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
To move C spaces to the left, add C to x wherever x appears:
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 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 = Cf(x) 

y = f(Cx) 

y = −f(x) 

y = f(−x) 

Examples
Example: the function g(x) = 1/x
Here are some things we can do:
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/(x−4) 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
Here are some things we can do:
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) = (x−4)^{3} − 4(x−4)  
Move 5 spaces to the left:  w(x) = (x+5)^{3} − 4(x+5) 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