precalculus

[clear glade]

do I change the base to anything or keep it as e?

[rigid sun]

let me see your graph

[rigid sun]

change your base to 4

[clear glade]

ok

can you explain why we changed (1/4) to 4 (or -4, forgot what it was turned in to )

which exponent ruiling is that

[rigid sun]

ok so to get from power to power, you mulitiply

ie x^2*x=x^3

right?

$x^2*x=x^3$

[clear glade]

right

[obsidian monolith]

maleb1964:

[clear glade]

yea

[rigid sun]

ok, so what if we wanted to go backwards?

then, we divide by x

this means

x^2/x=x^1

and then

[clear glade]

x^2/x=x^1$

[rigid sun]

x^1/x=x^0

[clear glade]

$x^2/x=x^1$

[obsidian monolith]

Waterblade:

[clear glade]

yep

[rigid sun]

$x^1/x=x^0$

[clear glade]

$x^1/x=x^0$

[obsidian monolith]

maleb1964:

Waterblade:

[rigid sun]

then

[obsidian monolith]

maleb1964:

[rigid sun]

ok well you get it

[clear glade]

yea

[rigid sun]

but you see

x^0=1

[clear glade]

yep

[rigid sun]

so we can replace x^0 with 1

and we get 1/x=x^-1

this proves that negative exponents are simply fractions but instead of multiplying we divide

[clear glade]

right

[rigid sun]

so b^-2= 1/(b^2)

[clear glade]

yep

[rigid sun]

ok

that's where we get 1/4^x= 4^-x

[clear glade]

I see now

[rigid sun]

ok

can change your base to 4

[clear glade]

ok

base is now 4, vertical shift 2, and reflected over y axis

[rigid sun]

ok

let me see it!

[rigid sun]

hey

it's almost there

but I think your vertical shift is off

[rigid sun]

change your vertical shift to zero

change your vertical stretch to 2

[clear glade]

why

oh

[flint vale]

Bc it’s coefficient

[rigid sun]

idk if i said that wrong or something

[flint vale]

No I think you fine

I think you confirmed a mistake when you said show

But I mean that’s fine

[indigo sandal]

Looking good partner

[rigid sun]

that's right

[flint vale]

Hohyah

[rigid sun]

[clear glade]

let's click that check answer

im sweating

not actually, but i'm nervous lol

[flint vale]

Just do it and move on

[clear glade]

👌

[rigid sun]

ok

[indigo sandal]

Nice!

[rigid sun]

lets goooo

[clear glade]

thanks so much

[rigid sun]

ok

[clear glade]

lol

[rigid sun]

things to pull away from this

1. the outside coefficient (2 in this case) will be the vertical stretch

[flint vale]

1

[rigid sun]

2. fraction exponents (1/b)^x are equal to b^-x, which is simply b^x graphed across the y axis

[flint vale]

WRT?

[rigid sun]

@clear glade I can give you a complete runthrough of how to graph these

instead of individual examples

[clear glade]

that would be nice

[rigid sun]

ok

[clear glade]

probably going to sleep soon

tho

[rigid sun]

it will be similar to what I told you earlier

don't worry, it will be fast

[clear glade]

ok nice

let's do it

[rigid sun]

so

[clear glade]

do u wanna take it to pm?

[rigid sun]

here is ok

[clear glade]

ok

[rigid sun]

1. reset your origin

this means to use vertical or horizontal shifts, if the equation has +2 at the end of it, then move the origin up 2, this will be your new origin

[clear glade]

I get it, and I'm guessing -2 at the end of it would be down 2, right?

[rigid sun]

yes

ill graph an example real quick

the grey one is x^2 +1

the blue is x^2

ok

imagine moving the origin up one

if you're graphing by hand, you can draw an imaginary line to simulate the x axis moved up 1

[clear glade]

Ok, undertandable. Since the +1 is at the end of the equation you moved it up by 1

[rigid sun]

yes

[clear glade]

Oh I see

[rigid sun]

step 2. make your horizontal stretches

this could mean flipping across the y axis, compressing, stretching

2x = twice as tight

1/2x= twice as loose

-x= flip across the axis

[clear glade]

alright, those operations make sense

[rigid sun]

lastly, do your vertical stretch

[clear glade]

wait

so the horizontal stretch is the exponent

[rigid sun]

it is the coefficient on the x part

[clear glade]

so if f(x) = 5^1/2x it would be 1/2

oh

[rigid sun]

yes

[clear glade]

oh ok

[rigid sun]

the stretch would be a 1/2, which means twice as long

step 3. vertical stretch

just multiply the height of each point by the coefficient

this is relative to the new origin

this will be the k in kb^x

it will be multiplying b

ill do a very fast example, then give you the steps one last time

[clear glade]

Alright

[harsh cipher]

hi guys

[rigid sun]

@clear glade

[harsh cipher]

the very left equation

[rigid sun]

1. move your origin

[clear glade]

I see @rigid sun

[rigid sun]

2. adjust for horizontal stretch

[harsh cipher]

how is sqrt -9(x-5) -7 equal to the left two equations

[rigid sun]

3. adjust for vertical stretch

[harsh cipher]

sorry to interrupt

[clear glade]

Althought what do you mean by move it upside down

[rigid sun]

it means flipped over the x axis

[clear glade]

Oh alright, that's what I thought

@harsh cipher it's ok, we're about to finish

[rigid sun]

ok i fixed it

[clear glade]

Lol nice

[rigid sun]

keep practicing more and you'll get the hang of it!

