CPhill

3)  The most difficult one

The parent graph is   y  =x^2

The  "-2"    shifts the parent  to the RIGHT  by 2  units

The "4"  shifts this  up  by 4 units

And finally....the " - "  out front  turns the  graph over

Here is the parent  and  the transformed graph : https://www.desmos.com/calculator/s4hc2ni1sk

Finally...your  teacher isn't expecting  "perfect"  points on the  graph....he/she  is more interested in your knowing how each  transformation looks...just  a general sketch on your part will likely be good enough

2.  The parent graph  is    y  =  l  x  l 

Here it is  :  https://www.desmos.com/calculator/atandsuqzf

The  function   y  =  l  x + 1  l    +  2

The  " +1"   shifts  the  graph to the LEFT  by 1  unit

And finally...the "2"  shifts  this  up  by  2  units

It's  helpful to see  that  the  "vertex"  of  the  first graph is at (0,0)

And the  "vertex " of  the second graph   is at  (-1, 2)

See both  graphs  here :  https://www.desmos.com/calculator/htgibomhib

The  easiest  way  to  learn this  is  to  know what  the "parent"  function  looks  like  and  then what  the  transform for each function is....this  is  easier  than "plotting points"

1)  Look  at the  graph of   y  =  ∛ x  here:  https://www.desmos.com/calculator/e9c8zn9cya

This  is  the "parent" graph

The  function    y =  ∛ ( x - 2)     just  shifts  the  graph to the RIGHT  by 2  units

Call the total  nuber of cherries . N

Nathan  started with  (9) /(9 + 2)   = 9/11  of  them

And Shane had (2/11) of them

So

(9/11) N  -  32                1

____________   =       ___        cross-multiply

    (2/11)N                      2

2 [ (9/11)N -  32 ] =  (2/11)N

(18/11 )  N  -  64  =  (2/11)N

(18/11)N - ( 2/11)N   =  64

(16/11)N   = 64

N = 64 (  11  / 16)    =    44   =  total number

Natan  started  with (9/11) (44)  =  36

Number of boxes =  60/(3/4)     =  60 (4/3) = 240 /  3   =  80

80  (3.50)  = $280

200  = v^2  / 16  *   sin (.75) cos (.75)

16 * 200   /  [  sin(.75) cos (.75) ]   =  v^2

3200  / .498  =  v^2

6416  = v^2

v  ≈  80 ft /s

75  =  ( 50^2  /16)  *  sinθ  cosθ

75 * 16   / 50^2   =   sinθ  cosθ

12/25  =   sinθ  cosθ

Note  that  sinθ  cosθ  =   sin 2θ / 2

So....mutiplying  both sides  by 2   we have

2 * 12  / 25   =  sin 2θ

24 / 25  =  sin  2θ

Note     arcsin  ( 24/25)   ≈  73.74°

Dividing  this angle in half  we get  that θ  ≈ 36.87°

 (cot x sin x)(sec x – cos x)    =

(cosx / sinx) (sinx)    ( 1/cosx  - cos x)   =

cos x   ( 1  - cos^2 x) / cos x ]   =

1 - cos^2 x  =   

 sin^2x

p(x)   = ax^2   +  bx +  c

The roots  are    4 + sqrt 17     and   4 - sqrt 17

By Vieta

sum of   the  roots  = 8  =    -b/a   =    -b/1     ⇒  b = -8

product of the  roots   =   16 - 17 = -1   = c /a  =  c /1  ⇒  c  =  -1

p(x)  =  x^2  -8x -  1

p(3) /  p(4)   =     -16 / -17  =  16 /  17

[7^(1/3)]^(5x)  =  14

[ [7^(1/3)]^(5x) ] ^2  = 14^2

[ 7^(1/3) ^ (10x)  ]  =  196

[ 7^ (1/3)] ^(10x  - 6]  =

[ 7^(1/3)]^(10x)

_____________  =

       7^(1/3)^6

[ 7^(1/3)]^(10x)                 196

_____________  =          ____  =          4

          7^2                           49