}}$

[obsidian monolith]

ramonov:

[grim sand]

To get the x by itself

U need to turn

No

[viscid thistle]

Use your logic here, remember what an sqrt CAN take instead

[grim sand]

It can take 2

[undone pawn]

weird this is something that shouldve been explained in class

[viscid thistle]

Generalise a bit more

[grim sand]

And turn it into 1

No

[viscid thistle]

Reposting it

[obsidian monolith]

Al𝟛dium:

[uncut mulch]

it can take a lot more numbers than 2, and rather than check infinitely many values,
set up an inequality

[grim sand]

How do I square root 4x

[viscid thistle]

"ask the function what values can it take" remembering it has an sqrt

[grim sand]

It can take the 4

But not the x or 2 cuz there is no square root of 2 or 1

[viscid thistle]

Try to generalise

[grim sand]

Yes?

[uncut mulch]

huh?

[grim sand]

4x+2>0

[viscid thistle]

But not the x or 2 cuz there is no square root of 2 or 1
@grim sand there is a solution for it, √2 is not a perfect square

4x+2>0
Almost

[grim sand]

4x+2<0

[viscid thistle]

Remember that $\sqrt{0}$ has a solution

[obsidian monolith]

Al𝟛dium:

[uncut mulch]

getting colder.

[grim sand]

Oh

4x+2> or = to 0

[undone pawn]

yep

[viscid thistle]

Uh yeah

$4x+2≥0$

[obsidian monolith]

Al𝟛dium:

[uncut mulch]

you can shorten that to >= in plain text

[grim sand]

Ok

[viscid thistle]

Try to solve for x

[grim sand]

-2/4 can be -1/2

So x >= -1/2

[viscid thistle]

Yes.

So NOW you know that the function $y=\sqrt{4x+2}$ can only take values of x such that x≥-1/2

[obsidian monolith]

Al𝟛dium:

[grim sand]

How do u know if it’s > or <
@grim sand I’m still a bit lost on this one

X has to be bigger than -1/2

So

[-1/2, INF)

[uncut mulch]

*greater or equal to

[grim sand]

Oh yea

But is that right tho

[-1/2, INF)
@grim sand this

[viscid thistle]

You have to use your logic, you are asking what values of can it take, which are positive or 0, so it's only ≥0 and not ≤0 because ≤0 would mean negative numbers or 0

[uncut mulch]

if you can describe it in words,
replace words with the appropriate mathematical symbol

[grim sand]

So if the equation was 4x-2

Instead of >= it would be <=

[viscid thistle]

You have to use your logic, you are asking what values of can it take, which are positive or 0, so it's only ≥0 and not ≤0 because ≤0 would mean negative numbers or 0
Addressing your question of ≤ or ≥

[grim sand]

Yes

Because it’s 4x+2 the symbol was >= but if it was 4x-2 would it be =<

[viscid thistle]

Nonono, that has nothing to do with the symbol

[uncut mulch]

$\sqrt{\blue{\text{this}}}$ \
i want "\blue{this}" to be greater or equal to $0$ \
$\blue{\text{this}} \geq 0$

[obsidian monolith]

ramonov:

[grim sand]

How would it look if you wanted it to be less or equal to 0

How would the equation look*

[viscid thistle]

$\frac{1}{\text{this}}$ i want to check for the values of this that make the denominator 0

[obsidian monolith]

Al𝟛dium:

[grim sand]

So make it = to or greater than 0

Or just make it equal to

[viscid thistle]

How would it look if you wanted it to be less or equal to 0
Use your logic about which values the thing can take or not

[grim sand]

It can take 0 if it’s square root

If it’s not it can’t

[viscid thistle]

Usually if its fractions and sqrt, you won't have to deal with <0 or ≤0

[grim sand]

So most of the time I only have to use > or >=

[echo wagon]

The inside of a square root always has to be $\geq 0$, but when you simplify the inequality, you could end up with $\leq$ or even $<$. Example: For $\sqrt{-x} $, we require $-x \geq 0$ which would become $x \leq 0$

[obsidian monolith]

Lunasong:

[grim sand]

Oh

Because it’s a negative (the X)

So u have to flip it

[echo wagon]

Yes

[grim sand]

Ah ok

So it always starts with >= or >

Unless it’s negative (the x)

Then u flip it

[echo wagon]

Yeah, you can still start with >= and then simplify inequalities as you always do

[grim sand]

Ah okay that makes sense

Can u give me another square root equation to try

[echo wagon]

$\sqrt{-\frac{1} {x} } $

[obsidian monolith]

Lunasong:

[grim sand]

-1/x>=0

Which will be

X<= 0

[echo wagon]

Almost, but one more thing is needed

[grim sand]

[0,INF)

[echo wagon]

No

You are also dividing by x, so what does that tell you?

[viscid thistle]

Okay well wait a sec

[echo wagon]

[0,INF)
@grim sand these are positive numbers, this does not correspond to x<=0.