[clear glade]

Thanks so much

Quick question

[rigid sun]

let me see if I can find you a practice website before I let you go

ok fire away

[clear glade]

Ok

So horizontal stretch is associated with side? The exponent or the leading coefficient

[rigid sun]

the exponent

[clear glade]

Alright

[rigid sun]

it will be a number on the x

like b^2x

[clear glade]

I see

[rigid sun]

yeah

uhhh

if you have a Khan academy account

here is some practice

[clear glade]

I think I do

[rigid sun]

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:exp-graphs/e/graphs-of-exponential-functions

[clear glade]

Thanks

Will try it tomorrow

Hopefully

[rigid sun]

if you finish that, you can always just make up examples, try to graph them, and check them with a graph on the computer

[clear glade]

Yeah

[rigid sun]

👌

[clear glade]

Well, thanks again. Appreciate you talking your time to help

[rigid sun]

gg

np

[clear glade]

Gg good night

[rigid sun]

peace

[clear glade]

@harsh cipher sorry to bury ur question, u could post it now so it can be more noticiable to others

[harsh cipher]

okay!

its okay! we are all here to learn and help each other

[clear glade]

Bye now

[rigid sun]

alright dude

lemme see

[harsh cipher]

hahaha

it's quite simple I just don't like to think.....

😦

[rigid sun]

lol

[harsh cipher]

lol

[rigid sun]

then we'll make thinking easier

mathematicians do that all the time

[rigid sun]

lets see if we can simplify ur problems

[harsh cipher]

awesome

so how did -9 get simplified to get 3 outside of the square root

[rigid sun]

they left the negative on it

you see the top right radical

[harsh cipher]

yes

[rigid sun]

they took -9, separated it into 9*-1

then they took the sqrt of 9=3, moved that to the outside, and left -1 on the inside

that's why on the top right green one, there's a negative on the outside of the parenthesis

thats the -1

you get that?

[harsh cipher]

so when I simplify bottom left. I should get the equation on the top right

[rigid sun]

other way around

[harsh cipher]

and if i expand that I get the one on the btm right

okay

[rigid sun]

$\sqrt{-9(x-5)}$

[obsidian monolith]

maleb1964:

[rigid sun]

$\sqrt{-1*9(x-5)}$

[obsidian monolith]

maleb1964:

[rigid sun]

$3*\sqrt{-(x-5)}$

[obsidian monolith]

maleb1964:

[harsh cipher]

I get it.

If we simplify Sqrt -9(x-5)

we get 3 sqrt -(x-5)

I don't get it when you say other way around though.

[velvet kelp]

Can i get help with this word problem

[uncut mulch]

how would you find the remainder when p(x) is divided by (x-2)?

[velvet kelp]

just put in x = 2 for the x value in p(x)

[uncut mulch]

yep

the same applies to dividing by (x+1)
you are told the remainders are equal and you should be able to figure the rest out

[velvet kelp]

but Im left with 2 undefined variables

[uncut mulch]

what do you mean 2?

[velvet kelp]

i mean idk the value of the remainder and C

[uncut mulch]

you are told the remainders are equal so what will you be able to do?

or how would it relate
p(2) and p(-1)

[willow bear]

god forbid you write a statement saying that two things are equal when you are told explicitly that they are equal

[velvet kelp]

ok but it doesn't make sense to me i don't know the remainder or c i know they're equal but how do I do the equation

[uncut mulch]

what does p(2) represent?

[velvet kelp]

the remainder

[uncut mulch]

what does p(-1) represent?

[velvet kelp]

the remainder

[uncut mulch]

you're told the remainders are equal, what would be your equation

[velvet kelp]

ohhhhhh

both the equations would be equal

i would just isolate for C

[willow bear]

ok but it doesn't make sense to me i don't know the remainder or c i know they're equal but how do I do the equation
do you know what an equation actually IS

[heady jewel]

loll ann nice name

have you heard from tuong tho

[shrewd urchin]

😂

[willow bear]

@frozen needle

[frozen needle]

lmao wut xd

[willow bear]

someone called me "discount tuong"

oh it was @viscid thistle

i might not have translated it correctly into french

[last linden]

that says reduced priced tuong?

[frozen needle]

[willow bear]

so evidently, i'm discount^2 SA

[royal gull]

No it was me

[last linden]

haha

[shrewd urchin]

Ex.10

I cant get the steps what they did?

[last linden]

thats the magnitude of the component of F in the direction of V

@shrewd urchin

do you know this formula
a dot b = |a||b|cos(theta)
they just rearranged it

@shrewd urchin helloo

[shrewd urchin]

Gotch @last linden

Gotcha*

[viscid thistle]

Who summoned... Oh XD

[ripe wave]

I'm a bit lost on how to show the work/solve

[willow bear]

do you know what function composition is?

[ripe wave]

Sort of

[willow bear]

there is no "sort of". either you know that $f \circ f \circ f \circ f(-2)$ refers to $f(f(f(f(-2))))$ or you do not.

[obsidian monolith]

Ann:

[ripe wave]

ok then I don't what function of composition is

[willow bear]

it's not "function of composition", it's "function composition", aka "composition of functions", and you should really go look that up.

[pseudo sonnet]

can someone explain what the hell is going on

https://i.imgur.com/wfgp6Iv.png

[serene heath]

t!wiki rational root theorem

rip bot 😔

its a theorem that helps you find rational roots of a polynomial

[patent beacon]

Natural question, do you know what the rational root theorem is?

