CPhill

No equation is shown in 1    and no expression is shown in 3.....

(-2,0)    and  ( 4,0)

These must be the roots of the quadratic...so we have that

y  = a ( x + 2) ( x - 4)      simplify

y  = a (x^2 - 2x - 8)

And since  54 is a max, then using symmetry, the point  ( [-2 + 4]/2, 54)  = (1,54)  must be on the graph

So

54   =  a ( 1^2 - 2(1) - 8)

54 = a (-9)

a = -54/9  = -6

So.....the function is

y = (-6) ( x^2 -2x  - 8)

y = -6x^2 + 12x  + 48

Here's a graph  :  https://www.desmos.com/calculator/s8dqi0vdcn

y asymptote  ⇒   CORRECT   !!!

x asymptotes 

x^2  + 2x  - 3                  ( x + 3) ( x - 1)               (x + 3)

___________  =            ______________ =   ________

2x^2  - x   - 1                  (2x + 1) ( x - 1)             (2x + 1)

x = 1/2   is also CORRECT!!!!!

GOOD JOB!!!

[We have a "hole"  at   x = 1    ]

1. No function shown 

2.  3x^2 - 7   =  0

Factors of p  = ±1, ±7

Factors of q =  ±1, ±3

Possible zeroes  =   ±1/3 , ±1, ±7/3, ±7

3.  No expression shown

Thanks, guys.....I really appreciate all the  nice comments!!!!

Heureka  is very good  !!!!

Very nice, heureka!!!!

This one completely stumped me  !!!!

Nice, Gadfly  !!!!....your Geometry skills are impressive  !!!

I have one question.....could you  explain  how you  know that  BOC is a right angle   ????

Yep....sorry....let me make an edit!!!

Thanks for  catching  that   !!!!

Note that the area  of a triangle  can be given by

(1/2) (side A) (side B)  ( sine of included angle  between sides A, B)

And note that   the sine  is greatest  at 90°  =  1

So.....the max area  is

(1/2) (y) (y) sin (90°)  =

(1/2)y^2 (1)   =

(1/2)y^2