[viscid thistle]

I'm gonna clarify it

[grim sand]

Since it’s -1/x

And ur flipping it

Don’t u multiply

Making it 0

But because it’s negative

It turns into a positive

So it’s still

[echo wagon]

X<= 0
@grim sand this i almost right, but something is off

Since you are dividing by x, x cannot be 0

[grim sand]

Oh

So x<0

[echo wagon]

Yes

[grim sand]

Ohhh that makes sense

Can I get another one

Please

[echo wagon]

Lol, I'm not that creative, but I'll try

[viscid thistle]

Okay i'm gonna generalise to clarify it as water: 3 examples $$\frac{1}{x-4}$$ A denominator can't be 0, so we are looking for the value of x that makes it equal to 0: $$x-4\mathbf{=}0\implies x=4$$ so x=4 is not on the domain

$$\sqrt{x-4}$$ We are gonna look at the values of x that make the sqrt to be positive or 0, because we know that an sqrt does not have a real solution for negative numbers. $$x-4≥0\implies x≥4$$ then for values less than 4, it will make the sqrt to be negative, see it yourself, take x=3 $\sqrt{3-4}=\sqrt{-1}$ no real solution.

$$\frac{1}{\sqrt{-2x+3}}$$ be careful of this one, we have a fraction AND a sqrt. Remembering which values a fraction can't take (so 0) and which values the sqrt CAN take (positive numbers and 0), we have to find the intersection of both, so it'd be ≥0 BUT we know that the denominator can't be 0 so it's only $>0$ $$-2x+3>0\implies -2x>-3\implies x<\frac32$$ so the domain will be (-inf, 3/2)

Okay PLEASE read everything

[echo wagon]

$\sqrt{x^2}$

[obsidian monolith]

Lunasong:

[echo wagon]

Read Al's message then try my question

[grim sand]

The picture is gone

[viscid thistle]

Now

[grim sand]

I don’t know how to do that ^2 yet

[viscid thistle]

Last fix

[obsidian monolith]

Al𝟛dium:

[viscid thistle]

Those are the 3 examples

[grim sand]

Yo

I understand it

🤯

It feels like a new world was just opened up to me

[viscid thistle]

Glad to hear

[grim sand]

Can u give me an example for me to solve rn

[viscid thistle]

Your problem lol

[grim sand]

Also @echo wagon I don’t know how to do that ^2 yet

[viscid thistle]

No7

[grim sand]

I thought we already solved that

[viscid thistle]

Try

Oh lol

[grim sand]

Well I forgot the answer to it

And didn’t put it in yet

So I’ll still do it

[echo wagon]

How are you doing square roots without knowing what squares are? Hmm

[grim sand]

Lmao

How are you doing square roots without knowing what squares are? Hmm
@echo wagon I don’t know 😔

[viscid thistle]

Oh yes we finished it true

[echo wagon]

@echo wagon I don’t know 😔
@grim sand Bizarre. $x^2 = x \times x$

[obsidian monolith]

Lunasong:

[viscid thistle]

A harder one $\frac{\sqrt{x}}{\sqrt{3x-2}}$

Focus on the denominator with sqrt first

And then the sqrt above

[grim sand]

Wait so my problem

I got

X>1/2

Because

[viscid thistle]

I don't remember lol

[grim sand]

Square root can take 0 and greater than 0

[viscid thistle]

It's been 2h

[echo wagon]

What was your problem?

[grim sand]

Square root can take 0 and greater than 0 but because of fraction

It can’t take 0

[echo wagon]

Yes

[grim sand]

So the equal to symbol is gone

[echo wagon]

Good job

[viscid thistle]

Yes

[echo wagon]

You little liar though

[grim sand]

So it’s x>1/2

[viscid thistle]

You successfully understood my big text

[echo wagon]

In question 8 I see a square

[grim sand]

Yes cuz it had examples

[viscid thistle]

So it’s x>1/2
@grim sand yeah

[grim sand]

We haven’t worked on it yet

We worked on question 7 for 2 hours

Lmao

Cuz I’m slow

[viscid thistle]

More like 2h to teach you domain and inequalities

Lol

[grim sand]

Yes

[echo wagon]

I mean if you're expected to answer that question next, then you should already know what squares aren't

[grim sand]

I only learned now how to use > and < correctly

I mean if you're expected to answer that question yet, then you should already know what squares aren't
@echo wagon I’m supposed to from last year

[viscid thistle]

Using logic

[grim sand]

But I haven’t had math since beginning of sophomore year

So I really dont know anything

[echo wagon]

Ah, okay

[grim sand]

Then since it’s x>1/2

I can turn that into

[viscid thistle]

The website is wrong if you remember

[grim sand]

(1/2,INF)

[viscid thistle]

Yep

And not [1/2, inf) as the web says

[grim sand]

I’m feeling so accomplished rn

It’s like a whole new world

[viscid thistle]

Glad to hear

[grim sand]

It feels amazing

Can u give me another example

For square root

[obsidian monolith]

Al𝟛dium:

[viscid thistle]

If you get this one, you are gonna feel super-accomplished

[grim sand]

X>2/3

[viscid thistle]

Focus on the denominator with sqrt first
And then the sqrt above

[grim sand]

Oh

We have to do the square root on top too?