Have you tried googling it, and are having difficulty with a part of it?

[pseudo sonnet]

yeah its basically P / Q then you do synthetic division till u get remainder 0

i watched some vid on it

if my teacher puts a factored form like this on the test to waste my time im gonna lit her ass up

😂

[patent beacon]

No, that's not the rational root theorem

[solid reef]

Can I get help for #2

Please good sirs

[proud sparrow]

so, what have you tried @solid reef

Do you understand what rationalising the numerator means?

[solid reef]

ive tried to simplify the denominator

but im not sure if i have to complete the square for it

and do the same to the top

i should know this but its been so long

kinda need a refresher

:PPP

[short sorrel]

@solid reef "rationalizing the numerator" means "get rid of the radical"

what's the way we usually get rid of a radical? well, we square it

but you might notice a problem if we just try and square the numerator

$(4 - \sqrt{x})^2 = (4 - \sqrt{x})(4-\sqrt{x}) = 16 - 8\sqrt{x} + x$

[obsidian monolith]

Namington:

[short sorrel]

we're still left with a sqrt(x) term

fortunately, we have a strategy to get rid of it:

rather than just squaring the numerator, multiply it by the conjugate

$(4 - \sqrt{x})(4+\sqrt{x}) = 16 - 4\sqrt{x} + 4\sqrt{x} + x = 16 + x$

[obsidian monolith]

Namington:

[short sorrel]

because one is positive, and the other's negative, the middle terms cancel out

leaving us only with the squared terms!

ok, thats great, but we can't just randomly multiply the numerator by (4 + sqrt(x))

what we can do is multiply a fraction by 1

this won't change the value

so it's fine

and note $1 = \frac{4+\sqrt{x}}{4+\sqrt{x}}$

[obsidian monolith]

Namington:

[short sorrel]

so

$\frac{4 - \sqrt{x}}{16x - x^2} \ \ = \frac{4 - \sqrt{x}}{16x - x^2} \cdot 1 \ \ = \frac{4 - \sqrt{x}}{16x - x^2} \cdot \frac{4 + \sqrt{x}}{4 + \sqrt{x}}$

[obsidian monolith]

Namington:

[short sorrel]

now multiply.

(and then simplify afterward, if applicable)

in general, "rationalizing" a binomial with a square root usually means "multiply by the conjugate"

[solid reef]

Thanks man

Helps a ton

Loving this discord

Appreciate everyone here ❤️

[half gyro]

Hello! Could anyone answer a graph question for me? I'm still poor at understanding graphs, and now I'm getting to one with no numerical points.

I don't really understand question c.

I figure with a and b I'm looking at the line and matching the x and y, but z isn't on the graph

[patent beacon]

Suppose f(b) = z
That's enough info to get a numerical value for z. What is it?

[half gyro]

b?

[patent beacon]

f(a)
Means to find a on the x-axis, go up until you hit the function, then find how high you are on the y-axis

So when I talk about f(b), I mean to find b on the x-axis, then go up/down to the function.

However, f(b) is a special case

No need to get too technical. The graph is doing something interesting around b. What is it?

[half gyro]

Ah... going through it?

[patent beacon]

Yep, precisely. Now, if we go to b and go up/down until we meet the function, you've recognized we don't need to go up/down at all

That is, the graph has zero height at x = b

Saying the same thing, f(b) = 0

That part make sense?

[half gyro]

Because b is at zero on the x axis?

[patent beacon]

Nope, because the function has zero height above/below b

This notation is important:
f(location on the x-axis) = height above/below x-axis

[half gyro]

Oh... so you're saying that because it's crossing through, it has zero height?

[patent beacon]

Yup

f(b) = 0
But we supposed that f(b) = z
So z = 0

Now, the question wants f(z), which is really just f(0) in disguise

[half gyro]

OH I see... so r, because it also has zero height, but on the y axis?

[patent beacon]

Remember, when I say f(0), I want you to find 0 on the x-axis and go up/down to the function. Then, use marks on the y-axis to see how high you went

You're right, it's r

[half gyro]

That makes so much sense now that you've explained it. So I just go up/down how many marks the x axis did, and find the corresponding y?

[patent beacon]

Yeah, your graph is essentially a line that matches x values to y values. If you've ever played battleship then you know how this works

[half gyro]

Thank you so much!! I appreciate your time and help.

[viscid thistle]

[lone prairie]

Hey guys I really could use some help on this question. More specially first on finding the function formula for g(x) and h(x)

[proud sparrow]

you don't need a formula for them

also, you are not told to find a formula

@lone prairie

[lone prairie]

Where would I start on question g. Do I just plug in 3 for x and then ..?

[proud sparrow]

yeah, then..?

[lone prairie]

Then multiply the Y of F and H when x = 3 ? Sorry I'm terrible at this

[stuck lark]

,rotate 270

[stuck lark]

,rotate 180

[lone prairie]

No good? @stuck lark

[stuck lark]

@lone prairie well done 👍🏽

[lone prairie]

Thanks!

My last question is how would I go about finding g(2) in this situation ?(trying to solve question h)

[stuck lark]

@lone prairie what did you try?

[proud sparrow]

@lone prairie read off the graph

[shrewd urchin]

Use magnifying glass

[lone prairie]

I think I would first need to find the slope which I did here. Slope = 3/1 ( if I did that properly?)

