 | 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) = x2, but it could be anything:
f(x) = x2
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 y-value:
g(x) = x2 + 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 x-value:
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) = x3 - x2 + 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 y-direction by multiplying the whole function by a constant.
g(x) = 0.35(x2)
- C > 1 stretches it
- 0 < C < 1 compresses it
We can stretch or compress it in the x-direction by multiplying x by a constant.
g(x) = (2x)2
- C > 1 compresses it
- 0 < C < 1 stretches it
Note that (unlike for the y-direction), bigger values cause more compression.
We can flip it upside down by multiplying the whole function by −1:
g(x) = −(x2)
This is also called reflection about the x-axis (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 y-direction.
We can flip it left-right by multiplying the x-value by −1:
g(x) = (−x)2
It really does flip it left and right! But you can't see it, because x2 is symmetrical about the y-axis. So here is another example using √(x):
g(x) = √(−x)
This is also called reflection about the y-axis (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 y-direction
- 0 < C < 1 compresses it
|
| y = f(Cx) | - C > 1 compresses it in the x-direction
- 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 y-direction:h(x) = 2/x
Compress it by 3 in the x-direction:h(x) = 1/(3x)
Flip it upside down:h(x) = −1/x
Example: the function v(x) = x3 − 4x
Here are some things we can do:
Move 2 spaces up:w(x) = x3 − 4x + 2
Move 3 spaces down:w(x) = x3 − 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 y-direction:w(x) = 2(x3 − 4x)
= 2x3 − 8x
Compress it by 3 in the x-direction:w(x) = (3x)3 − 4(3x)
= 27x3 − 12x
Flip it upside down:w(x) = −x3 + 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 left-right
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 y-direction
- 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 non-rigid 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.