[viscid thistle]

X>2/3
Okay yes

[grim sand]

I just completely ignored the top

[viscid thistle]

We have to do the square root on top too?
Yeah

[echo wagon]

Both square roots have to be defined. You're right, but I think only by accident, haha

[viscid thistle]

Yeah lol

[grim sand]

Yeah I think I got it right by accident

How do I do the top square root

I just went straight to the bottom

[echo wagon]

When is it defined?

[viscid thistle]

Lo solve for x≥0

Done

Lol

[grim sand]

When it’s greater than 0

[echo wagon]

Or equal

[grim sand]

But it’s a fraction?

[echo wagon]

But the numerator can be 0

[viscid thistle]

Bc it's not in the deno anymore

[grim sand]

Or does that only apply for the bottom

Oh

That makes sense

[echo wagon]

0/ 1 is just 0. 1/0 is not defined.

[grim sand]

So there’s a lot of logic

[viscid thistle]

You'd have to find the intersection between both

[grim sand]

I see I’m learning

@echo wagon ur helping out a lot

[viscid thistle]

Intersection between x≥0 and x>2/3 is just x>2/3 lol

@echo wagon ur helping out a lot
...

[grim sand]

No no

Ur doing an amazing job

I wouldn’t get anywhere without u

[viscid thistle]

Im kidding lol

[grim sand]

But Luna made it easier to understand

Lol

Dumbed it down for me

[viscid thistle]

Just been here for >2h...

[echo wagon]

Yeah, wtf, my five minutes here was way more helpful than your two hours. Get outta here, Al

[viscid thistle]

Jk lol

Lmao

[grim sand]

😂😂

But yes u two are amazing

And all the other peeps who helped me

[viscid thistle]

Honorable mentions: Ramonov

[grim sand]

So If the top was for example x>0 and bottom is x>2/3

How would I combine them

[viscid thistle]

Finding the "intersection"

[grim sand]

What’s that

[viscid thistle]

What i do with those is to imagine a real number line

Like the video did

Know what it is?

[echo wagon]

Let me try to dumb it down again: You need both $x \geq 0$ and $x > \frac{3} {2} $ to be true. But if $x > \frac{3} {2} $ then it is also true that $x \geq 0$. So you combine those by just saying $x > \frac{3} {2} $

[viscid thistle]

Let me just link a vid lol

[grim sand]

Yes

A number like

The one with V at the end

[obsidian monolith]

Lunasong:

[grim sand]

It keeps disappearing

[echo wagon]

Because I am changing my mistakes lol

[grim sand]

Oh so u only combine the end

U leave x alone

[viscid thistle]

https://youtu.be/WP0np2o7KVk

What i'm interested in for you are the last seconds of solving the inequalities

Not the process

The real number line thing

[obsidian monolith]

Al𝟛dium:

[grim sand]

I’m sorry but

I can’t pay attention to her

She moves too much and my attention drifts

And I forget what she says

[viscid thistle]

That is the real number line

You can find the intersection pretty easier by drawing it

Okay ig ill find another vid

[grim sand]

Ok I saw the end part

[echo wagon]

Combining inequalities can get pretty complicated with more inequalities, and sometimes they can't be combined, so it's difficult to explain the whole process over text. A picture is definitely the way to go.

[grim sand]

What if it doesn’t have anything in common

[viscid thistle]

https://youtu.be/Ww7xtG2S7IM

[grim sand]

I like that guy

He’s so thorough

[echo wagon]

Like x >= 4 and x <= 2? Then there are no solutions.

[grim sand]

So it’ll just be called no solution?

[viscid thistle]

For more grasping on notation

[grim sand]

And I can’t do the x y thing with it

[viscid thistle]

So it’ll just be called no solution?
No intersection between them

So yeah no solution

[grim sand]

Oh but I can still do the domain

But it’ll be 2 domain

Is this right

I’m pretty sure it is right cuz they both end at 5 yes

[echo wagon]

Yes, good job

[viscid thistle]

Yep

[grim sand]

Yus

[novel cargo]

yeey

[grim sand]

I’ve been learning domain the past 2 hours

Now

Range

🙃

[novel cargo]

I would take a break if I were you

just my opinion

[grim sand]

I can’t

[novel cargo]

oh

[grim sand]

I lose enthusiasm if I take breaks

[viscid thistle]

So for range, do you know what the inverse of a function is?

[grim sand]

I haven’t even eaten breakfast lol I’m so intrigued by this

[viscid thistle]

LOL

[grim sand]

I think I do

Inverse functions

[viscid thistle]

Oh that's great

[grim sand]

Would this be right

[viscid thistle]

https://youtu.be/Cf4rxWLShuU

I'm like too exhausted to write lol

[grim sand]

What does QED mean

And I don’t blame u

You were helping me for two hours

[viscid thistle]

Indeed

[grim sand]

I don’t get tired when I’m this intrigued

I can go all day

[viscid thistle]

That's what i call passion

[echo wagon]

QED means you are smart

[viscid thistle]

[grim sand]

My mans talking faster than sonic can run

[viscid thistle]

Ikr lmao

The guy above is so good at explaining

Go watch it

[grim sand]

I watched him

I understood the range

It’s just inverse functions

I love those

[viscid thistle]

Great

Anything else?

