Families of functions and transformations

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[bright panther]

Which problem do you need help with exactly?

[cunning cape]

I am up now

[tepid plover]

yay

[cunning cape]

So still need help ?

[tepid plover]

yep

this chapter is cas too

[cunning cape]

Oo okie

Wait imma change in a min, just came back from prayer

[tepid plover]

all good

I'm not even up to q4

im up to 3b.

[cunning cape]

Okie I am back

[cunning cape]

So we doing b rn ?

[tepid plover]

yep

I got a and b

oh ai and aii

[cunning cape]

Oo b too, noicee

[tepid plover]

I didn't get b

[cunning cape]

Found derivative ?

[tepid plover]

yeah

1 - 2ax

[cunning cape]

Wha

Oh

Oh yee

Okie now find derivative at x=1/a

Equals to not plus ☠️

[tepid plover]

so sub that in?

1 - 2a(1/a)

[cunning cape]

Yup

1-2 then

[tepid plover]

yep

[cunning cape]

Now we have found slope at (1/a,0)

But that's not all

We gotta find equation

Remember the 1 point form of straight line ?

[tepid plover]

y = mc + c ?

otherwise no

[cunning cape]

No no

That is intercept form

y - y1 = m(x-x1)

[tepid plover]

ok

[cunning cape]

Try putting it in that

What's m ?

What slope did you find

[tepid plover]

-1?

[cunning cape]

Yup

Now place that, with the points

What's y1 ?

[tepid plover]

0?

[cunning cape]

And x1 ?

[tepid plover]

1/a

[cunning cape]

Yup

What is the equation then

[tepid plover]

y - 0 = -1(x + 1/a) ?

[cunning cape]

-1/a right ?

[tepid plover]

yeah but isn't it - -

[cunning cape]

Hmm where 2 -

[tepid plover]

already 1 in the equation

nvm

lmao

[cunning cape]

Oooo oki

So now what is the final equation

[tepid plover]

y = -x + 1/a

[cunning cape]

Yup, that's EQ of tangent

And normal is just slope changed to -1/m

[tepid plover]

hmm

what u mean -1/m

[cunning cape]

If slope of tangent = m

Slope of normal = -1/m

[tepid plover]

so -1 / -1

[cunning cape]

I gtg for lunch , but ig you can do next question too by this

[cunning cape]

Yws

Yes

[tepid plover]

kk

im up to question 4 now

I found the derivative to = (x-a)(3x-a-2)

than solve to = 0

and got x = a+2/3 and x = a

but than I tried subbing those into the derivative equation and for some reason it = 0

wait

[cunning cape]

Because you did just that here right

Subbed it in same equation ?

[tepid plover]

im supposed to sub into origional

aint I

[cunning cape]

Ye

Idk what you are doing here actually

But you don't put same thing in same equation

[tepid plover]

how would I know which 1 if local max and min

don't I have to find the 2nd derivative?

6x - 4a - 2

[cunning cape]

I am back

I am fast eater it seems finished in 8 mins :D

[cunning cape]

Well ye that's the traditional way

But since last time I think I have a better way

We are 0 at to positions in this

a and 1

Now usually whenever we are at 0 we switch signs, go from positive to negative or negative to positive

But we have (x-a)²

Due to it being squared

We don't switch signs here

So we reach a local maximum point then go back to being negative

Atleast if a less than 1

If a greater than 1 then it reaches local minimum point

[tepid plover]

wait im lost

lmao

@cunning cape can we stick w 2nd derivative

[cunning cape]

Ahh okie

Let's stick to 2nd derivative

[tepid plover]

so wouldn't it be 6x - 4a - 2?

[cunning cape]

2nd derivative ?

[tepid plover]

yea

[cunning cape]

Oo idk I didn't find it

But

[tepid plover]

I used my cas

[cunning cape]

I trust you found the right 2nd derivative

[tepid plover]

but it makes 0 sence

[cunning cape]

What's cas

[tepid plover]

calc

calculator

[cunning cape]

Ohh oki

Let's use this then

[tepid plover]

it said 6x - 4a - 2

but than if u sub a in

[cunning cape]

Wait

[tepid plover]

2a - 2

[cunning cape]

,w differentiate (x-1)(x-a)^2

[hasty rampart]

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[cunning cape]

,w differentiate -(a-3x+2)(x-a)

[hasty rampart]

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Upgrade to WolframAlpha Pro!

Requested by akki22

[cunning cape]

Yup

[tepid plover]

welp

[cunning cape]

Your calc was right

[tepid plover]

yay

[cunning cape]

Now we out in the values

[tepid plover]

yeah but if u sub in a?

