Building Functions
Sequences
An example of sequence is
2,4,6,8,10,12,14,16,18,20,....
The recursive formula for this sequence is
tn=tn-1+2
The explicit formula for this sequence is
tn=t1+2(n-1)
*NOTE: the letters and numbers in italics are meant to be subscripts.
2,4,6,8,10,12,14,16,18,20,....
The recursive formula for this sequence is
tn=tn-1+2
The explicit formula for this sequence is
tn=t1+2(n-1)
*NOTE: the letters and numbers in italics are meant to be subscripts.
Function Operations
There is a way to add, subtract, multiply, and divide functions. To show each of these operations we are going to use the two functions
f(x)=2x²-3x+4
AND
g(x)=x+3
f(x)=2x²-3x+4
AND
g(x)=x+3
Adding
To add the functions listed above we start by putting the whole function in paranthesis and showing them like this
(2x²-3x+4)+(x+3)
and then we have to combine like terms. The 2x² can't be combined with anything so we bring it down and it stays the same. The -3x can be combined with the x to make -2x and we can combine the 4 and 3 to make 7. The final result will look like this
2x²-2x+7
(2x²-3x+4)+(x+3)
and then we have to combine like terms. The 2x² can't be combined with anything so we bring it down and it stays the same. The -3x can be combined with the x to make -2x and we can combine the 4 and 3 to make 7. The final result will look like this
2x²-2x+7
Subtracting
To subtract the functions listed above we start exactly the same way we did before except we will put a subtraction sign in between the functions. It will look like this
(2x²-3x+4)-(x+3)
and then we combine like terms just like before. The 2x² cant be combined with anything so it just comes down and stays the same. We can combine the -3x and the x to get -4x. We got -4x now because we have to subtract -x from -3x. Finally 4 and 3 can combine to get 1 because when you subtract 3 from 4 you get 1. The final result will look like this
2x²-4x+1
(2x²-3x+4)-(x+3)
and then we combine like terms just like before. The 2x² cant be combined with anything so it just comes down and stays the same. We can combine the -3x and the x to get -4x. We got -4x now because we have to subtract -x from -3x. Finally 4 and 3 can combine to get 1 because when you subtract 3 from 4 you get 1. The final result will look like this
2x²-4x+1
Multiplying
To multiply the functions we do the same thing as before but we put a multiplication sign between the functions. It will look like this
(2x²-3x+4)*(x+3)
Multiplying functions is the longest function operation to perform. We simplify the function first by multiplying, then adding, and then you get the answer. So in this problem we use the double distributive and multiply x by everything and then 3. First we multiply x by 2x², -3x, and 4. After that we have
2x³-3x²-4x
After that we have to multiply 3 by everything. So after we multiply everything by 3 we have
6x²-9x+12
We now combine the two functions we created by adding because we have x+3. if we had x-3 we would subtract. Our function combination will look like
(2x³-3x²-4x)+(6x²-9x+12)
Now we add this and we get
2x³+3x²-5x+12
That will be our final answer.
(2x²-3x+4)*(x+3)
Multiplying functions is the longest function operation to perform. We simplify the function first by multiplying, then adding, and then you get the answer. So in this problem we use the double distributive and multiply x by everything and then 3. First we multiply x by 2x², -3x, and 4. After that we have
2x³-3x²-4x
After that we have to multiply 3 by everything. So after we multiply everything by 3 we have
6x²-9x+12
We now combine the two functions we created by adding because we have x+3. if we had x-3 we would subtract. Our function combination will look like
(2x³-3x²-4x)+(6x²-9x+12)
Now we add this and we get
2x³+3x²-5x+12
That will be our final answer.
Dividing
At the level I am at now all we do is put f(x) over g(x) and get
2x²-3x+4
x+3
2x²-3x+4
x+3
Applying Function Operations to Linear Functions
We are going to use the two functions
f(x)=2x+8
AND
g(x)=x-4
f(x)=2x+8
AND
g(x)=x-4
Adding
(2x+8)+(x-4)=3x+4
Subtracting
(2x+8)-(x-4)=x+12
Multiplying
(2x+8)*(x-4)=2x²+16x-32
Dividing
(2x+8)÷(x-4)=2x+8
x-4
x-4
Vertical and Horizontal Translations
For each type of function there is a parent function. The parent function of the linear function is f(x)=x and the parent function for the exponential function is f(x)=2^x. Each child function has something added onto the parent function to change the location of the graph. These are called translations and depending on what and where you put a little add-on you change the location of the parent function. Translations move a graph up, down, left, and right. These are types of transformations. The two other transformations are reflections and dilations. We are only going to talk about translations today.
Input
When you have a function and you add or subtract a number from the input it will move left or right depending on the symbol.
When you add to the input it moves to the left.
When you subtract from the input it moves to the right.
When you add to the input it moves to the left.
When you subtract from the input it moves to the right.
Output
When you have a function and you add or subtract a number from the output it will move up and down based upon the symbol.
when you add to the output it moves up.
when you subtract to the output it moves down.
when you add to the output it moves up.
when you subtract to the output it moves down.
Examples
The two graphs above are different because the location of the origin has shifted because of the add-ons to the original function. After we subtracted 4 and added 6 to the parent function the origin shifted from (0,0) to (4,6).
The two graphs above are different because the location of the origin has shifted. The origin went from (0,1) to (0,4) because all we did is add 3 to the output.
The difference between these two graphs is the location of the origin. The original absolute value function is at the location (0,0) and the child absolute value function is at the location (-5,4)