[grim sand]

So for domain I only knew how to do it on growth

Graph*

But now I think I understand equations

But range

I know how to do equations how do I do graphs

[echo wagon]

You just did it on a graph

[viscid thistle]

https://youtu.be/KirGQOwjBVI

lmao

At this point, whatever

[grim sand]

Lmao u just got every vid in a folder or something

[viscid thistle]

Lol i just look range by looking at the graph literally

[grim sand]

So u just graph it

@echo wagon

Can u dumb this down

So since it’s fraction

It can’t be equal to 0

[viscid thistle]

Yeah

[echo wagon]

Do you know what the U means?

[grim sand]

And so it would be

X>0

[viscid thistle]

Nono

[grim sand]

Union

[viscid thistle]

Its a fraction

[grim sand]

Or something like that

[viscid thistle]

Without sqrt

[echo wagon]

Do you know what union means?

[grim sand]

No

Still confused on what it does

I just think it flips it

[echo wagon]

Union means x can be in the first thing or in the second thing

[grim sand]

So it just flips it

😃

[viscid thistle]

Verbally, union means "or"

[grim sand]

Verbally, union means "or"
@viscid thistle yes this is what I mean by dumbing it down

[viscid thistle]

Or it is used to glue things lol

[grim sand]

I understand that

Wait so u said it’s a fraction so it can’t be x>0?

[echo wagon]

So $x \in (1,2) \cup (5,6)$ means $x \in (1,2)$ or $x \in (5,6)$ so $1 < x < 2$ or $5 < x < 6$

[viscid thistle]

No one said that

[echo wagon]

Do you understand that example?

[viscid thistle]

He doesn't know what in means maybe

[grim sand]

What’s that E

[obsidian monolith]

Lunasong:

[grim sand]

Equals

Ok

[viscid thistle]

Nope

[grim sand]

Oh

[echo wagon]

The e means x belongs to that set

[viscid thistle]

You need to dumb it down more luna

[grim sand]

Yes

[viscid thistle]

He's not that level yet

[echo wagon]

Dumbing down is my specialty

[grim sand]

😂

[viscid thistle]

[echo wagon]

Do you know what sets are and that domains are sets?

[grim sand]

The pairs

Or the group of numbers together

[echo wagon]

A collection of numbers (or other things) is a set

The set (3, 5) is the collection of numbers between 3 and 5

[grim sand]

So 4 works in there

[echo wagon]

So when you say $x \in (3,5)$ it means x is in that set so it is a number between 3 and 5

[obsidian monolith]

Lunasong:

[grim sand]

So what’s the U for

[echo wagon]

Union means you are putting two sets together, so you are combining all the things that are in either set into one collection

[grim sand]

Lmao

So for example x E (3,5) can be shown as 3<x>5

And if u put U and a random set

It’ll be the same thing

[echo wagon]

So (3,5) is the numbers between 3 and 5, and (8,9) is the numbers between 8 and 9 so $(3,5) \cup (8,9)$ is all the numbers that are between 3 and 5 or that are between 8 and 9

[viscid thistle]

Almost

[obsidian monolith]

Lunasong:

[grim sand]

But with OR in the middle

Oh

[viscid thistle]

So for example x E (3,5) can be shown as 3<x>5
3<x<5

Not 3<x>5

[grim sand]

So meaning X can be 3 4 5 or 8 8.5 9

Oh yeah

Got confused for a second

[echo wagon]

Or 3.5 or 3.111 or 8.999 etc

[grim sand]

Yeah

I just use 8.5 cuz u only gave 8 and 9

[echo wagon]

But yes, all those numbers belong to the set

[grim sand]

Oh that’s easy to understand

So now I know what U means

So now back to the fraction

It’s not x>0?

[viscid thistle]

Remember my big text

[echo wagon]

So in your question, they want you to write the domain as a union

[viscid thistle]

May i repost?

[grim sand]

I saved ur big text

[viscid thistle]

LMAO

[viscid thistle]

Look at the first example then

[echo wagon]

That is because you can't write it as one interval. So you have to use two intervals and combine them into one with union

[grim sand]

Oh

So it’s just X=0

[viscid thistle]

x can't be 0

x=0 is not on the domain

[grim sand]

Lost

Back to basics 😔

[viscid thistle]

$\frac20$ does this exist?

[obsidian monolith]

Al𝟛dium:

[grim sand]

No

So denominator can’t be 0

So it can’t be equal to 0

[viscid thistle]

Then it is not on the domain

:)

[grim sand]

So it’s x>0

[viscid thistle]

No, why you are going back to inequalities

[echo wagon]

So it’s x>0
@grim sand why?

[viscid thistle]

This is not sqrt

No sqrt

[grim sand]

Oh so it only works for square root

[echo wagon]

Can x be -1?

