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Write a rule for the function if the d:{-2,-1,0,1} r:{15,12,9,6}

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I like to teach this problem with the guess my rule. If we started out, I'd have you tell me what goes in and I'd tell you what comes out. You'd say 0 and I'd say 2, and you'd say 1 and I'd say 3 and you'd say 2 and I'd say 4 and either you would answer plus 2, keep geussing, or give me a look like the answer is so painfully obvious that my eyes might pop out of their sockets.


And eventually we'd get to this rule, where -2 goes in and 15 comes out, and -1 goes in and 12 comes out.  You could tell that x goes up by 1 each time and y goes down by 3, we have a linear relationship and that all we need for that is a starting point (or x-intercept) and a rate of change (y goes down 3 every time x goes up 1) and then you'd think in your head it must by y = -3x + 9 based on the evidence but you'd have to go back and check each answer before you guessed b/c if you didn't and you made a mistake you would lose and I would win and I told you that the loser has to wash the winner's car b/c I knew you were too young to actually have a car.

Assume there is a linear relationship between the 

Domain and range values. Choosing (-2,15)and (-1,12) 

We calculate the slope m=(15-12)/(-2-(-1))= -3 so y= -3x+b now we let x= -2 and y=15,so we get 15=6+b and so b=9. Then by substituting othecoordinates we 

find that the line y= -3x+9 fits the domain and range data points.

We need two things to write a linear function  - a rate of change, and a starting value (the y-value when x = 0)

Well, you can see that the x-values in the domain are increasing by 1. (-2, -1, 0, 1, ....)
And the y-values in the range are decreaasing by 3 (15 - 3 =12. 12 - 3 = 9. 9 - 3 = 6... )
So that means our rate of change (slope) = -3 / 1 = -3.
The starting value/y-intercept (when x = 0) is y = 9.
(Since 0 is the third x-value and 9 is the third y-value in the list)

So your equation is y = -3x + 9

As a check, use the first values in your list: if x = -2, y should be 15. -3(-2) + 9 = 6 + 9 = 15. √
As a double-check, use the second values in your list: if x = -1, y should be 12. -3(-1) + 9 = 3 + 9 = 12. √√