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Day 2 Activity Name ___________________________ Heronβs method of approximating square roots can be used as a basis for a computer program to find square roots. In programming, a process of sequential approximations such as this is called iteration. When a function undergoes iteration, the initial input is called the seed, and the sequence of outputs created by iterating (repeating) the function is called an orbit. For example, the iteration of π (π₯ ) = π₯+ 21 π₯ 2 can be used to find an approximation of β21. Start by choosing any number for the seed. Then substitute the seed for x to find the first orbit. Suppose you choose 4 as the seed. The first orbit is 4+ 21 4 2 = 4.625. 1. Use the first orbit as your new seed, and find the second orbit. 2. Use the second orbit as your new seed, and find the third orbit. 3. Repeat the process two more times. What is your final orbit? 4. Describe any patters you notice. How is the final orbit related to β21. 5. Find the first four orbits of π(π₯) = What happens to the orbits? π₯+ 16 π₯ 2 π₯+ by choosing an initial seed that is less than zero. 7 6. Find the first four orbits of β(π₯) = 2 π₯ by starting with 2 as the initial seed. Explain the relationship between the orbits and the square root of 7. Algebra1Teachers @ 2015 Page 1 Day 2 Activity Name ___________________________ Answer Key: 1. 4.58277027 2. 4.582575699 3. 4.58257555; 4.582575695; 4.582575696 4. Possible answer: the orbits get closer and closer to the same number; they begin to repeat. The final orbit is an approximation for the square root of 21. 5. Answers will vary. For example, if the seed used is -4, the approximation of the square root of 16 is -4. The orbits are all negative numbers. 6. 2.75; 2.6477272; 2.645752048; 2.645751311; the orbits approximate the square root of 7. Algebra1Teachers @ 2015 Page 2