[viscid thistle]

You can see it yourself x=1 exists perfectly

[grim sand]

Yes but Itd just be a negative

[viscid thistle]

$\frac21=2$

[obsidian monolith]

Al𝟛dium:

[grim sand]

And negative can exist

So

Infinite

[echo wagon]

So -1 has to be in the domain

So it can't be x > 0

[viscid thistle]

You are getting confusing Luna lol

x>0 is greater

Positive numbers not negative

[grim sand]

So any number but 0 works

[viscid thistle]

Yes

[grim sand]

That’s where the U comes in

OHHHHHH

[viscid thistle]

So all real numbers but 0

Remember that (3,0) does not include 0

[grim sand]

So [INF,0) U (0,Inf]

[viscid thistle]

Bc it's parens

[echo wagon]

Yes, but always use round brackets for infinity

[viscid thistle]

Wait no

[grim sand]

Oh

[viscid thistle]

Yeah ^

[grim sand]

Then my teacher did it wrong

Cuz it got [ this on infinite

[viscid thistle]

Infinity is a concept not a number

Yeah another one that they did wrong

[grim sand]

Now I understand U

Cuz I was like

“If it can’t be 0 but It can be positive and negative then how tf do I do everything but 0”

Then that’s where the U came in my mind

I was like

Oh yea

[echo wagon]

So you can't always combine inequalities, sometimes you have to end up using union because there are two different parts to the domain

[grim sand]

I understood the first half

I’ll pretend I understood the second

[echo wagon]

Of what I said?

[grim sand]

Yes

[echo wagon]

Lol, like what you did here

You couldn't write the domain as one interval

So you have to use two intervals and take their union

[grim sand]

Oh

I understand now

Now how do I do the squares

[echo wagon]

Do you know what 3^2 means?

[grim sand]

Question 8

Yes

That means

3x3

[viscid thistle]

Yeh

[echo wagon]

Okay just checking

Have you worked with parabolas?

[viscid thistle]

Do you know the identity (a+b)²?

[grim sand]

For range don’t I just have to do inverse functions

Have you worked with parabolas?
@echo wagon I probbaly have

Probably*

But I don’t remember the names for them because there’s so many names

[echo wagon]

It's the graphs that almost look like the union sign lol

[grim sand]

Yes

[viscid thistle]

,w plot x²+2x+1

Look lol

[grim sand]

Yes I work with that

[viscid thistle]

Or even this

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Al𝟛dium#4084

[echo wagon]

And do you know what the turning point of a parabola is?

[grim sand]

Yes

The vertex

[echo wagon]

Yes

What's that?

[grim sand]

The highest/lowest point of the parabola

[echo wagon]

Right

[viscid thistle]

Ding ding ding

[echo wagon]

Okay, so you know the parabola can go up or down right?

[grim sand]

Yes

[echo wagon]

If it goes up, then all the y values above the y coordinate of the vertex is in the range

Does that make sense?

[grim sand]

No?

I thought domain was X values only

[echo wagon]

I meant range

[grim sand]

Oh

Then yes

[echo wagon]

And if it goes down, then all the y values below the y coordinate of the vertex are in the range

So to find the range of a parabola, you need the vertex and you need to know if it's going up or down

[grim sand]

It’s an equation

So I have to graph it

[echo wagon]

Graphing it works, but how would you find out what the vertex is to draw it?

[grim sand]

Oh

I don’t know

I just graph it using desmos

[echo wagon]

Ah

Okay, quick recap on parabolas

The standard form is $y = ax^2 + bx + c$

[obsidian monolith]

Lunasong:

[echo wagon]

Do you know that?

[grim sand]

Yes

[echo wagon]

If a > 0 then it's going up, and if a < 0 then it's going down

Do you know that?

[grim sand]

I didn’t

But now I do

[echo wagon]

Okay

Now, the form they gave it to you in is not the standard form, it's called the vertex form because it automatically gives you the vertex

If $y = a(x-p)^2 + q$ then the vertex is $(p, q) $

[obsidian monolith]

Lunasong:

[echo wagon]

Does this ring a bell?

[grim sand]

So In this case

Vertex is -3,-3

[echo wagon]

No

[viscid thistle]

Almost

[echo wagon]

Remember there is x - p already in the bracket, and just p is the x coordinate

So if it's x - 3, then what is p?

[grim sand]

3

[echo wagon]

Yes

[viscid thistle]

Yesyesyds

[grim sand]

So they don’t touch the -

[echo wagon]

The what?

[grim sand]

Negative

Or minus sign

[viscid thistle]

And if you see a +7 then the p is just -7

[grim sand]

Oh

So it’s the opposite of whatever I see

[echo wagon]

Yes

[grim sand]

Oki

[viscid thistle]

Because -(-7)=+7

[echo wagon]

A short explanation: anything squared is always greater than or equal to 0. Do you agree?

[grim sand]

Yes

It can never be negative

[echo wagon]

So if you have a(x-p)^2 then the smallest it can be is 0

[grim sand]

Yes

[echo wagon]

So if you have a(x-p)^2 + q then the smallest it can be is q

Does that make sense?

[grim sand]

Why q

If Q was 5 and P was 2

2^2 would be 4

[echo wagon]

Well the smallest the first term can be is 0, so if you add q the smallest it can be is q

[grim sand]

[viscid thistle]

0+q=q

[grim sand]

Oh

[echo wagon]

If x > 0 then x + 5 > 5, right ?

[grim sand]

Yes

[echo wagon]

So if a(x-p)^2 >= 0 then a(x-p)^2 + q >= q

Does that make sense?

