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Unformatted text preview: Math 148 - Winter 2009 Lab Assignment 2 (v0.2029) