Question 470844
y= π-3sin^-1 (1-4x) from y=sin^-1 x
<pre>
To find out the operations, we must first "route out" the path
from  y = sin<sup>-1</sup>(x) to y = <font face = "symbol">p</font> - 3sin<sup>-1</sup>(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 - {{{1/4}}})

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 - {{{1/4}}})

which means we go from the graph of

y = sin<sup>-1</sup>(x) 

to the graph of

y = sin<sup>-1</sup>(-x)

to the graph of

y = sin<sup>-1</sup>(-4x)}}})

to the graph of

y = sin<sup>-1</sup>[-4(x - {{{1/4}}})]

to the graph of

y = -sin<sup>-1</sup>[-4(x - {{{1/4}}})]

to the graph of

y = -3sin<sup>-1</sup>[-4(x - {{{1/4}}})]

to the graph of

y = -3sin<sup>-1</sup>[-4(x - {{{1/4}}})] + <font face = "symbol">p</font>

 
The first operation in going from the graph of

y = sin<sup>-1</sup>(x) 

which is this

{{{graph(200,650,-2,2, -5,8, asin(x))}}}

to the graph of 

y = sin<sup>-1</sup>(-x)

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

{{{graph(200,650,-2,2, -5,8, asin(-x))}}}
   
Next we go from that graph to the graph of

y = sin<sup>-1</sup>(-4x)

That involves replacing x by 4x which is a 
horizontal shrinking of the graph by a factor of {{{1/4}}}:

{{{graph(200,650,-2,2, -5,8, asin(-4x))}}}

Next we go from that graph to the graph of

y = sin<sup>-1</sup>[-4(x - {{{1/4}}})]

That involves replacing x by x - {{{1/4}}}
which is a right shift of {{{1/4}}} a unit:

{{{graph(200,650,-2,2, -5,8, asin(1-4x))}}}

Now we go from that graph to the graph of

y = -sin<sup>-1</sup>[-4(x - {{{1/4}}})]

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

{{{graph(200,650,-2,2, -5,8, -asin(1-4x))}}}

Now we go from that graph to the graph of

y = -3sin<sup>-1</sup>[-4(x - {{{1/4}}})]

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

{{{graph(200,650,-2,2, -5,8, -3asin(1-4x))}}}

Finally we go from that graph to the graph of 
  
y = -3sin<sup>-1</sup>[-4(x - {{{1/4}}})] + <font face = "symbol">p</font>

which involves adding <font face = "symbol">p</font> to the right side,
which shifts the graph vertically <font face = "symbol">p</font> units upward:

{{{graph(200,650,-2,2, -5,8, pi-3asin(1-4x))}}}


Edwin</pre>