[grim sand]

What if P is smaller than Q

What happens

[echo wagon]

It doesn't matter

[viscid thistle]

getting out of road

[grim sand]

So I won’t ever see it in an exam or something

[echo wagon]

Oh, it could happen

I mean it does not affect the fact that the smallest value is q

I'm not saying the smallest value of p and q is q

[grim sand]

Okay so what ever number Q is

It’s the smallest

[viscid thistle]

No

[echo wagon]

I should clarify a little. I made a small over simplification

[grim sand]

Huh

[viscid thistle]

Okay i think this is getting out of road

[grim sand]

[echo wagon]

So if a(x-p)^2 >= 0 then a(x-p)^2 + q >= q
@echo wagon This is true if a is greater than 0

If a is less than 0 then q is the biggest value

[viscid thistle]

Like, out of what we need you to know to solve the problem

[echo wagon]

That's true, but I think motivation for why the vertex form gives the vertex is good

[viscid thistle]

Yeah maybe

[grim sand]

I’m a man with a lot of questions

So A smaller = Q bigger

[echo wagon]

Let me give one more attempt, and then you can ignore me if it doesn't make sense

[grim sand]

A bigger = q smaller

A bigger/smaller than 0

[echo wagon]

Anything squared is >= 0

[viscid thistle]

Let me give one more attempt, and then you can ignore me if it doesn't make sense
Okay gl

[echo wagon]

So (x-p)^2 >= 0

If a > 0, then we get a(x-p)^2 >= 0

Then a(x-p)^2 + q >= q

[grim sand]

No

[echo wagon]

Does that make sense or not at all?

[grim sand]

Made sense till the end

Where did Q come from

[echo wagon]

We are adding q to both sides of the inequality

In the same way if x > 0 then x + 5 > 5

[grim sand]

Oh

That makes sense

OH

That makes sense

If A>=0

[echo wagon]

So if a > 0, then q is the smallest value of a(x-p)^2 + q

[grim sand]

Yes I understand now

[echo wagon]

If a < 0, then the sign of the inequality flips and the biggest value is q

Also, when do you get that biggest/smallest value? When the first term is zero. So when x - p = 0 or x = p

So the vertex is (p, q) because p is the value of x that makes the first term 0, and then gives you the biggest/smallest value q

[viscid thistle]

,w 4x²+x-1 & -4x²+x-1

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Al𝟛dium#4084

[viscid thistle]

Look how a>0 and a<0 changes

[grim sand]

It flips

[echo wagon]

Okay, so the vertex is (3,-3)

And the parabola is going downwards

And we said if it goes downwards, then all the y values below -3 is in the range

So what's the range?

[grim sand]

Yes

Range would be

[viscid thistle]

That looked like the meme

[grim sand]

(INF, -3]

[echo wagon]

Yes!

[viscid thistle]

,w plot -3(x-3)²-3

You can see it yourself

But yes

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Al𝟛dium#4084

[echo wagon]

And if the parabola was going upwards and the vertex was at (7, 6), what would the range be?

[viscid thistle]

We want to have a visual perspective here

[grim sand]

6,INF

[viscid thistle]

Yessir

Parens

But yes

[echo wagon]

Good

And the domain?

[grim sand]

7,INF

[echo wagon]

No

[viscid thistle]

No

[grim sand]

Oh

[echo wagon]

For what values of x is (x-7)^2 + 6 defined?

[grim sand]

0

[viscid thistle]

Defined

No

[echo wagon]

Only for 0?

[grim sand]

Infinite

[echo wagon]

What does that mean?

[grim sand]

Any number

[viscid thistle]

All real numbers or (-inf, inf)

[echo wagon]

Yes!

[viscid thistle]

Better

[echo wagon]

Good

So it's defined for all numbers, so the domain is all numbers, like Al said

[grim sand]

Since it’s a parabola

It doesn’t end

Yes

The graph just keeps going

So both the X values will be infinite?

[echo wagon]

So the domain of a parabola is always all of R (unless the domain is restricted, and if you don't know what this means, don't worry)

What do you mean both the x values?

[grim sand]

There’s 2 lines

So isn’t there multiple X’s?

[viscid thistle]

Nono

I was showcasing 2 functions

Not one

,w plot 4x²+x-1

[grim sand]

No look see it’s two lines going down from the vertex

[viscid thistle]

Yeah that's how parabolas are, the vertex acts like the symmetry line

[proven marten]

point*

sorry for butting in lol

[viscid thistle]

Nah it can represent the symmetry lime

Lol

[echo wagon]

Fight

[grim sand]

Lmao

Don’t fight

It confuses me more

[proven marten]

Isnt that the axis of symmetry though

[viscid thistle]

Symmetry point doesn't exist

Lol

[grim sand]

What is a symmetry point

[viscid thistle]

Isnt that the axis of symmetry though
Whatever

Same stuff

[echo wagon]

The line through the vertex is the axis of symmetry

[proven marten]

I agree

Luna making us fight >:(

[viscid thistle]

Indeed

We friends

[grim sand]

So the tip is the axis of symmetry

[echo wagon]

The line through the vertex is the axis of symmetry
@echo wagon the vertical line

[grim sand]

Smart people arguing and I’m just here like
👁️👄👁️

[proven marten]

The x coordinate yes is the axis of symmetry

[echo wagon]

You say a point can't be an axis, but now you want an x coordinate to be an axis. Hmmm

[proven marten]

What

[grim sand]

I’m lost here

[echo wagon]

Smart people arguing and I’m just here like
👁️👄👁️
@grim sand anyway, you understand the range now?

