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Function Rule Calculator

 


Function Rule Calculator

May a times in real life we have situations as follows, for example: In a theater showing a play, 300 tickets are sold for  3000 dollars and 500 tickets are sold for 5000 dollars. What is the rule that defines the amount of money received for x number of tickets sold?

Another example: The following table gives  the population of a particular species of an organism:
Time(weeks)    0    1     2     3      4     5    6
Number           2    7    10    11    10    7    2


Find a formula that relates the time to the number of organisms.
In such situations we need to use the function rule calculator. 
For the purpose of this article we shall limit ourselves to understanding how to calculate functions rules of linear and quadratic type only.
 
We shall understand better with the help of an example.

Example 1: Find a function rule that relates x to y as given in the table below:

 X    Y
-5    0
-3    3
1     9

Solution: First let us plot those points to check what kind of a function we need. That is whether it has to be a linear or a quadratic function.
 
From the above graph we see that we are looking at finding a linear function. 
The general form of a linear function is y = mx + b, where m is the slope and b is the y intercept.
From the table (or the graph) we have three ordered pairs of the form (x,y) which are (-5,0), (-3,3) and (1,9).
So if we plug any two of these three values into the general form of a linear function we have:
0 = m (-5) + b and 3 = m (-3) + b
Solving those two equations simultaneously for m and b we get, 
 m = 3/2, and b = 15/2
So our required function would be: y= mx + b = (3/2)x + 15/2

Example 2: Find a function to relate x and y as shown in table below:
X    Y
-2    5
-1    0
3     0
4     5

Solution: On plotting the points we have:
 
So this time we see that it is a quadratic curve. The general form of a quadratic curve is:
Y = ax2 + bx + c. Now if we plug in any of the three ordered pairs into that equation we have:
5 = a(-2)2 + b(-2) + c = 4a – 2b + c 
0 = a(-1)2 + b(-1) + c = a – b + c and
0 = a(3)2 + b(3) + c = 9a + 3b + c
Solving those three equations for a, b and c we get, a = 1, b = -2 and c = -3. So the required function would be: y = x2 – 2x - 3