CPhill

y = 32    bounded above and below since this is a horizontal line

y =  2^x     bounded below by the x axis.....unbounded above

y = 2 - x²  this is an inverted parabola with a vertex at (2,0)....thus...it is bounded above and unbounded below

y = √ [ 1 - x² ]       this is the upper part of a circle with a radius of 1.....thus....it is bounded below and above

y = x² √ [ x + 4 ]  ......this one is hard to see......look at the graph here :

https://www.desmos.com/calculator/oblelcc1ie

Notice that it is bounded below but not above

The number of total possible sets....choose any 5 people from a total of 42  =   C(42,5)   (1)

The total sets  consisting only of women......choose any 5 of 24   = C(24,5)      (2)

Probability  =  (2) / (1)  =  C(24,5) /  C(42,5)  ≈ .05 =  5 percent

ab  = 7

a^2b + ab^2  + a + b  = 80

What  is     a^2 + b^2   ???

So we have :

a^2b + ab^2  + a + b  = 80         factor  the first two terms

ab (a + b)  + (a + b)  = 80        factor again

( a + b) (ab + 1)  = 80

(a + b) (7 + 1)  = 80

(a + b) (8)  = 80     divide by 8 on both sides

a + b  =   10

Square both sides

a^2 + 2ab + b^2  = 100

a^2 + 2(7) + b^2   = 100 

a^2 + 14 + b^2   = 100       subtract 14 from both sides

a^2 + b^2   =  86

Nice, heureka.....!!!!

Just key it in like normal and hit  " = "

Excellent, NSS !!!

-√67  ≈  -8.185

√67  ≈  8.185

So....were looking for the sum of the integers between   - 8  and 8  inclusive

But........  the sum of all the integers from  -a to a   inclusive must just be 0

For f(x), write y

y  = ax^2 + bx + c         subtract c from both sides       

y - c  = a  [ x^2  + (b/a)x   ]       

Complete the square on x....take (1/2)  of (b/a)  =  b/[2a]....square it  =  b^2/[4a^2 ]........add to both sides.....don't forget that we're actually adding  ab^2/[4a^2 ]  to the left side

y - c + ab^2 / [4a^2]  =  a [ x^2 + (b/a)x  + b^2/(4a^2 )  ]       

Factor the right side........simplify the left

y - c + b^2 / (4a)  =  a [ x  +  b /(2a) ] ^2 

Add c, subtract b^2 / (4a)  to both sides

y  = a [ x  +  b/(2a) ] ^2  - b^2 / (4a) + c

y  = a [ x + b / (2a) ]^2   +    (  - b^2  / (4a)  + c  )

So.....the vertex   is given by     [ -b / (2a) ,     - b^2  / (4a)  + c  ]

Note that we can prove that this is correct.........

The x coordinate of the vertex is  -b / (2a)

Putting this into   ax^2 + bx + c   for x, we  can find the y coordinate of the vertex thusly :

y  = a [b^2 / (4a^2) ] + b [ -b/(2a)] + c

y  =  [b^2 / (4a)] - b^2 /(2a)  + c

y =  [ b^2  - 2b^2 ] / (4a)  + c

y  =  [-b^2 / (4a) ]  + c

Last one :

A photographer offers a photo shoot for a $85 flat fee. Customers may purchase prints for $5 per sheet.

How many sheets can a customer purchase and spend at most $150?

What linear inequality with variable x represents this situation?

What is the solution to that inequality? Enter the solution as an inequality using  x.

For the first one....we have this inequality

85 + 5x ≤  150         where x is the number of sheets

Subtract 85  from both sides

5x ≤ 65

Divide both sides  by 5

x ≤ 13     .....so the customer can by 13 sheets at most

Two cars leave the same location traveling in opposite directions. One car leaves at 3:00 p.m. traveling at an average rate of 55 miles per hour. The other car leaves at 4:00 p.m. traveling at an average rate of 75 miles per hour. Let  x represent the number of hours after the first car leaves.

How many hours after the first car leaves will the two cars be 380 miles apart?

We want to find this : distance traveled by first car  + distance traveled  bt second car  = 380

And distance  = rate * time

So...the first car travels x hours and the second car travels one hour less = x - 1

And we have

55x + 75(x - 1)  = 380     simplify

55x  +  75x - 75  = 380   add 75 to both sides

130x  =  455     divide both sides by 130

x = 3.5 hrs

Jason started a hat-making business. He spent $750 to purchase supplies to get started and he uses about $3.50 worth of supplies per hat made. Jason charges $15 for each hat. Let  h represent the number of hats.

What is the minimum number of hats Jason will need to sell to make a profit?

He will make a profit when Sales > Costs  ...so

15h   >  750  + 3.50h         subtract   3.50h from both sides

11.5h > 750      divide both sides by  11.5

h > ≈  66 hats

m  =   [ a + b] / 2   multiply both sides by 2

2m  = a + b       subtract b from both sides

2m - b  = a

That's all I have time for....!!!!