Nevermind, haha

[grim sand]

I understand range for that

But now I don’t understand domain for it

Just when I thought I understood domain

[echo wagon]

You said it's defined for all values

That's right

[grim sand]

So it’s just always infinite for a parabola?

[echo wagon]

So what does the set of all real numbers look like as an interval?

[proven marten]

Could you understand that the vertex x coordinate is either the maximum or minimum point of a parabola

[grim sand]

Infinite

[echo wagon]

Write it as an interval

[undone pawn]

oh woah this is still going on

[grim sand]

They’re confusing me with their arguing

[echo wagon]

Not on the same question

[proven marten]

We're not arguing !

[undone pawn]

oh nice

[grim sand]

With their debate

[echo wagon]

We're not really arguing, and we aren't 'arguing' about the range

So focus on the range

[proven marten]

[echo wagon]

How do you write the set of all real numbers as an interval?

[grim sand]

I don’t know

...

What are real numbers

[echo wagon]

(something, something)

[undone pawn]

ah philosophy

What are real numbers

[proven marten]

TOK

[echo wagon]

Real numbers are just numbers as you are familiar with them

[proven marten]

Lol

[echo wagon]

So what does the set of all numbers look like?

[proven marten]

^

[grim sand]

They always either look like (1,1) or x=0

[echo wagon]

What do you mean by that?

[grim sand]

Yes 😃

Umm

Sets of numbers are those groups right

So it would just be (number, number)

[proven marten]

Its going to be hard for him to determine the interval for real numbers without him knowing the defintion

[undone pawn]

I dont think they're comfortable with set notations and what "infinity" means in this context

[grim sand]

What does that even mean

[proven marten]

Do you know what

[undone pawn]

cause a while back we were discussing about domains of functions and similar issues cropped up

[proven marten]

infinity is

[grim sand]

Yes

[echo wagon]

@grim sand Okay, you know (1, inf) is all the numbers greater than 1, right?

[grim sand]

Oh

I thought it was X and Y

[proven marten]

cause a while back we were discussing about domains of functions and similar issues cropped up
@undone pawn harvest

[viscid thistle]

#chill for the procrastinators here

[echo wagon]

And (-inf, 1) is all the numbers less than 1

[proven marten]

No, that is not how intervals work?

[undone pawn]

right see they're getting confused by set notations and cartesian points

[grim sand]

Oh

[uncut mulch]

i knew it

[undone pawn]

which ramonov tried to clear up earlier

[uncut mulch]

hah

[undone pawn]

lmao

good call

[grim sand]

Confused here with all the big words..

[undone pawn]

understandably so

[proven marten]

Imma head out, too many people explaining

[undone pawn]

someone needs to explain to you the whole idea of sets

[grim sand]

Luna explained it

[undone pawn]

it's unfortunate that they use the same notation as points though

[grim sand]

It’s the numbers that come together as groups

Like (3,5) or (4,5)

Right

[undone pawn]

no wait.

do you know what it means???

what does the set interval (3,5) mean?

[uncut mulch]

it depends on context really. with proper context, there shouldn't be a huge issue distinguishing whether parentheses notation is being used to represent a point or interval

[undone pawn]

as in if I say, x belongs to (3,5)

what does that tell me about x?

[echo wagon]

@grim sand What they mean is, on the curve y = x^2, (2, 4) is a POINT on the curve. This refers to a specific point with two coordinates. But (2,4) as an interval is the set of all numbers between 2 and 4. These are written in the same way, but are not the same thing. Do you understand that?

[grim sand]

It means

Anything between 3 and 5

[undone pawn]

ok yep that's right

[proven marten]

goodu

[undone pawn]

I thought it was X and Y
not sure why you got confused in the middle then

[grim sand]

@grim sand What they mean is, on the curve y = x^2, (2, 4) is a POINT on the curve. This refers to a specific point with two coordinates. But (2,4) as an interval is the set of all numbers between 2 and 4. These are written in the same way, but are not the same thing. Do you understand that?
@echo wagon yes

[echo wagon]

Okay, good

[grim sand]

So when used on a graph it’s a point

But right now we using it as a set

[echo wagon]

Now, for the domain of a parabola, I'll just tell you what it looks like because maybe you haven't seen it before. The domain is (-inf, inf). Can you see how this would mean x can be any number?

[grim sand]

Yes x can be any

Since the parabola is endless

[viscid thistle]

Good

[echo wagon]

Yes, so the set containing all numbers is written as (-inf, inf)

So that's the domain

[grim sand]

Why is the second one a positive

But the first is negative

[proven marten]

(smaller, bigger)

[grim sand]

If the parabola was facing up

Would the negative and positive change

[viscid thistle]

Only 2 ppl max here please.

