Lesson Given a point on a plot of a function, find the corresponding point on the plot of transformed function

Given a point on a plot of a function, find the corresponding point on the plot of transformed function

Problem 1

The fact that (-9,4) is a point on the graph of f(x) means that

    4 = f(-9).


Now, you need to have -9 = 5x,  so for it you take x = .


It provides you f(5x) =  = f(-9) = 4.    (1)


Next, according to the definition  g(x) = -2f(5x) - 4,

you should multiply the '4'  in  (1)  by (-2) and subtract 5 from the product.


It gives you the ordinate y = (-2)*4 - 5 = -8 - 5 = -13.


Thus the required point  (x,y)  is  (,).


This point is the transformed point on the graph of g(x).


At this point, the problem is solved completely.

Problem 2

First, we want the argument 2x-3 be 3.


So, we write this equation

    2x - 3 = 3.


Then we solve it  and find  x =  =  = 3.


Next,  we want to calculate  y = -5f(2x-3) + 1.

We just know that 2x-3 is 3, because we found x in this way.
So, we write

    y = = -5f(3) + 1 = (at this point, we know and use that f(3)=4, so we continue) = -5*4 + 1 = -20 + 1 = -19.


Thus the point  (x,y) = (3,-19)  is on the graph  y = -5f(2x-3) + 1.


ANSWER.  If f(3) = 4,  then the point which must be on the graph  y = -5f(2x-3) + 1  is  (x,y) = (3,-19).

Problem 3

(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 for both its parts,
     as well the instruction on what to do in a general case.