[stuck lark]

no need to find slope at all, just find g(2) and h(2)

[lone prairie]

Well I know h(2) = 7 according to the table

I'm struggle with g(2) since it's on the graph but I don't know the equation of the line

[stuck lark]

you don't need the equation of g(x) if you already have the graph of g(x)

[lone prairie]

I see but g(x) is only shown on the left side of the graph so it never visually crosses to g(2)

[stuck lark]

for g(2), what x value are you looking at?

[lone prairie]

Oohh maybe LOl I think I got it. I was looking at the wrong line. G(2) = 4 ?

[stuck lark]

g(2) = 4 👍🏽

[lone prairie]

God I'm an idiot thanks. So (g + h)(2) = 7 + 4 = 11?

[stuck lark]

there you go

[lone prairie]

Thanks a lot for the support 🙂🙃

[stuck lark]

no prob

[harsh cipher]

hey guys

nm. the answer doesn't use spacing so it confused me.

I answered my own question. 🙂

[stuck ivy]

I'm having trouble simplifying this fraction it goes through many steps

I relearnt all the laws and stuff and I got a final answer of - a - 4a/ 4b+a

I made sure to use difference of cubes and factored out a negative to make (4b - a ) into (a - 4b)

[stuck ivy]

We had a good run

[formal siren]

Can someone please explain how to simplify cos(2tan^-1(x))?

[rocky bison]

@formal siren You still there?

[formal siren]

What’s up

@rocky bison

[proud raven]

stelios you get it figured out

i know its been a while

[formal siren]

Nah I haven’t yet

[heady jewel]

@formal siren what have you tried

[formal siren]

I literally don’t know how

I think it’s already at its simplified term

[pseudo sonnet]

https://i.imgur.com/gr3XAFt.png

[stuck lark]

what'd you try

[proud sparrow]

so what does that give you?

[pseudo sonnet]

(3x+1)(x-1)/(x-1)

[pseudo sonnet]

https://i.imgur.com/skLSZBH.png

does this solution satisfy Question 2?

[willow bear]

this is a straight line

with a hole

[pseudo sonnet]

😂

[willow bear]

it could be said to be its own asymptote, but then again that would be dumb

try adding 1/x to it

[pseudo sonnet]

like that?

[willow bear]

yes

[pseudo sonnet]

https://i.imgur.com/VB2PGMg.png

anyone have any ideas

completely lost tbh

[restive knoll]

Guys, how to find x?

[willow bear]

@pseudo sonnet P(x) = ax^3 + bx^2 + cx + d

[proper kindle]

if you have positive reals x,y,z satisfying xy+yz+xz = 1

is it true that the maximum value of x+y+z is sqrt(3) ?

[willow bear]

idk, is it?

[proper kindle]

I'm not sure, that's why I'm asking @willow bear

Do you want me to rephrase my question?

[willow bear]

how did you get sqrt(3)

[proper kindle]

Through computer

If you want, here is what I've tried:

you can substitute tan A/2, tan B/2 and tan C/2 to make use of the identity

[languid dust]

help

[proper kindle]

Then it remains to maximise tan A/2 + tan B/2 + tan C/2

But using CS I seem to get the inequality flipped

@willow bear https://www.wolframalpha.com/input/?i=maximise+a%2Bb%2Bc+if+ab%2Bbc%2Bac%3D1

[willow bear]

@languid dust no.

in all your time here, you've failed to demonstrate sufficient reading comprehension to properly react to people's attempts to help you, choosing instead to ghost them without even letting them know whether you're still stuck or are able to continue the problem.

not once have you responded to a message longer than a dozen words.

[proper kindle]

(I hope that's not me?)

[willow bear]

it's not you.

[proper kindle]

For some reason, I seem to have proven that x+y+z >= sqrt(3)

Even though Wolfram says that's the maximum

[pseudo sonnet]

@willow bear im still confused, how does the ax^3 + bx^2 + cx + d , help me in figuring out what coefficients are of the equation in order to solve P(-1)

[languid dust]

bruh

im literally TRYING TO work this shit out

[oblique meteor]

@pseudo sonnet you plug in the values in the equations

[languid dust]

but you guys tell me stuff i dont understand

like it's another language

[willow bear]

i've attempted to help you at least twice before and have not received any feedback at all

[willow bear]

if there's something you don't understand then just ASK ffs

[languid dust]

bc idk what you are trying to tell me

bc

whenever you do

[willow bear]

but don't leave us all in the dark

[pseudo sonnet]

@oblique meteor i just tried that

[languid dust]

sorry

[pseudo sonnet]

not sure what to do from here

[languid dust]

i dont get the notification

[pseudo sonnet]

ohhh

[languid dust]

if someone does respond bc im working on the question

[pseudo sonnet]

wait

1 sec

i think i know

[oblique meteor]

P(x) = ax^3 + bx^2 + cx + d
P(0) = d
P(3) = 27a+9b+3c+d

There is a useful hint that all coefficients are integers

[proper kindle]

Ok, I'm going to ask my question and ping helpers in another channel

[oblique meteor]

Which means a, b, c, d are all integers

So P(0) AND P(3) are both integers and factors of 139

[pseudo sonnet]

i got that far lol

[oblique meteor]

Now I believe 139 is a prime number

[willow bear]

,w factor 139

[obsidian monolith]

Display results online and refine query

Requested by Ann#0413

[pseudo sonnet]

yeah i looked it up 139 has no factors

[oblique meteor]

it indeed is, so you can only write it as 139 × 1

So either P(0) = 1 and P(3) = 139 or its the other way round

Now since a, b, c, d are non negative, so which one do you think is larger? P(0) or P(3)? @pseudo sonnet

[pseudo sonnet]

p(3)

[oblique meteor]

so P(0) = 1

You got your d

@pseudo sonnet Can you guess what to do next

[pseudo sonnet]

im plugging d = 1

into 27a+9b+3c + 1 = 139

then subtracting 1 on both sides

[oblique meteor]

That's good

[pseudo sonnet]

i got this far

[oblique meteor]

That's nice. That's your first equation.