[proven marten]

I count 3 without me anyways

[grim sand]

There’s like 10 ppl explanation

I’m trying to understand all

[undone pawn]

lol

[viscid thistle]

Everyone wants to help, but we can confuse him

[undone pawn]

I'll just lurk

[viscid thistle]

So only me and Luna as always

[grim sand]

Yes

[echo wagon]

Remember how (-inf, 1) means everything less than 1 but there is no limit to how small it can be? And (1, inf) means everything bigger than 1 but there is no limit to how big it can be. Now (-inf, inf) means there is no limit to how big or how small it can be. It's just the way we write intervals. If there is no limit to how small it can be, we have a -inf first, and if there is no limit to how big it can be, we have a +inf second

[viscid thistle]

(0, inf) would only mean x>0

[grim sand]

This isn’t very accurate

But

[viscid thistle]

We want to include negative numbers too

[grim sand]

The domain

Is only that line?

[echo wagon]

I don't know what line you are talking about

[grim sand]

And so -INF would be the left and inf would be the right?

The one that’s pointed on by the arrow

[echo wagon]

That's the parabola

[grim sand]

So the domain

[echo wagon]

The domain isn't the parabola, it is the collection of all the x values on the parabola

[grim sand]

Okay so the range is just from the vertex to the bottom

[echo wagon]

Yes

[grim sand]

The domain is

Oh

[echo wagon]

Can you remember your previous domain exercise where it was like [-3, 3]?

[grim sand]

So the domain is all the X on the parabola

[echo wagon]

Yes

[grim sand]

So it can be any number from 0 to infinite

And 0 to -infinite

[echo wagon]

Yes

And 0

[grim sand]

So why is the domain -INF and INF

Actually

That makes sense

Yes

[echo wagon]

That's just how we write it, really. Because it fits how we write other intervals

[grim sand]

Can I get an example 😀

[echo wagon]

(-inf, inf) is the interval containing all numbers

Example of what?

[viscid thistle]

^

[grim sand]

Like this

[echo wagon]

All parabolas have the same domain, so can't exactly give a different example

[grim sand]

Question 8

So every parabola domain is just (-INF,INF)

[echo wagon]

Yes

[viscid thistle]

y=4(x+3)²+3

[grim sand]

Oh that’s easy

[viscid thistle]

What'd be p and q

(-inf, inf) also called R or all real numbers

[grim sand]

P would be 3

Q is 3

[viscid thistle]

No

[grim sand]

Negative 3

[viscid thistle]

See you forgot

Yes

[grim sand]

P is negative 3

😃

[echo wagon]

And the domain of every straight line is also (-inf, inf) because x can be any number. Any function where x can be any number, the domain is (-inf, inf).

[grim sand]

(-inf, inf) also called R or all real numbers
@viscid thistle what’s the exact definition of real numbers

[viscid thistle]

Also explained by "x² doesn't give any problems in the domain"

[grim sand]

y=4(x+3)²+3
@viscid thistle
So vertex is (-3,3)

Range is [3,INF)

Domain is (-inf, INF)

Yes

😀

[echo wagon]

Real numbers are just all the numbers you know at this point. There are things called complex numbers, but you don't need to worry about it yet. Just know the reason people specify real numbers is to make sure people know they are talking about the normal numbers that you know and not the complex numbers.

[grim sand]

Oh ok

What are complex numbers

[echo wagon]

Complex numbers allow you to take square roots of negative numbers.

[grim sand]

Oh thats confusing

[viscid thistle]

don't worry yet about complex numbers

[echo wagon]

But let's not digress too far now

[grim sand]

Ok

So did I do it right tho

The equation u gave me

[viscid thistle]

Btw the set of real numbers is represented as $\bR$

[obsidian monolith]

Al𝟛dium:

[grim sand]

Oki

[viscid thistle]

The equation u gave me
@grim sand yeah it seems correct

,w plot y=4(x+3)²+3

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Al𝟛dium#4084

[echo wagon]

So $\bR = (-inf, inf) $. They are the same thing.

[obsidian monolith]

Lunasong:

[grim sand]

How would I graph these tho

Without an auto grapher

[echo wagon]

What?

[viscid thistle]

Nvm

[echo wagon]

Step 1) Find the vertex

Step 2) Determine if it goes up or down

[viscid thistle]

You'd graph it getting key points

[willow bear]

bad tex!!!!!!!!!!!

there's a dedicated infinity symbol!!!!!!!!

[viscid thistle]

Lol i knew it

[willow bear]

$\bR = (-\infty, +\infty)$

[obsidian monolith]

Ann:

[undone pawn]

lmao

[echo wagon]

Haha, I know. I use inf, because that's what they have to type in for their solutions

[grim sand]

To determine if it goes up or down

It’s A rifht

Right*

[echo wagon]

Yes

[grim sand]

Wether it’s -A or positive A

[viscid thistle]

Yeah the coefficient of x²

[echo wagon]

Step 3) If there are x or y intercepts, find them

And then draw just connect the dots in the right shape

Whether a is positive or negative

[grim sand]

How do I keep it going

[echo wagon]

You can't keep going forever obviously

Just add arrows at the end to show that it keeps going

[viscid thistle]

You can draw an arrow at the end of each branch

[grim sand]

Oki