CPhill

a)   y = (1/8) ( x - 6)^2 + 4

We are working backwards, here

(y - 4) = (1/8) ( x - 6)^2

8(y - 4) = ( x - 6)^2           the vertex is ( 6, 4)

4p = 8   ⇒    p = 2

The parabola turns upward  so  the focus is  ( 6 , 4 + p)  =  ( 6 , 4 + 2) = (6, 6)

And the directrix is   y = 4 - p  =  4 - 2   = 2   ⇒  y = 2

This is the reflection of the "floor" function across the x axis  and shifted to the left by 2 units

Here is the graph:

The correct graph is at the top left

See the answer here :

https://web2.0calc.com/questions/amount-of-real-solutions

Any product of a non-zero real and an irrational is irrational   ( I believe)

c.) focus (–2, 1) and directrix y = 13

turns downward

vertex =   ( - 2 ,  (13 + 1) / 2 )   =  ( -2, 14/2)   = ( -2, 7)

p =   6

4p ( y - 7) = - (x - - 2 )^2

4(6) ( y - 7) = - ( x + 2)^2

24 ( y - 7) = - ( x + 2)^2

y - 7=  - (1/24) ( x + 2)^2

y = - (1/24)( x + 2)^2 + 7

b.) focus (3, 5) and directrix y = 7

Again....the directrix is above the focus....so this turns downward

The vertex is hafway between the focus and directrix = (3, 6)

p =   distance between focus and vertex = 1

So we have

4p (y - k) =  -(x - h)^2             (h, k) = (3, 6)   and p = 1

4(1) (y - 6) = - (x - 3)^2

4 ( y - 6) = - ( x - 3)^2

y - 6 =  - ( 1/4)(x - 3)^2

y =   - (1/4) ( x - 3)^2 + 6

a.) focus (0, –2) and directrix y = 2

The directrix is above the focus so this parabola opens downward

The vertex is   halfway between the focus and the directrix....so it must be (0, 0)

"p" is the distance between the vertex and the focus [ or the vertex and the directrix ] =  2

So.....we have this form

4p ( y - k) = - ( x - h)^2         where (h, k) is the vertex and p = 2

So we have

4(2) ( y - 0) = - ( x - 0)^2          simplify

8y = -x^2         divide both sides by 8

y = -(1/8)x^2

The graph is vertically stretched by a factor of   [ 2 / (1/2)  ]  = [ 2 * (2/1) ]   =  4

And the graph is shifted up by     8 - 2  =   6 units

We have that

2200 =  500 +  cost per foot * feet of fencing      ......subtract 500 from both sides

1700 =   cost per foot * feet of fencing

1700 = cost per foot * 200

1700 /200 =  cost per foot = 8.5   =  $8.50

f(x) =  2cos (x) - 4

We have the form

f(x) =   Acos [ Bx ] + C

Note that

B = 1    =   2pi / period

This implies that the period must be 2pi

The frequency  =    1 /period =    1 / [ 2 pi ]