SOLUTION: (1,2) is a point on the graph of y=f(x) Find the corresponding point on the graph of each of the following functions. (a) y=2+f(-x) (b) y=3f(x-5)-1 (c) y=f(3x-5)-1

When we perform an operation on the entire right side, it causes a VERTICAL
transformation on each y-coordinate.

When we perform an operation on x, it causes a HORIZONTAL transformation on each
x-coordinate.

When the entire right side is multiplied by -1, each y-coordinate is reflected
VERTICALLY across the x-axis.
When only x is multiplied by -1, each x-coordinate is reflected HORIZONTALLY
across the y-axis. 

When the operation is on the entire right side, the shift, stretch, or shrink is
as you would expect. That is:
Adding to the entire right side shifts VERTICALLY UPWARD. 
Subtracting from the entire right side shifts VERTICALLY DOWNWARD.
Multiplying the entire right side by a positive number greater than 1 causes a VERTICAL
STRETCH. 
Multiplying the entire right side by a positive number less than 1 causes a VERTICAL
SHRINK.

However, when the operation is on x only, the shift, stretch, or shrink is
OPPOSITE from what you would expect. 
Adding to x shifts LEFT. 
Subtracting from x shifts RIGHT.
Multiplying x by a positive number greater than 1 causes a HORIZONTAL
SHRINK. 
Multiplying x by a positive number less than 1 causes a HORIZONTAL
STRETCH. 
--------------------------------------------------------------------------------
(a) y=2+f(-x)

The transforming steps are: f(x), f(-x), 2+f(-x)

We start with 
y=f(x)
Then first we replace x by -x
y=f(-x)
Replacing x by -x reflects the x-coordinate of each point HORIZONTALLY across
the y-axis.
So (1,2) is reflected to (-1,2)
Then we add 2 to the entire right side.
y=2+f(-x)
Adding 2 to the entire right side shifts the y-coordinate of each point 2 units
VERTICALLY UPWARD.
So (-1,2) is shifted VERTICALLY UPWARD to (-1,4)
Answer: (-1,4)
--------------------------------------------------------------------------------
(b) y= 3f(x-5)-1

The transforming steps are f(x), 3f(x), 3f(x-5), 3f(x-5)-1

We start with 
y=f(x)
Then we multiply the entire right side by 3
y=3f(x)
Multiplying the entire right side by 3 stretches VERTICALLY the y-coordinate of
each point to 3 times its distance from the x-axis.
So (1,2) is shifted UPWARD to (1,6)
Then we subtract 5 from x
y=3f(x-5)
Subtracting 5 from x shifts each point 5 units HORIZONTALLY RIGHT.
So (1,6) is shifted HORIZONTALLY RIGHT to (6,6)
Then we subtract 1 from the entire right side.
y=3f(x-5)-1
Subtracting 1 from the right side shifts the y-coordinate of each point DOWNWARD 1
unit.
So (6,6) is shifted DOWNWARD to (6,5)
Answer (6,5)
--------------------------------------------------------------------------------
(c) y=f(3x-5)-1

The transforming steps are f(x), f(3x), f(3x-5), which must be written as
, then , or f(3x-5)-1 

We begin with 
y=f(x)
Then we replace x by 3x
y=f(3x)
Multiplying x by 3 shrinks the x-coordinate of each point HORIZONTALLY to 1/3 of
its distance from the y-axis.
So the point (1,2) gets its x-coordinate shrunk to (1/3,2)
Now we must calculate how much shifting we must do to x to get f(3x-5)
So we factor out 3 and rewrite y=f(3x-5) as

So now we can see that x has been replaced by x-5/3.
Subtracting 5/3 from x shifts the x-coordinate of each point 5/3 units
HORIZONTALLY RIGHT.
So (1/3,2) is shifted HORIZONTALLY RIGHT to (1/3+5/3,6) or (6/3,6) or (2,6)
Finally we must subtract 1 from the entire right side which shifts each y-coordinate VERTICALLY DOWNWARD.
So (2,6) is shifted VERTICALLY DOWNWARD to (2,5)
Answer: (2,5)

Edwin