 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 will get it when you are 21. Adding 4 made it happen earlier.
 C > 0 moves it left
 C < 0 moves it right
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)
An easy way to remember what happens to the graph when we add a constant:
add to y to go high
add to x to go left
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   C > 0 moves it up
 C < 0 moves it down

y=f(x+C)   C > 0 moves it left
 C < 0 moves it right

y = Cf(x)   C > 1 stretches it in the ydirection
 0 < C < 1 compresses it

y = f(Cx)   C > 1 compresses it in the xdirection
 0 < C < 1 stretches it

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 right:h(x) = 1/(x−4) graph
Move 5 spaces 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 right:w(x) = (x−4)^{3} − 4(x−4)
Move 5 spaces 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 transformations on f()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
Play with this graph
7260, 7261, 7262, 7267, 7268, 555, 556, 557, 558, 1191
What is a Function? Algebra Index
FAQs
There are four main types of transformations: translation, rotation, reflection and dilation. These transformations fall into two categories: rigid transformations that do not change the shape or size of the preimage and nonrigid transformations that change the size but not the shape of the preimage.
What is the transformation rule for function? ›
There are different formulas for different rules of transformation. For vertically transformation the function f(x) is transformed to f(x) + a or f(x)  a. For horizontal transformation the function f(x) is transformed to f(x + a) or f(x  a). Further for stretched or compressed transformation is it f(cx) or cf(x).
What are the 3 main types of transformations? ›
Translation is when we slide a figure in any direction. Reflection is when we flip a figure over a line. Rotation is when we rotate a figure a certain degree around a point.
In what order should I do transformations? ›
If a function has multiple transformations, they are applied in the following order: 1. Horizontal translation 2. Reflection, Stretching, Shrinking 3. Vertical Translation.
How do you identify a transformation? ›
Identifying Transformations on a Graph
This point is typically the origin (0,0) of the graph or a point on the figure, but it can also be another point on the graph. Identify if each point on the original figure is in a different orientation in the transformed figure, and if the figure appears to be turned.
What are the possible ways to transform a function? ›
Well, a function can be transformed the same way any geometric figure can: They could be shifted/translated, reflected, rotated, dilated, or compressed. So that's pretty much all you can do with a function, in terms of transformations.
What are the basic transformations? ›
There are four common types of transformations  translation, rotation, reflection, and dilation.
How do you describe transformations in math? ›
Transformations change the size or position of shapes. Congruent shapes are identical, but may be reflected, rotated or translated. Scale factors can increase or decrease the size of a shape.
What is the formula for rotation? ›
90° clockwise rotation: (x,y) becomes (y,−x) 90° counterclockwise rotation: (x,y) becomes (−y,x) 180° clockwise and counterclockwise rotation: (x,y) becomes (−x,−y) 270° clockwise rotation: (x,y) becomes (−y,x)
How to tell if a function is even or odd? ›
To know whether or not a function is even or odd, compare f(x) with f(x). If f(x) = f(x), f is even; if f(x) = f(x), it is odd.
The equation for calculating Transformation Efficiency (TE) is: TE = Colonies/µg/Dilution. Efficiency calculations can be used to compare cells or ligations.
How to find the domain of a function? ›
The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number. The domain of a function can be determined by listing the input values of a set of ordered pairs.
What are the four 4 types of transformation process? ›
Types of Transformation Processes
The transformation process can be of four types: materials, information, customers, and services. Each type involves different inputs, transformation processes, and outputs. Materials Transformation Process: Involves converting raw materials into finished products.
What are 4 different types of linear transformations? ›
Most common geometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear.
What are the 4 four operations of function? ›
Functions can be added, subtracted, multiplied, and divided just like the individual terms of a function itself. The notation for these is as follows: Addition f ( x ) + g ( x ) means to add two functions together and can also be written as.
What are the four 4 ways to represent the functions? ›
There are four representations of functions: mapping diagram, graph, table, and equation. Mapping diagrams are best for discrete functions with a small domain and range. Tables are best for discrete functions with unrelated input values. Tables can be used to help graph equations of functions.