Now similarly, you can factorise 689 and get two more equations in terms of a b c

[pseudo sonnet]

like that?

wait

1 sec i know something else i can do

[oblique meteor]

d=1

[pseudo sonnet]

😂 1 sec

[oblique meteor]

What, no

8a+4b+2c+1 is not equal to 689

[pseudo sonnet]

yeah i was thinking about it and made no sense

[oblique meteor]

It is some factor of 689.

And so is (a+b+c+1)

Both factors are greater than 1 because a b c are all greater than 0

So 1 and 689 aren't the factors at play here.

So factorise 689

[pseudo sonnet]

nvm

13, 53

[oblique meteor]

Yep

You should know which one is gonna be 13 and which one 53

[pseudo sonnet]

a + b + c + 1 = 13

and 8a + 4b + 2c + 1 = 53

[oblique meteor]

Yes.

Simplify them as much as you can and you have 2 more equations

[pseudo sonnet]

im guessing now i pick 2 equations

systems of equations?

[oblique meteor]

Method of elimination

Try doing some operations and eliminate c to get two equations in terms of a and b

[pseudo sonnet]

the professor's brain to make the solution of P(-1) = -1

ultimate troll

thanks @oblique meteor , @willow bear ♥

[oblique meteor]

Welcome :)

[serene heath]

why is ur screen so reflective lol

[rare zephyr]

I thought it was an edit

[analog owl]

😂

[serene heath]

anyway thatd just be x<=-2 union x>=4

[willow bear]

$(-\infty, -2] \cup [4, +\infty)$

[obsidian monolith]

Ann:

[rare zephyr]

I think it's x<=-2 and x>=4

[willow bear]

you are specifically asked for interval notation

are you able to write the blue and the green sets individually as intervals? @analog owl

[analog owl]

I think it can

[willow bear]

what's it

[analog owl]

What now

[willow bear]

no

you aren't asked for this

do you even know what interval notation is

[analog owl]

Yes

[willow bear]

what is it then

according to you

[analog owl]

(-inf,2)U(4,inf)

[willow bear]

ok first off

the problem

asks you

to write

your answer

in

interval notation

and yet

AND MOTHERFUCKING YET

you kept making attempts to answer this problem NOT in interval notation

going RIGHT FUCKING AGAINST ONE OF THE PROBLEM'S REQUIREMENTS

come ON

and second, no, that's wrong too

[analog owl]

Oh

I noticed that

The thing is that I take math in spanish

Thats why Im sorry

[fiery wave]

Calm down Ann

[analog owl]

I tried to translate my book

[willow bear]

no fuck you @fiery wave i'm not calming down

[serene heath]

that was uncalled for

[viscid thistle]

ok

[slender prawn]

"a is equal to"

How do I start

[oblique meteor]

Try finding x2, x3 and you should see a pattern.

[slender prawn]

I got x2= 1/(a-1) and x3= 1/(a-2)

So x23 should be 1/(a-22)

Which is equal to 1, so a would be 23, but that isn't a possible answer

[proud sparrow]

read the question carefully

@slender prawn

x24

[slender prawn]

Ow, I took x23 from f(x23)... Thankd

[oblique meteor]

:+1:

[autumn marsh]

reeee i have a precalc test tmrw

and i dont have all my formulas memorized

oops

[stuck ivy]

For this question the answer I get is 2

it's composing functions

[wind igloo]

How do you get 2?

[flint vale]

yeah how

we just plugged x=0 into g then the result into the x of f

and did not get 2

[obsidian monolith]

Digicat195:

[trail badge]

Answer 2 is interesting, but how? 🤔

[bleak lance]

x-9/x+67 determine domain of x

I made x+67=0

And I got (-infinity,-67)U(-67,infinity)

I got that wrong tho

[remote citrus]

Why does this have no reference triangle?

[wind igloo]

What sort of triangle would you draw?

I mean, all of the other reference triangles are right triangles.

With the reference angle as one of the acute angles.

3pi/2 is like pi/2

It'd be 90 degrees.

And you can't draw a triangle with two 90 degree angles.

[harsh cipher]

hi guys

[viscid thistle]

hey

[harsh cipher]

and girls

The square root function, on the right side

the range is y greater than or equal to zero

is that correct?

I think the range should be y is less than or equal to root 2?

[compact raft]

Heyhaving some problems with my math he can someone walk me through the step of solving the quadratic word problem

[harsh cipher]

Is anyone here?

no one is around for help today 😦

[compact raft]

same

[glossy jetty]

Can someone help me understand interval notation

[willow bear]

what about it is giving you trouble exactly

@glossy jetty

[glossy jetty]

I’m confused about the ( and [ and the orders

[willow bear]

"the orders"?

( and [ mean "exclude the endpoint" and "include the endpoint" respectively.

@glossy jetty?

[neon drum]

i dont understand part c

[last linden]

what's your understanding of the meaning of f o g or f(g(C))?

