Function Transformations
Just like Transformations in Geometry, we can move and resize the graphs of functions 
Let us start with a function, in this case it is f(x) = x^{2}, but it could be anything:
f(x) = x^{2}
Here are some simple things we can do to move or scale it on the graph:
We can move it up or down by adding a constant to the yvalue:
g(x) = x^{2} + C
Note: to move the line down, we use a negative value for C.
 C > 0 moves it up
 C < 0 moves it down
We can move it left or right by adding a constant to the xvalue:
g(x) = (x+C)^{2}
Adding C moves the function to the left (the negative direction).
Why? Well imagine you will inherit a fortune when your age=25. If you change that to (age+4) = 25 then you would get it when you are 21. Adding 4 made it happen earlier.
 C > 0 moves it left
 C < 0 moves it right
An easy way to remember what happens to the graph when we add a constant:
add to y: go high
add to x: go left
BUT we must add C wherever x appears in the function (we are substituting x+C for x).
Example: the function v(x) = x^{3}  x^{2} + 4x
Move C spaces to the left: w(x) = (x+C)^{3}  (x+C)^{2} + 4(x+C)
We can stretch or compress it in the ydirection by multiplying the whole function by a constant.
g(x) = 0.35(x^{2})
 C > 1 stretches it
 0 < C < 1 compresses it
We can stretch or compress it in the xdirection by multiplying x (wherever it appears) by a constant.
g(x) = (2x)^{2}
 C > 1 compresses it
 0 < C < 1 stretches it
Note that (unlike for the ydirection), bigger values cause more compression.
We can flip it upside down by multiplying the whole function by 1:
g(x) = (x^{2})
This is also called reflection about the xaxis (the axis where y=0)
We can combine a negative value with a scaling.
Example: multiplying by 2 will flip it upside down AND stretch it in the ydirection.
We can flip it leftright by multiplying the xvalue by 1:
g(x) = (x)^{2}
It really does flip it left and right! But you can't see it, because x^{2} is symmetrical about the yaxis. So here is another example using √(x):
g(x) = √(x)
This is also called reflection about the yaxis (the axis where x=0)
Summary
y = f(x) + C 

y = f(x + C) 

y = C·f(x) 

y = f(Cx) 

y = f(x) 

y = f(x) 

Examples
Example: the function g(x) = 1/x
Move 2 spaces up: h(x) = 1/x + 2
Move 3 spaces down: h(x) = 1/x  3
Move 4 spaces to the right: h(x) = 1/(x4) (play with the graph)
Move 5 spaces to the left: h(x) = 1/(x+5)
Stretch it by 2 in the ydirection: h(x) = 2/x
Compress it by 3 in the xdirection: h(x) = 1/(3x)
Flip it upside down: h(x) = 1/x
Example: the function v(x) = x^{3}  4x
Move 2 spaces up: w(x) = x^{3}  4x + 2
Move 3 spaces down: w(x) = x^{3}  4x  3
Move 4 spaces to the right: w(x) = (x4)^{3}  4(x4)
Move 5 spaces to the left: w(x) = (x+5)^{3}  4(x+5) (play with the graph)
Stretch it by 2 in the ydirection: w(x) = 2(x^{3}  4x) = 2x^{3}  8x
Compress it by 3 in the xdirection: w(x) = (3x)^{3}  4(3x) = 27x^{3}  12x
Flip it upside down: w(x) = x^{3} + 4x
All In One ... !
We can do all transformation in one go using this:
a is vertical stretch/compression
 a > 1 stretches
 a < 1 compresses
 a < 0 flips the graph upside down
b is horizontal stretch/compression
 b > 1 compresses
 b < 1 stretches
 b < 0 flips the graph leftright
c is horizontal shift
 c < 0 shifts to the right
 c > 0 shifts to the left
d is vertical shift
 d > 0 shifts upward
 d < 0 shifts downward
Example: 2√(x+1)+1
a=2, c=1, d=1
So it takes the square root function, and then
 Stretches it by 2 in the ydirection
 Shifts it left 1, and
 Shifts it up 1