CPhill

3.

y  = 2 ( x + 1/4)^2  -  25/8

Remember.....the  y intercept point has an x value  = 0.....so...let   x  = 0....and we have

y  =2 (0 + 1/4)^2  - 25/8

y  = 2 (1/4)^2  - 25/8

y  = 2 (1/16) - 25/8

y  = 1/8  - 25/8

y   = -24/8    =   -3   ....just as you found, LF   !!!!

2. 

To find the number of people who use the fitness suite only...we can  take 57 - 36  = 21

To find the number of people who only use the swimming pool, we have 49 - 36 = 13

To  find those that use neither, we have :   100 - 36  - 21 - 13  = 30

Here's the diagram :

"Range" refers  to  the y values covered by the graph....these are   [ -3, 6 ]

The y values  - not "points" -  in this range  are  ( -3, 0, 1, 5 )

1.

We have  a total of 10 students  who have either  a brother or a sister....

To find the number that goes  in the middle, x,  we can solve this

B  +  S   -  x  = 10

9 + 7  - x  =  10

16  - x  = 10

16  - 10  = x 

6  = x

Here's the diagram :

Let's look at the first few terms

-4 +  2  - 1 +  1/2.......

The common ratio, r,  =  -1/2   .....since  l r  l  < 1....this will converge

And the first term is  -4

So...the sum  is given by

[ first term  ]  /  [ 1  - r ]  =

-4  /  [ 1 - (-1/2) ] =

-4  /  [ 3/2]   =

-8 / 3

We have the equation

y  = Asin (Bx)

The amplitude  =   A   = 4   

Finding B   is a little tricky.....we want to  solve this  :

2pi / Period  = B

2pi  / [ 3pi / 2 ]  = B

4pi / 3pi  = B

(4/3)  = B

So...the equation becomes

y  = 4sin (4/3 x)

Here's the graph for one period   :  https://www.desmos.com/calculator/ncw51ksw5x

Here's the graph  :

https://www.desmos.com/calculator/7aqynw02yn

52

1/x  + 1  / (x + 2)   =   9/40    get a common denominator on the left

[ (x + 2 )  +  x ]  /  [ x (x + 2) ] =    9/40        cross-multiply

40 [ x + 2 + x   ] =  9 [ x(x + 2) ]      simplify

40  [ 2x + 2]   =  9 (x^2 + 2x )

80x + 80  = 9x^2  + 18x

9x^2  + 18x   =  80x  + 80        subtract  80x, 80  from  both sides

9x^2  - 62x  - 80  = 0       factor

(9x + 10) (x - 8)  = 0     set each factor to 0  and solve for x

9x  + 10  = 0            x - 8  = 0

9x  = -10                   x  = 8

x  = -10/9

Since we must have interger answers.....x  = 8    and  x + 2  =  8 + 2  = 10

50.  2sinθcosθ + √2 cos θ   = 0          factor out cosθ

cosθ (2sin θ + √2 )  = 0

So...either  cosθ  =  0      and this happpens at  pi/2  and  3pi/2     ......or....

2sin θ + √2   = 0       subtract √2  from both sides

2 sinθ  =  -√2    divide both sides by  2

sinθ  =   -√2 / 2     and this happens a 5pi/4    and at 7pi/4

So...the answers are    pi/2 , 3pi/2./  5pi/4 and  7pi/4   ⇒   the third answer

51.     √(7x) [ √x - 7√7  ] =

 √(7) x- 49√x