2a - 2

a = 1

so a is local min

[cunning cape]

Why a = 1 ?

[tepid plover]

don't u do

2a = 2

a = 2/2

or no?

[cunning cape]

Nahh

[tepid plover]

u don't make it = 0

[cunning cape]

2a-2

[cunning cape]

Nope we don't make it 0

[tepid plover]

I thought that only because what does 2a - 2 even mean

[cunning cape]

We make derivative 0

[tepid plover]

is it > or < 0

[cunning cape]

But that's for something else

[tepid plover]

so is 2a - 2 > or < 0 ?

[cunning cape]

Yes

Oh

That's the thing

If a<1 it's negative

If a>1 it's positive

[tepid plover]

what is a?

[cunning cape]

a is something we do not know

And a is not equal to 1 either

One thing we know for certain tho

At x=1 it will change signs right ?

[tepid plover]

umm

x = 1 into 2nd derivative?

[cunning cape]

We can try that, but we won't get a satisfactory result

[tepid plover]

what do u mean at x = 1 it will change signs

[cunning cape]

In a polynomial

Whenever we hit 0, and total power is 1 or odd

We change from negative to positive or positive to negative at that point

Wait lemme show you example

Let's say we have

(x-1)(x-2)(x-3)

,w plot (x-1)(x-2)(x-3)

[hasty rampart]

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Requested by akki22

[cunning cape]

There

Look at 1

What was the sign of function before 1 m

?

[tepid plover]

< ?

[cunning cape]

Wha

< ?

[tepid plover]

sign of function?

[cunning cape]

Positive or negative

[tepid plover]

<

negative?

[cunning cape]

Yup

After 1 ?

[tepid plover]

positive

[cunning cape]

At 1,2,3 we hit 0 right ?

What about before and after 2?

[tepid plover]

yep

before 2 pos after 2 neg

[cunning cape]

And before and after 3

[tepid plover]

before 3 neg after pos

[cunning cape]

Okie so you see how sign changes when we hit 0 ?

[tepid plover]

yep

[cunning cape]

Now imma show you when it doesn't change sign

[tepid plover]

kk

[cunning cape]

,w plot(x-2) (x-1)^2

[hasty rampart]

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Requested by akki22

[cunning cape]

See at 1

Before 1 negative

After 1 negative again

[tepid plover]

yep

[cunning cape]

Because (x-1)²

[tepid plover]

yep

[cunning cape]

,w plot (x-3)(x-2) (x-1)^3

[hasty rampart]

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Requested by akki22

[cunning cape]

Better

See @tepid plover it changed sign at 1 ?

[tepid plover]

to pos?

[cunning cape]

Yes from negative to positive again

So that means

If the power is odd

It changes sign

[tepid plover]

yep

[cunning cape]

If it's even it goes back to same sigm

And another thing

So (x-a)²(x-1)

We know that at x-1 we change sign and at (x-a)² we stay wherever we already are

[tepid plover]

yep

[cunning cape]

Now what options do we have in the question ?

[tepid plover]

max or min?

or poi?

is that what u mean

[cunning cape]

No no I mean in question

It has some options to choose from right ☠️

[tepid plover]

what u mean options to choose from

[cunning cape]

Is it multiple choice question 👀

[tepid plover]

no

[cunning cape]

Ah ok

[tepid plover]

it says to state the nature of each of the turning points'

[cunning cape]

Okie which turning point

a or 1

[tepid plover]

the two we found

from the previous question

[cunning cape]

We can't say for a because we have no idea if it's > or < 1

[tepid plover]

so we try the other 1

[cunning cape]

Okie this, so x=1 is a turning point

x=a is maxima or minima or 1

[cunning cape]

Write it whole

Those are it's 3 possibilities

[tepid plover]

idk what ur asking

[cunning cape]

Oo it's the answer

[tepid plover]

well it says this

[cunning cape]

Hmmm

How do they know if a greater than or less than 1

[tepid plover]

maybe they graph it?

[cunning cape]

We don't know value of a

So we can't graph it

AHHHH

They have it written there

a>1

We didn't read after the full stop

[cunning cape]

In this

Remember if a>1 we found it to be minima too ?

[tepid plover]

yeah

[cunning cape]

And question says a>1

[tepid plover]

6x - 4a - 2

[cunning cape]

Wha

Oh 2nd derivative

[tepid plover]

yeah

2a - 2

so u could sub in 2

4 - 2

[cunning cape]

Yup we get 2

Positive

Therefore minima

[tepid plover]

finally lmao

[cunning cape]

If only we read the whole question

[tepid plover]

next 1 :0

[cunning cape]

At x= 1

What's derivative ?

