math

inverse of y=log4(x-3)

4^x=y-3

y=3^x/4

4x=3^y y=log3(4x)

logs getting minused log3(x)-log3(y)

x/y

graph sum like this: 2/x-2 -1

the asmyptotes arw (0,-1) (2,0) then find x & y-int then graph the graph

inverse of the previous one y=2/x-2 -1

x+1=2/y-2 to y-2=2/x+1 so y=2/x+1 +2

graph y=log6(x+5)-3

asymptote is (-5,0) then know that log6(1)=0 so then log6(-4+5)-3=0 so log6(1)-3 So 0-3=-3 so (-4.-3) . then log6(6)=1 so log6(1+5) so 1-3=-2 so (1,-2)

logx(n+7)=y so

n+7=x^y

so find sin^-1(3.7/7)

so first fine the decimal of 3.7/7 then do the sin^-1(0.5286)

t find triangkes with one angle missing or side

lke p/sin(127) = 10/sin(28)

find measurement , cos euation

b²=a²+c²-2ac*cos(b)

area eq

½ absin©

what ot mean when find all roots

also imaginary

so if 1+i

(x-(1-i))(x-(1+i)) is (x-1-i)(x-1+i) is (x-1)²+1

dicrminant equation

b²-4ac

discrminant:

(> 0) and perfect square

Discriminant	Number of Solutions	Type of Solutions
(> 0) and perfect square	2	Real, rational

convert to radian from degree

145*pi/180= -29pi/36

how to fine the radious to then determine the 6 trig functions

r=sqroot(x²+y²)

the 6 trig functions

sintheta=y/r costheta=x/r tantheta=y/x cstheta=r/y sectheta=r/x cottheta=x/y

hoe to do arithmetic sum

Sn=((a1-a30)/2)*n

how to do the geometric sum finite

Sn=(a1(1-r^n)/(1-r))

how to geomtric infite

1/1-r

hpw t the cobinations yk the nCo (u+2v)^4

4Co(u)^4(2v)^0

comound interest

A=the first money(1+rate/the amoutn fof the comound)^amunt of compund * time

continous

a=th amlunt bege^rate*t

sum of cubes q

x^3+y^3=(x+y)(x^2-xy+y^2

dif of squares

a²−b²=(a−b)(a+b)

perfct square trinomial

a2+2ab+b2=(a+b)2

sum of cubes

a3+b3=(a+b)(a2−ab+b2)

diffrence of cubes

a3−b3=(a−b)(a2+ab+b2)

powers of 4

a4−b4=(a2−b2)(a2+b2)

How do I know the end behavior from the degree?

• Odd degree (x,x3,x5x, x^3, x^5x,x3,x5) → ends go opposite directions

• Even degree (x2,x4,x6x^2, x^4, x^6x2,x4,x6) → ends go same direction

What if the leading coefficient is positive? odd degree

Odd degree:

y=x3y=x^3y=x3Left ↓, Right ↑

Say:

"Positive odd = smile rising to the right."

What if the leading coefficient is positive? even degree

Both ends ↑

Say:

"Positive even = bowl."

What if the leading coefficient is negative? odd degree

y=−x3Left ↑, Right ↓

Say:

"Negative odd = falling to the right."

What if the leading coefficient is negative?

Even degree:

y=−x4Both ends ↓

Say:

"Negative even = upside-down bowl."

actual tan grah

period= pi and asymtptoes at pi/2 and half always 0 where the line starts from. then the first pont is (pi/4,1) and (-pi/4,-1) as ¼ is pi/4

arithmetic with first term formula like explicit

An=a1+r(n-1)

geometric expliit formula

An= a1r^n-1

for examp;e determine number of terms (n) in each artihmetic series

ex: (-23)+(-30)+(-37)+(-44)…., Sn=-459

-459=n/2(2(-23)+(n-1)(-7)) so so 7n²+39n-918 so the quad eguation and 9

graph y=2e^0.5x

so first k=o ,

y-int= (0,2) then test out diffrent x values like -2 & 2 so (2,5.44) & (-2,0.736)

discrimiant

(> 0) but not a perfect square

(> 0) but not a perfect square	2	Real, irrational

discriminant =0

Discriminant	Number & Type of Solutions
(= 0)	1 real solution (a repeated/double root)

discrminant less than 0

Discriminant	Number & Type of Solutions
(< 0)	2 complex (non-real) solutions

unite circle coordinates for 0-90

Angle	Coordinates ((x,y))
(0^\circ)	((1,0))
(30^\circ)	(\left(\frac{\sqrt3}{2},\frac12\right))
(45^\circ)	(\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right))
(60^\circ)	(\left(\frac12,\frac{\sqrt3}{2}\right))
(90^\circ)	((0,1))

coordinates 210

210∘−180∘=30∘

So the numbers are those of 30∘30^\circ30∘:

(32,12)\left(\frac{\sqrt3}{2},\frac12\right)(23​​,21​)

But Quadrant III means both coordinates are negative:

(−32,−12)\boxed{\left(-\frac{\sqrt3}{2},-\frac12\right)}(−23​​,−21​)​

find t : 2=(1.037)t

2P=P(1.037)t

Divide by PPP:

2=(1.037)t2=(1.037)^t2=(1.037)t

Use logarithms:

t=log⁡(2)log⁡(1.037)t=\frac{\log(2)}{\log(1.037)}t=log(1.037)log(2)​

t=log⁡(2)log⁡(1.037)t=\frac{\log(2)}{\log(1.037)}t=log(1.037)log(2)​

t≈19.08t \approx 19.08t≈19.08