@neon drum

[neon drum]

so you get g(C) then whatever you get there you plug into f() @last linden

[last linden]

yes

you replace the variable for the function f with g(C)

[neon drum]

right I understand that, I'm just not sure what c is asking

f(g(c/(2pi))?

[willow bear]

no

f(C/(2π))

[last linden]

g(C) = C/(2π)
f(g(C)) = f(C/(2π))

[last linden]

you havnt finished the question yet

thats not the answer

[neon drum]

oh so 2pi((c/(2pi)))

?

[last linden]

yes

[neon drum]

thanks

[crude hemlock]

What would be the definition of a “general term” in terms of sequences?

[tame wedge]

Can someone quickly explain how to simplify exp(ln(y)/2)

[willow bear]

y^(1/2)

[tame wedge]

I just forgot that a^mn = (a^m)^n... yikes :/

Thanks

[smoky fog]

Could someone point me in the right direction to get the basics of calculus down? I think it's something I want to learn next, so where do I start?

Like, the most basic, foundation level stuff, thanks

[flint vale]

,rotate 270

[serene heath]

rearrange then factor out

[patent beacon]

@smoky fog
Paul's online notes are awesome and cover pretty much everything

Download a pdf. Stewart's calc is good for the learner

And, calc is probably the most covered subject on YouTube. I've heard professor Leonard is good

[clear glade]

can someone help me with d.)

I got it to x = -2^y+1 + 1

[flint vale]

solve for y

you know how to do that

well y here is f inverse of x but you get the idea

[clear glade]

Hmm

ok

[small cedar]

how do i do this question:

Which of the following has the greatest value: log 2 (3), log 3 (5) or log 5 (11)

i need help, thanks

[flint vale]

well theres two things, directly seeing how many times base goes into result or rearranging

so 2 to the power of what is 3, 3 to the power of what is 5 etc, ur just comparing the what

[ripe dust]

I don't understand why the denominator is x-4 instead of 4-x

what is that kind of sorcery

[rose locust]

how do I get angle C? 43° isn’t correct

[ripe dust]

500*sin(20)/250 ??

more precisely arcsin(500*sin(20)/250)

[rose locust]

Let me show you the bigger question one second

[ripe dust]

43... have you seen the angle...

it's more than 90... why would it be 43

wait that's easy lmfao

[rose locust]

Okay yeah I know I’m dumb whatever you don’t need to rub it in.

[ripe dust]

find the angle D first there is a big triangle bro

you have two sides with one known angle you can use law of sine

after finding d since it's an isosele triangle (idk the word in english but the two are the same lengths) you can find the angle A

and after you do 180 degrees - A

43.16 is the angle D @rose locust

the angle DA(CoastGuard) is also 43.16

so 180-43.16 = 136 degrees

[rose locust]

so I just put the angle in the wrong place i should have put 43 in angle D instead of A

[flint vale]

@ripe dust they just factored out a -1

[ripe dust]

thanks im retard

[flint vale]

Peep the numerator

It’s fine dw

Idk the rest of the question but it’s kinda weird why they’d do that

[ripe dust]

it's an iscosceles triangle so it's the same angle depending on if it's the angle a from the right or the left @rose locust

@flint vale it's some limit stuff

because the limit is 4

so x-4

[flint vale]

Oh

Cool

[pine temple]

https://cdn.discordapp.com/attachments/269573202018041856/623340915393822753/image0.jpg

for this transformation

would it be

(1/2y+2,-x)?

[pine temple]

<@&286206848099549185>

help pls

[pseudo sonnet]

can someone explain why my x intercepts arent on the graph

after i solved them and set them equal to 0

i graphed it on my calculator and there arent any x-ints

[tight geyser]

what calculator u using

if ur math is correct then it should be calculator settings

use wolfram to see if its there, if yes, then that means ur calc settings wrong

[pseudo sonnet]

nah dude my settings are right

how come the x ints arent plotted

nvm my factoring was wrong

there arent any x-ints

[river kindle]

Your graph looks upset.

[idle dust]

hahhaa

[harsh cipher]

hi guys

[smoky fog]

@patent beacon thanks

[hoary valley]

Guys how to know if both lines are Identical or not? I know how to get the slopes and then what ?

[stuck lark]

@hoary valley what else makes up the equation for a line besides slope?

[hoary valley]

Depends on the equation form ?

[stuck lark]

that's true. let's go with this one: y = mx + b... what do m and b represent?

[hoary valley]

m = slope , b is y-intercept

[stuck lark]

yep, so how can you tell if the two lines are identical?

[hoary valley]

If they have the same y intercept and slopes ? 😅

[stuck lark]

there you go

[hoary valley]

@stuck lark In the answer sheet it says that problem is "True" they're identical,, but when I solved it I got the same slope but the y-intercept had different signs but the same number

[stuck lark]

@hoary valley show us your work

[hoary valley]

Omg

Nvm 😓

I got it right the 2nd time, Thanks you much ❤ @stuck lark

[stuck lark]

no problem man

[hoary valley]

@stuck lark If I got the same slope but different y-intercept that means the lines are Parallel, right ?

[stuck lark]

yep 👍🏽

[hoary valley]

@stuck lark What about Perpendicular ? I know their slopes should be like reversed with different signs but what about the y-intercept?

[stuck lark]

y-int doesn't matter. what makes them perpendicular is that one line's slope is the negative reciprocal of the other

[hoary valley]

Oh Gotcha..!! ❤ ❤ @stuck lark

[hoary valley]

Can someone explain to me what is this question ? I know how to calculate the distance between 2 points, but I don't understand what is (a,b) nor (b,a)

[river kindle]

I imagine those are the same points, but going from a to b, then b to a in the distance formula.

