dagpc_251120_completing_matrix_reduction__finding_various_fns_given_form_and_points___comments_74 скачать в хорошем качестве

dagpc_251120_completing_matrix_reduction__finding_various_fns_given_form_and_points___comments_74

We reduce a matrix starting with its upper-triangular form, obtaining a matrix from which we can simply read off the solution of the original system. We then begin an overall survey of the course, starting with the most important toolkit functions and their general transformed forms. We review how to use simultaneous equations dictated by the form of the transformed function to find a linear function given two graph points, a quadratic function given three points, an exponential function given its asymptote and two points, and a polynomial function given its zeros and their multiplicities as well as a single point distinct from these. More specifically: We find the linear function y = mx + b using two estimated points to obtain the specific linear function whose graph passes through those points. Using the form y = a x^2 + b x + c and three points we write three linear equations for the parameters a, b and c. The solution allows us to write down the corresponding quadratic function. We note that the simultaneous equations could be solved using matrix methods. Using the form y = a b^x with two known or estimated points we obtain the equation of an exponential function asymptotic to the x axis (using two simultaneous equations, solved by substitution),. We find a polynomial function given three zeros and their multiplicities as well as the y intercept, obtaining the factored form of the polynomial with an unknown constant coefficient. We easily evaluate the coefficient using the coordinates of the y-intercept; (note that the process works even if the known point is not the y-intercept) and write down the equation defining the function. We then construct the graph of the polynomial, seeing it as a power function.with wiggles close to the y axis when we zoom out, and graphing it in the vicinity of its zeros (using the power function to determine its long-run behavior).