[tepid plover]

1 sec

[cunning cape]

Oki

[tepid plover]

?

[cunning cape]

Wha

[tepid plover]

what u mean

[cunning cape]

Derivative

Of the function

1st derivative

[tepid plover]

(x-a)(3x-a-2)

sub in 1?

[cunning cape]

Yes

[tepid plover]

(1-a)(3-a-2)

[cunning cape]

(1-a)²

[cunning cape]

Yee

Now we put it in point slope form

[tepid plover]

waht is slope form

[cunning cape]

Remember y-y1 = m(x-x1)

That's point slope form

The normal y= mx+c is slope intercept form

[tepid plover]

ok sorry

im back

how do I put (1-a)(3-a-2) into that

[cunning cape]

(1-a)(3-a-2) = (1-a)² right ?

What is this ?

(1-a)²

Why did we find it ?

[tepid plover]

wait what

how do we know it = (1-a)^2

[cunning cape]

3-2 = 1

(1-a)(1-a)

Just wrote it like that cuz easier to write

[tepid plover]

where is 3 - 2

OH

yeah

yeah

nvm

(1-a)^2

[cunning cape]

(3-a-2)

(3-2-a)

[cunning cape]

Yesh that

What was it ?

How did we find it ?

[tepid plover]

by subbing 1 into the derivative

?

[cunning cape]

So that means it's the slope at 1 ?

[tepid plover]

y

[cunning cape]

Derivative = slope

We found derivative at x=1

[tepid plover]

yeah

y = (a-1)^2

[cunning cape]

Now

Place it in the slope point form

[tepid plover]

I don't get how to do that

y - y1 = m(x - x1) ?

[cunning cape]

m=slope

Slope=(1-a)²

Now at what point does question ask you to find this ?

x=1 ?

[tepid plover]

tangent

[cunning cape]

Tangent where

[tepid plover]

at x = 1

[cunning cape]

Yes

So X1=1

What's y1 ?

[tepid plover]

0

[cunning cape]

Yup

What's EQ of tangent then

[tepid plover]

y - 0 = (1-a)^2(1)?

[cunning cape]

Hmm

Read the formula again once

[cunning cape]

This

[tepid plover]

what does x1 =?

[cunning cape]

[tepid plover]

isn't m (a-1)^2

[cunning cape]

x1=1

Not x=1

[tepid plover]

what does x =

[cunning cape]

x is a variable

In equation you have variables right ?

[tepid plover]

y - 0 = (a-1)^2(x + 1)

there lmao

[cunning cape]

Yes

That's tangent at that point then

Hmmm

X+1 ??

[tepid plover]

• 1

x-1

[cunning cape]

Yes

[cunning cape]

Now ii

Find slope at x=a

[tepid plover]

we already did that didn't we

a was a point

so isn't it 0

[cunning cape]

Yup you are right

Now point slope form

[tepid plover]

so for next we sub a+1/2 into derivative

[cunning cape]

Now point slope form,not not

[cunning cape]

Ye

[tepid plover]

so

(a+1/2 - 2)(2a - 1/2)

[cunning cape]

Hmm 1/2 right ? In right one

Nahh

You are right

[tepid plover]

hmm

now times thoes?

-(a-1)(2a-1/2) / 2

[cunning cape]

Multiplying then is gonna be messy ☠️

Let's keep it this way until the end

Point slope form now

[tepid plover]

y - 0 = (a+1/2 - a)(2a - 1/2)(x - a+1/2)

[cunning cape]

Is y=0 at that point ?

[tepid plover]

idk 💀

[cunning cape]

Find y at that point

Original function

[tepid plover]

it's equal to

(a+1/2 - 1)(a+1/2 - a)

y - (a+1/2 - 1)(a+1/2 - a)^2 = (a+1/2 - a)(2a - 1/2)(x - a+1/2)

holy fuck its messy

@cunning cape

[cunning cape]

Yeshh ☠️

[tepid plover]

its probs wrong too

[cunning cape]

That's why it's not a good number

[cunning cape]

Y was a+1/2-1 ?

[tepid plover]

yeah

wait

[cunning cape]

It could be right

I didn't check it whole tho

[tepid plover]

(a+1/2 - 1)(a+1/2 - a)^2

[cunning cape]

I mean steps were correct

[tepid plover]

that was y

[cunning cape]

☠️

The calculation just keeps on getting worse and worse

[tepid plover]

I give up

ngl

fuck that

[cunning cape]

☠️

Well atleast first 2 were easy peasy

Anyways it's not the answer that I want atm

Did you understand how we get to the answer ?