Well, maybe the distance formula isn't relevant in this

[quick dagger]

can tangent exceed the value 1?

[serene heath]

what

[uncut mulch]

are you referring to the function?

[clear glade]

How do I do $2(4/x+2) + 4$

[obsidian monolith]

Waterblade:

[uncut mulch]

is that meant to be 4/(x+2) or 4/x + 2

[clear glade]

(4/x+2)

$(4/x+2)$

[obsidian monolith]

Waterblade:

[uncut mulch]

$2 \left(\frac{4}{x + 2}\right) +4$

[clear glade]

+2 is down there with that x

[uncut mulch]

thats literally what i was asking

[clear glade]

sorry

[obsidian monolith]

ramonov:

[clear glade]

yep

[uncut mulch]

and what are you supposed to do?

[clear glade]

it's part of a function composition problem but I forgot how to deal with the fractions so I'm unsure to how to just simplify that expression

[uncut mulch]

can you post the original problem

[clear glade]

yes

[uncut mulch]

for this question i would recommend splitting f(x) into 2 fractions before doing the composition

[clear glade]

oh ok

2x/x + 4/x?

[uncut mulch]

yeh

[clear glade]

ok ill give it ashot

I think I got it, thanks.

domain would be x is not equal to -4 if the answer is x + 4, right?

wait nvm

$3^3x = 1/9$

[obsidian monolith]

Waterblade:

[clear glade]

how would I solve x for this one?

oops, that x should be up there with the 3 exponent

[uncut mulch]

$3^{3x}=\frac19$

[obsidian monolith]

ramonov:

[clear glade]

yep

[uncut mulch]

can you rewrite 1/9 as an exponent?

[clear glade]

oh yea

3^-2

[uncut mulch]

can you continue from here?

[clear glade]

I believe so, thanks

will try it now

Hmm, I forgot what I have to do now actually

[trim flame]

so when u turn

1/9 to an exponent

[clear glade]

oh wait

if it's same base i just set exponents to each other i think?

[trim flame]

yea

[latent goblet]

i don't get these problems

[short sorrel]

do you know how to set up the equations?

[latent goblet]

nope

thats the only thing i don't know

[short sorrel]

well, let's start by looking at our unknowns:

we don't know the volume of B, and we don't know the volume after they mix together (A+B)

but the volume after mixing them together will be 1500 mL + B

so really, the only unknown we have is the volume of B

I'll use the letter B to represent that

let's try and find something we can compare between the three solutions: how about amount of alcohol? (not concentration, amount)

do you know how to calculate the amount of alcohol in solution A?

[latent goblet]

i don't think so

[short sorrel]

solution A is 6% alcohol by volume

[latent goblet]

yes

[short sorrel]

so the amount of alcohol in solution A

is $0.06 * 1500$

[obsidian monolith]

Namington:

[latent goblet]

o

[short sorrel]

6% of 1500 mL

we can do the same thing with solution B

except we don't know solution B's volume

it's known

so I'll just write $0.2 \cdot B$

[obsidian monolith]

Namington:

[short sorrel]

with B representing solution B's volume

now, the reason i calculate like this is that:

when we mix these solutions together

the new solution should surely have all the alcohol of the two solutions combined

i.e., if we add up the alcohol of solution A and solution B

we get the amount of alcohol in the mixture

so $0.06 \cdot 1500 + 0.2 \cdot B = $ amount of alcohol in mixture

[obsidian monolith]

Namington:

[short sorrel]

so next we want to represent "amount of alcohol in mixture"

this'll be similar to what we did in solution A

[latent goblet]

wait where did u get 0.2?

[latent goblet]

ok

[short sorrel]

anyway, the desired concentration of the mixture is 10%

and the volume of the mixture

[latent goblet]

i aint reading the problem correctly lol

[short sorrel]

is gonna be the volume of A + the volume of B, right?

thankfully, we know the volume of A

so we can write that as just

$0.1 \cdot (1500 + B)$

[obsidian monolith]

Namington:

[short sorrel]

and thus, we have $0.06 \cdot 1500 + 0.2 \cdot B = 0.1 \cdot (1500 + B)$

[obsidian monolith]

Namington:

[short sorrel]

solve this equation for B.

[latent goblet]

ok thanks

[tame wedge]

Hey I’m learning Parametric equations and I’m not sure on this: (x(t),y(t))=(tan(t),cos^2(t)), the questions asks to eliminate the parameter to obtain y=1/(1+x^2). I’ve tried manipulating the equations for x and y and can’t seem to come up with anything. Any help? Thanks.

[uncut mulch]

do you know your trig identities?

[heady jewel]

lol

have you heard of sin^2x + cos^2x=1

you should know it based on your pfp

[tame wedge]

Yeah, but I’m not sure where to use it though?

Nvm just figured it out, thanks for the hints

Didn’t need that one but it got my brain going I guess

[lyric cliff]

Can anyone help me out? I have a problem that says:
John has $99 in different bills ranging from $1 to $10. Theres 26 of them in total and the amount of $1 is double of the amount of $5. How many does each bill class have?

I seem to have made an equation: 1x+5y+10z=26 // 1x+5y+10z=99

but dont know where to go from there

[rigid sun]

you have the same equation for both

[willow bear]

how can 1x+5y+10z be equal to both 26 and 99 at the same time?

