CPhill

Domain  = x values covered by the graph

Range = y values covered by the graph

4 looks good

5   Domain is correct  

     Range is   (-inf, 0 )

7. Domain   [ -3, 5 ]      ⇒  -3 ≤ x ≤ 5

   Range    [-4, 4]  ⇒  -4 ≤ x ≤ 4

8.  Domain   (-inf, inf )     ⇒    -inf < x < inf

     Range  [-3, 3 ]   ⇒   -3 ≤ x ≤ 3

No prob, skye....!!!!

(x + 2)(x - 4)     expand

x^2 + 2x - 4x - 8

x^2 - 2x - 8

The x coordinate of the vertex is  -(-2) / (2 * 1)  =   2 /2  = 1

The y coordinate of the vertex is  (1)^2 - 2(1) - 8  =  1 - 2 - 8  =  -9

So...the vertex is (1, - 9)

A second point can be found if we let x = 0

(0)^2 - 2(0) - 8  =

-8

So  (0, - 8)  is on the graph

Here's the graph :  https://www.desmos.com/calculator/tgpdpinlno

The verex here is    (0, 8)

To find a second point....let x = 1  and we have

-(1)^2 + 8  =   -1 + 8  = 7

So....a second point is  (1, 7)

Here's the graph :  https://www.desmos.com/calculator/ayaokwy4vm

The x coordinate of the vertex is given by    - (-12) / (2 * - 2)  =  12/ -4  = - 3

Putting this back into the function to get the y coordinate of the vertex we have

-2(-3)^2 - 12(-3) - 24  =   

-18 + 36 - 24 = 

- 6

So....the vertex is  (-3, -6)

And we can find another point by letting x = 0

-2(0)^2 - 12(0)  - 24 =

-24

So...the second point is  (0, -24)

Here's the graph :  https://www.desmos.com/calculator/synw1wfvuo

Let the coordinates of this point be ( 4, y)

The point on the y axis closest to this point is (0, y)

So... we have this equation

Distance from (4, y) to (4, 1)   = Distance from (4, y) to (0, y)

( 4 - 4)^2 + (1 - y)^2  =  (4 - 0)^2 + ( y - y)^2          smplify

y^2 - 2y + 1   =  16

y^2 - 2y - 15  = 0      factor

(y - 5) ( y + 3) = 0

Since y  must be negative......y = -3

So.....the point is  ( 4, -3)

Exterior angle of Mary's polygon =    360 / m

Interior angle of George's polygon  =   (g - 2)*180 / g

Since these are congruent....then

360 / m  =   (g - 2)  * 180 / g         cross - multiply

360g  =  180m * (g - 2)

360g = 180 gm - 360m

360m  = 180gm - 360g

2m = gm - 2g

2m =   g ( m - 2 )

g =  2m / ( m - 2)

We have the form

y = Asin (Bx + C) + D

A = the amplitude = 4

To compute "B"  we have

B =  2pi / period    =   2 pi / 4   =    pi /2

C = phase shift    (none)  so    C = 0

D = midline  =  1

So.......we have

y =  4 sin (  pi/2 * x ) + 1

We only have these to choose from   [ 2, 3, 5 , 7 ]

We have 4  choices for each digit.....so

4⁴ =     256   positive integers

We have the form

y = Asin (Bx + C) + D

A = the amplitude =  2

The period =  4pi

Thus....in 2pi....there is (1/2) of a period.....so "B" =  (1/2)

There is no phase shift....so......C = 0

Since the midline is 2, then D =  2

So  the equation is 

y =   2sin ( 1/2 * x) + 2    =  2sin (x/2) + 2

Here:  https://www.desmos.com/calculator/o4e9ngaqie