[tepid plover]

yeah

great

[tepid plover]

yeah

[cunning cape]

Noicee

[tepid plover]

to find the gradient do I just find derivative

[cunning cape]

Gradient = slope = derivative

[tepid plover]

so it's just 2(x-2)

[cunning cape]

Yup

At x=a tho

So 2(a-2)

[tepid plover]

ah yeah

what does it mean express in terms of a

[cunning cape]

The variables in answer should be a

Not x or m or anything else

[tepid plover]

the 2nd question asks express m in terms of a

[cunning cape]

yup

so your answer shouldn't have any other variables

[tepid plover]

so m = 2(a-2)?

[cunning cape]

well yee it should be just that

[cunning cape]

for b we can sub in original equation

[tepid plover]

a?

sub a into origional

so x = a and y = (a-2)^2

[cunning cape]

yee

thats the y coordinate for p

[tepid plover]

idk how to do c or d tho

[cunning cape]

hmm

whats the process for finding eq of tangent ?

[tepid plover]

derivative?

[cunning cape]

yes

derivative

[tepid plover]

2(x-2)

[cunning cape]

then y - y1 = m(x-x1)

[tepid plover]

2(a-2)

I don't get when we use that formula

[cunning cape]

when they ask you for equation of tangent at any point

in all of last questions they asked equation at tangent at those points right ?

[tepid plover]

yeah

kk

but to find y do I sub what in

a

[cunning cape]

you don't find y or x in the equation

y and x remain variables

you find x1 and y1

[tepid plover]

thats what I meant

y1

is it y - (a-2)^2 = 2(a-2)(x - a)?

[cunning cape]

yes

expand it

expand everything in this

[tepid plover]

y=2ax−x−a^2

[cunning cape]

okie then

also how is it just -x

shouldn't it be -2x ?

[tepid plover]

probably

[cunning cape]

well anyways that would be equation of tangent

[tepid plover]

[cunning cape]

,w simplify (a(2a-4))

[hasty rampart]

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Requested by akki22

[cunning cape]

bruh

oh okie works

2a^2 - 4a - a^2 - 4 + 4a

yes the simplification is correct

[tepid plover]

so I just calculated it wrong

[cunning cape]

a^2 - 4a

[cunning cape]

ye

[tepid plover]

so if I want to find the x intercept of it do I just make y = 0

[cunning cape]

yup

[cunning cape]

found a ?

[tepid plover]

about to attempt

1 min

no idea even how to start @cunning cape

[cunning cape]

f(x) = x^3

f(1+h) = 27

(1+h)^3 = 27 ?

[tepid plover]

1 + h = 3

h = 2

[cunning cape]

yup

[cunning cape]

try 2nd one

x = 1

f(ax) = 27

ax = 27

[tepid plover]

f(ax)^3 = 27?

ax = 3

than what

oh x = 1

[cunning cape]

yes

a = ?

3 :D

[tepid plover]

for next do I just sub 1 and 8

into equation?

8 = a(1)^3 - b(1)^2

8 = a - b

did smth wrong didn't I

[cunning cape]

they also told you its a turning point

[cunning cape]

nope

what happens at a turning point ?

to derivative ?

[tepid plover]

umm

slopes another way

[cunning cape]

hmmm

at a turning point slope becomes 0

[tepid plover]

ok

[cunning cape]

find its derivative and make it 0

[cunning cape]

you know why right ?

[tepid plover]

idk what u mean by slope becomes 0

[cunning cape]

derivative = slope ??

[tepid plover]

yeah

it's 0

[cunning cape]

whats derivative of it ?

[tepid plover]

im so lost

lmao

[cunning cape]

1. at turning point derivative of function is 0

[cunning cape]

f(x) = ax^3 - bx^2 has turning point at (1,8)

find its derivative equate it to 0 at x = 1 ?

[tepid plover]

3ax^2 - 2bx

3ax^2 - 2bx = 0 ?

but also sub in 1 for x?

[cunning cape]

yes

[tepid plover]

3x^2 - 2b = 0?

[cunning cape]

yup

[tepid plover]

3x^2 = 2b

[cunning cape]

wait

no wait

x^2 = 1 ??

3a

not 3x^2

[cunning cape]

from here

[tepid plover]

oh

3a - 2b = 0

[cunning cape]

yup

[tepid plover]

b = 3a/2

[cunning cape]

and you have this too

find a and b from these 2 eq

[tepid plover]

8 = a - 3a/2

wait

i just use my calc

a = -16 b = 024

-24

[cunning cape]

yup

[tepid plover]

I'm fucking doen

after doing maths for 10 hours

your actually a life saver @cunning cape

[cunning cape]

aha thank you

I need life saving of my own tho

[tepid plover]

+close