[lyric cliff]

26 is amount of bills

99 is amount of dollars

[willow bear]

what are your x, y and z?

[lyric cliff]

its the same

[short sorrel]

no it isnt

[willow bear]

what are your x, y and z?

[short sorrel]

some bills are worth $5

some are worth $10

[lyric cliff]

thats what the 1, 5 and 10 are for..

[short sorrel]

so the amount of bills should be different from the amount of dollars.

[willow bear]

what are your x, y and z?

@lyric cliff what are your x, y and z?

[lyric cliff]

x,y and z are random variables i created

[short sorrel]

what do they represent?

[willow bear]

so they don't mean anything in the context of the problem?

ok

fine

[lyric cliff]

the amount of bills

forget about the equation = 99

well

[short sorrel]

if they're the amount of bills

then why are you multiplying the amount of bills by 5 and 10

when adding up to 26?

after all, 26 should just be the total amount of bills

where does the multiplication come in?

[lyric cliff]

ugh

lemme explain myself

[short sorrel]

there's nothing to explain

[willow bear]

so x is the amount of ones
y is the amount of fives
z is the amount of tens
what is the total number of bills?

[short sorrel]

you have that 26 = 99

[lyric cliff]

Yess ann, 26 is total amount

99 is amount in dollars

$99

[short sorrel]

so how are their equations the same

[willow bear]

no

[short sorrel]

if they represent different things?

[lyric cliff]

split into 26 bills

[willow bear]

the total AMOUNT OF BILLS is not x + 5y + 10z, that's for sure.

[lyric cliff]

yeah i see. x+5y+10z is for total amount of dollars.

[short sorrel]

yes

so what's the equation for the amount of bills?

[lyric cliff]

x+y+z=26

🙂

So now I have
x + y + z = 26 (amount of bills)
x + 5y + 10z = 99 (total $ amount)

[short sorrel]

you should be able to make one more equation

"the amount of $1 is double of the amount of $5"

[lyric cliff]

still cant figure out how to get an answer to the original question

[willow bear]

"the amount of $1 is double of the amount of $5"

you should be able to turn this into an equation too

[short sorrel]

you don't have enough equations right now

[willow bear]

to get 3 equations in 3 variables

[short sorrel]

but you can use that information

to make one more

[lyric cliff]

oh

2x= y

[short sorrel]

sounds good

now solve the system

[willow bear]

um

no

[short sorrel]

oh wait

[willow bear]

it's not y = 2x

[short sorrel]

ugh this is why we use descriptive variables

@lyric cliff you've got it a bit backwards

the amount of $1 is double the amount of $5

amount of $1 = 2 times the amount of $5

so x = 2y

(thanks for correcting my idiocy, ann)

[lyric cliff]

ok

got it.

thanks for the help

[azure hamlet]

Can you help me on polynomials?

How do I find a solution set of an equation using given roots?

I have an example

<@&286206848099549185>

[knotty spear]

i mean if you know that sqrt(5)/2 is a root you can factor it out

so your polynomial = (x-root)* some other polynomial

[azure hamlet]

So it will be 2x-sqrt(5)?

I don't understand how to make solution sets

[knotty spear]

it should be possible to factor out an (x- sqrt(5)/2)

and then you are left with a quadratic polynomial

that you can solve

[azure hamlet]

Oh, I get it

Thanks

[knotty spear]

[slate scroll]

How would one solve x^4 + 2x^2 = 3

[willow bear]

substitute u := x^2, get a quadratic equation in u

[serene heath]

double posting

[ionic beacon]

My brain small please help

I'm not even sure what the rule is

[willow bear]

what rule

[ionic beacon]

arithmetic

idk never done this before

ok i found it

just dont know how to prove it

[next willow]

$a_n=a+dn$

[obsidian monolith]

Mat:

[next willow]

$54=\sum_{n=0}^8 a+dn = a\sum_{n=0}^8 1+ d\sum_{n=0}^8 n$

[obsidian monolith]

Mat:

[next willow]

Maybe like this you can proceed @ionic beacon?

[ionic beacon]

where did the an= a + dn come from

o

i see

so is a = a0?

o nvm

[rigid sun]

hey so i just was juggling around with i

and

$i^2=\sqrt{-1}^2$

[stuck lark]

\sqrt{}

[rigid sun]

k

[obsidian monolith]

maleb1964:

[rigid sun]

then

$\sqrt{-1}^2=\sqrt{-1*-1}$

[obsidian monolith]

maleb1964:

[kindred mica]

No.

[rigid sun]

yeah i was like tf

[kindred mica]

First of all $\sqrt{x^2} = |x|$, not $x$. Second of all, $\sqrt{ab} = \sqrt a \sqrt b$ only works for $a,b \ge 0$.

[rigid sun]

that's where i thought it was supposed to fall apart

i just wasn't sure

[kindred mica]

nice

[rigid sun]

if it was a domain error or me being sped

[kindred mica]

inb4 mathematicians are wrong

[rigid sun]

i knew it was wrong

i just wasn't 100% sure why

[viscid thistle]

inb4 -1=1

[kindred mica]

Yeah I get it. I was just making fun of the cranks who think they've found a killing mistake in some foundational concept in mathematics.

[rigid sun]

well

[kindred mica]

Lol.

[viscid thistle]

@ionic beacon I can already tell the paper is by edexcel

[lament garden]

hi

i need help with my homework

and its due on midnight

[forest canopy]

just ask

[night spear]

Yeah

What specifically?

[lament garden]

pre calculus