Science:Math Exam Resources/Courses/MATH307/December 2010/Question 04 (a)
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Question 04 (a) 

In this question we are once again given 4 points (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) and (x_{4}, y_{4}) in the plane. This time we want to find a quadratic function that comes closest to going through the points by doing a least squares fit. (a) The least squares equation you need to solve to find the coefficients a, b and c has the form A^{T}Aa = A^{T}b. Write down expressions for A, a and b. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Write the conditions that q(x) goes through all four points in matrix notation. The values x_{i} and y_{i} are known, and the values a, b and c are unknown. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. If the quadratic polynomial could fit all four points, then these conditions would be satisfied: this is equivalent to:
However, we only have 3 unknowns and 4 equations, hence we ask for the best solution (in the least square sense), which is found by solving . Therefore,
