math 172 final
basic graph of sin
0,a,0,-a,0
draw graph for say 2/ 9-x^2
1. domain : d(x) =/ 0 2.simplify 3. VA: d(x)=0 4.HA: highest power from both D and N 2/ x^2 so if N(x) < D(x) y=0 5. x intercept : N(x)=0 6.y intercept: x=0 plug in 0 as x 7: key points: # you get from simplify. you get # up and down from both.
y= asin(bx+c) or y= acos(bx+c)
1.amplitude /a/ 2. period: 2pie/b 3. displacement: -c/b 4. 1/4 of period: you add 1/4 to the period answer 5. points: 1) x=0 2)0+ step4 answer 3) 2answer with step4 answer keep going until 5 points
Law of Sines
ASA/SSA(two side and angle opp from them) if
Cartesian plane
Q1,2,3,4
Law of Cosine
SSS/ SAS(two sides and include an angle) a^2= b^2+c^2-2bc*cos(A) same for b and c just switch the letters accordingly
Soh Cah Toa
Sin= opp/hyp (csc) Cos= adj/hyp (sec) Tan= opp/adj (cot)
basic graph of cos
a,0,-a,0,a y=cos(x)
y= 2x^2-16x+24 graph
a=2 b=-16 c=24 do axis of sym -b/2a -(-16)/2(2) 16/4= 4 thats the middle # in table use 2 # above and 2 # below 6,5,4,3,2 plug in and make graph label either minimum or max
Quadrant 4 C
cos is positive. sin and tan negative
cos(15)=cos(60-45)
cos(60) cos(45)+ sin(60) sin(45) 1/2. radical 2/2 rad 3/2. rad 2/2 rad3/4+ rad 6/2= rad 2 and rad 6/4 rad2+rad6 over 4 **
f(x+h)-f(x)/h
f(x)= 7x^2+6 you plug f(x+h) into the x then write f(x) so its 7(x+h)^2+6-7x^2+6/h watch out for factoring and signs when distributing. can cancel. last if you can factor h do it so you can cancel that as well.
a=5 b=10 <c= 60degrees find length c
law of cosine bc of SAS c^2= a^2+ b^2-2ab * cos(c) 5^2+ 10^2- 2(5)(10)cos(60) 25+100-100(1/2) 125-50 c^2= 75 radical 75 radical 25 * 3 c= 5 radical 3
price*quantity
list price and quantity (starting with # either - or + #x) like (100-10x) (30+15x) multiply those to get equation then axis of sym -b/2a to get a # and put it as x in say price equation to get answer.
give two points on graph (3,0) (0,-15)
m= y2-y1/ x2-x1 then use y-y1=m(x-x1) plug in points and solve m= -15-0/0-3= -15/3=5 y-0=5(x+3) move zero and distribute y=5x-15
y= 2-3/2x (4,-2)(x1,y1)
parallel -3/2 perpendicular 2/3 plug in point to equation. y-y1=m(x-x1)
graph y=2-/x-5/
remember to find a # to make 0 so 2-/7-5/=0 then go down 2-/6-5/= 1 2-/5-5/= 0 2-/4-5/= 1 2-/3-5/= 0 plug in table and then graph
45
sin and cos = radical 2/2 tan 1
Quadrant 2 S
sin is positive. Cos and tan negative
sin ( a+b) = sin(a) cos(b) + cos(a) sin(b) use this to prove sin(a+ 3pie/2)= -cos(a)
sin(a+3pie/2)= sin(a)cos(3pie/2)+cos(a) sin (3pie/2) sin(a)*0 + cos(a)* (-1) cancel sin(a) * 0 cos(a) * (-1) = -cos(a) -cos(a)
Quadrant 1 A
sin, cos,tan are positive
30
sin= 1/2. cos= radical 3/2 tan= 1/radical 3
60
sin= radical 3/2 cos= 1/2 tan= radical 3
unit circle
sin= y cos= x tan= y/x 360= 2pie
Quadrant 3 T
tan is positive. cos and sin negative
y= radical x-7
we need a # to make it 0 so we need perfect squares radical 7-7= radical 0= 0 rad 8-1= rad 1 = 1 rad 11-7 = rad 4= 2 rad 16-7 = rad 9 = 3 plug in table and make graph
vertex and point
y= a(x-h)^2+K plug in and find a. then label a,h,k in new equation.
g(x)=18e^.07x
y=18e^.07x x=18e^.07y divide by 18 x/18=e^.07y apply natural law. x/18= lne^.07y lne cancels so lnx/18=.07y divide by .07 1/.07ln(x/18)=y y=1/.07ln(x/18) g^-1 = 1/.07 ln (x/18) domain is x/18 > 0/1 = x>0 so (0,infinte)
f(x)= { 8-4x if X </ 5 , 4x-8 if x>5}
you do two points for each starting with 5 bc its the # they gave you </ means #less than 5 so use 5 then 4 > means #more than 5 so use 5 then 6 plug those in and make graph open point is </
usually a time and population question 100 people that triples every 6 years
you list time and population 0 6 12 18 100 300 900 2700 100*1 100*2 100*3. 100*4 100*3^0. 100*3^1. 100*3^3. 100*3^4 only thing that changed was power so a) P(t)= 100*3^t b) 1300=100*3^t -100 -100 13= 3^t ln13=ln3^t power to front ln13=tln3 divide by ln3 ln13/ln3=t 6(ln13/ln3)****
add fractions
you multiply both top and bottom by the same amount, to keep the value of the fraction the same 1/3+1/6 so 1/3*2= 2/6 2/6+1/6= 3/6= 1/2 1/5+1/3= 3/15+5/15= 8/15 1/5*3= 3/15 1/3*5=5/15