SOLUTION: Consider the parent function {{{f(x) = sqrt(x)}}}. What parameters (a, k, d, c) affect the domain of g(x) = af[k(x-d)]+c? Please explain.

What is done to the variable only possibly affects the domain.
What is done to the entire function only possibly affects the range. 

Only d and k are things done to the variable x only, so they possibly affect the
domain.
Only a and c are things done to the entire function, so they possibly affect the
range.

Explanation:

We will assume that a, k, d, c are all positive.

If we do something to the VARIABLE ONLY, it's a HORIZONTAL change.
If we do something to the ENTIRE FUNCTION, it's a VERTICAL change.

We must always do all HORIZONTAL CHANGES first. That is, we must do something to
the VARIABLE first if there is anything to do to it.  Only when we have done
everything HORIZONTAL to the VARIABLE ONLY, will we proceed to do something
VERTICAL to the ENTIRE FUNCTION.  

Horizontal changes do the OPPOSITE to "what you would expect":

Horizontal changes by ADDING to the variable move the curve LEFTWARD horizontally.
Horizontal changes with SUBTRACTING from the variable move the curve RIGHTWARD
horizontally.
Horizontal changes with MULTIPLYING the variable by numbers > 1 SHRINK the
graph horizontally.
Horizontal changes with MULTIPLYING by numbers < 1 STRETCH the graph
horizontally.

VERTICAL changes do "what you would expect":

Vertical changes with ADDITION to the entire function move the curve UPWARD
vertically.
Vertical changes with SUBTRACTION from the entire function move the curve
DOWNWARD vertically.
Vertical changes with MULTIPLICATION of the entire function by numbers > 1
stretch the graph vertically.
Vertical changes with MULTIPLICATION of the entire function by numbers < 1
shrink the graph vertically.

Here is the order in which we will proceed from f(x) to g(x)

Notice that the usual order of operations are followed, multiplication is done
before addition and subtraction.







Start with the parent function   

We do something to the variables first, which will result in horizontal changes.

First we do something to the variable x only by multiplying x by k, so
it will do the opposite of what you might think: 



That shrinks the graph horizontally by a factor of 1/k if k > 1 or stretches it
by a factor of k if k < 1.

Next, we again do something to the variable x only by subtracting d from the
variable x, so again it will do the opposite of what you might think: 



That shifts the graph horizontally TO THE RIGHT by a distance of d units.

We have done everything to the variable.  So now we do something to the ENTIRE FUNCTION.

Now we multiply the ENTIRE FUNCTION by "a".  Now what we do will be "what you
would expect":



This stretches the graph by a factor of 'a' if a > 1 and skrinks it vertically
by a factor of 1/a if a < 1.

Finally we add 'c' to the ENTIRE FUNCTION.  This does "what you would expect".
It shifts the entire graph vertically UPWARD by 'c' units.



Edwin