Function Transformations

Hey guys ! got something here for the class..Hope this helps.. :p

FUNCTION TRANSFORMATION

Let's start this by knowing what function is....

      *A Function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

means that...

     " In each value of x, there is only one value of y"

sample??..

   IN  f(x) = x² 

     ^{*note: f(x) = y}

 ^{IF x = 2, THEN y = 4}

 IF x = 3, THEN y = 9

 IF x = 4, THEN y = 16 ...and so on...

If we graph it, it should look like this...

    

f(x) = x²

^{NOW THAT'S A FUNCTION...}

   ^{*SO IS A CIRCLE A FUNCTION??..}

      

 

...well its

HOW TO KNOW IF A GRAPH IS A FUNCTION??.. :D

  Well there's a TEST to know if a GRAPH is a FUNCTION....

ITS V. L. T OR VERTICAL LINE TEST.. :D

If we draw a vertical line in a circle, the line INTERSECTS TWICE. So it means, its NOT a function.

NOTE: If the vertical line intersects the graph only once, the graph is a FUNCTION.

NOW THAT WE KNOW WHAT FUNCTION IS.

*Can we move or resize the graph of a function?..YES!!..

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 you can do to move or scale it on the graph:

You can move it up or down by adding a constant to the y-value:

g(x) = x² + C

Note: if you want to move the line down, just use a negative value for C.

• C > 0 moves it up
• C < 0 moves it down

 

You 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 are going to 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 you add a constant:

add to y: go high
add to x: go left

BUT you must add C wherever x appears in the function (you are substituting x+C for x).

Example: the function v(x) = x³ - x² + 4x

Move C spaces to the left: w(x) = (x+C)³ - (x+C)² + 4(x+C)

You 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

 

You can stretch or compress it in the x-direction 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 y-direction), bigger values cause more compression.

 

You 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)

You can combine a negative value with a scaling.

Example: multiplying by -2 will flip it upside down AND stretch it in the y-direction.

 

You 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 = C·f(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)	Reflects it about x-axis
y = f(-x)	Reflects it about y-axis

All In One ... !

You 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 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

Thats it...

Sources:

GOOGLE :D

http://www.mathsisfun.com/sets/function-transformations.html