SOLUTION: If points (3,1) are on the graph of y=f(x), what points must be on the following graphs? a) y=-1/2f(x-1)+4 b) y=-3(f(3x+6))-2 please show all work of finding points,

(a)  In order for to be more clear, I will designate new function -1/2f(x-1)+4 as g(x) = -1/2f(x-1)+4.


     The fact that the point  (3,1)  lies on the graph y=f(x) means that

         1 = f(3).

     In other words, x-coordinate "3" of the point is the argument of the function f(x), 
     and the output of f(x) at x=3 is 1.


     It means that when x=4, then x-1=3 and we can calculate f(3) = 1.

     So, when x=4, then new function  y= g(x) = -1/2*f(4-1)+4 = -1/2*1+4 = 4 - 1/2 = 3.5.

     Thus we find that if (3,1) lies on the plot y=f(x), then the point (4,3.5) lies on the graph y= g(x) = -1/2*f(x)+4.


     ANSWER.  If (3,1) lies on the plot y=f(x), then the point (4,3.5) lies on the graph y=-1/2*f(x)+4.




(b)  In order for to be more clear, I will designate new function -1/2f(x-1)+4 as h(x) = -3f(3x+6)-2.


     Same as in (a), the fact that the point  (3,1)  lies on the graph y=f(x) means that

         1 = f(3).

     In other words, x-coordinate "3" of the point is the argument of the function f(x), 
     and the output of f(x) at x=3 is 1.


     It means that when 3x+6=3, then 3x = 3-6 = -3, x= -3/3 = -1.

     So, when x=-1, then new function  y= h(-1) = -3*f(3*(-1)+6)-2 = -3*f(-3+6)-2 = -3**f(3)-2 = -3*1-2 = -3 -2 = -5.

     Thus we find that if (3,1) lies on the plot y=f(x), then the point (-1,-5) lies on the graph y= g(x) = 3f(3x+6)-2.


     ANSWER.  If (3,1) lies on the plot y=f(x), then the point (-1,-5) lies on the graph y= h(x) = -3f(3x+6)-2.

    In other words, if you are given that the point (3,1) lies on the plot y=f(x)

    and they ask which point is on the graph of new function g(x) = A*f(Bx+C) + D,

    where A, B, C and D are some given constants, you do as follows:


          (1) you find new value of x from  Bx+C = 3, which gives you  the value of the argument x =   for the new function.

              This value of x provides you f(Bx+C) = f(3).


          (2)  After that you calculate  g(x) = A*f(Bx+C)+D = A*f(3)+D.


          (3)  So, the ANSWER in this GENERAL CASE is that the point  (,) lies on the plot of the new function g(x).


     Thus you have the solution to the given problem in its two versions,
     PLUS the instruction on what to do in very general case.