To find out the operations, we must first "route out" the path
from  y = sin-1(x) to y = p - 3sin-1(1 - 4x)

To find the "route", we change the expression in

y = f(x) to  y = A*f[B(x+C)] + D

First we change the expression in parentheses to the form B(x+C)
in parentheses:

     1 - 4x

Write it in descending order

    -4x + 1

Factor out the coefficient of x

   -4(x - )

So to build that expression from x
one operation at a time of replacing x:

We go from x to -x to -4x to -4(x - )

which means we go from the graph of

y = sin-1(x) 

to the graph of

y = sin-1(-x)

to the graph of

y = sin-1(-4x)}}})

to the graph of

y = sin-1[-4(x - )]

to the graph of

y = -sin-1[-4(x - )]

to the graph of

y = -3sin-1[-4(x - )]

to the graph of

y = -3sin-1[-4(x - )] + p

 
The first operation in going from the graph of

y = sin-1(x) 

which is this



to the graph of 

y = sin-1(-x)

involves replacing x by -x which is a 
reflection across (or into) the y-axis:


   
Next we go from that graph to the graph of

y = sin-1(-4x)

That involves replacing x by 4x which is a 
horizontal shrinking of the graph by a factor of :



Next we go from that graph to the graph of

y = sin-1[-4(x - )]

That involves replacing x by x - 
which is a right shift of  a unit:



Now we go from that graph to the graph of

y = -sin-1[-4(x - )]

which involves multiplying the whole right side by -1,
which reflects the graph across (or into) the
x-axis:



Now we go from that graph to the graph of

y = -3sin-1[-4(x - )]

which involves multiplying the whole right side by 3,
which stretches the graph by a factor of 3



Finally we go from that graph to the graph of 
  
y = -3sin-1[-4(x - )] + p

which involves adding p to the right side,
which shifts the graph vertically p units upward:




Edwin