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EXPLICIT RULES: INPUT-OUTPUT FORMULAS Function Machine Model 1. If you input 4 into the function machine, the output is 6. If you input 5 into the function machine, then the output is 7 What is the input-output rule for the function machine? OUTPUT = INPUT + 2 ___________________________________________ 10 Therefore, if the input is 8 the output would be __________ 2. If the input is 1, the output is 4. If the input is 2, the output is 7. If the input is 3, the output is 10. What is the input-output rule for the function machine? OUTPUT = INPUT + 3 _____________ When looking at a sequence, an INPUTOUTPUT formula connects the term number to the term, rather than connecting the term to the previous term like NOW-NEXT formulas. +4 INPUT-OUTPUT FORMULA OUTPUT = INPUT + 4 x +4 1 +4 2 +4 3 4 y 5 6 7 8 NOW-NEXT FORMULA +1 +1 +1 Start = 5 NEXT = NOW + 1 Examples x 1 2 3 4 y -2 -1 0 1 INPUT-OUTPUT: OUTPUT = INPUT - 3 NOW-NEXT: Start = -2 NEXT = NOW + 1 x 1 2 3 4 y 3 6 9 12 INPUT-OUTPUT: OUTPUT = INPUT(3) NOW-NEXT: Start = 3 NEXT = NOW + 3 You Try! x 2 3 4 5 y 1 1.5 2 2.5 x 1 2 3 4 y 3 5 7 9 NOW-NEXT: NOW-NEXT: x 1 2 3 4 y 1 8 15 22 NOW-NEXT: Start = 1 Start = 3 Start = 1 NEXT = NOW + 0.5 NEXT = NOW + 2 NEXT = NOW + 7 INPUT-OUTPUT: OUTPUT = INPUT(0.5) INPUT-OUTPUT: OUTPUT = INPUT(2) + 1 INPUT-OUTPUT: OUTPUT = INPUT(7) - 6 What similarities do you see between the Now-Next rule and the Input-Output rule? Repetitive addition can be written as multiplication resulting in the coefficient in the input-output rule Then adjust with addition/subtraction. Comparing Rules Complete with your table Comparing Rules Comparing Rules Complete with your table Comparing Rules Comparing Rules Input-Output Machines -8 -8 +12 +12 Find the change in both the x column and the y column. The rate of change is the change in y over the change in x −8 −8 12 12 Both rates simplify to 1 so the coefficient is 1, then find the “adjustment factor” by figuring out what you add or subtract to get to the y value. 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑖𝑛𝑝𝑢𝑡 − 5 Input-Output Machines -9 -3 -6 -2 Find the change in both the x column and the y column. The rate of change is the change in y over the change in x 1 1 −3 −9 −2 −6 Both rates simplify to so the coefficient is , then find the “adjustment factor” by 3 3 figuring out what you add or subtract to get to the y value. 1 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑖𝑛𝑝𝑢𝑡 3 Input-Output Machines -3 +12 +6 -24 Find the change in both the x column and the y column. The rate of change is the change in y over the change in x 12 −3 −24 6 Both rates simplify to -4 so the coefficient is-4, then find the “adjustment factor” by figuring out what you add or subtract to get to the y value. 𝑜𝑢𝑡𝑝𝑢𝑡 = −4𝑖𝑛𝑝𝑢𝑡 Input-Output Machines Work on your own to find the rest!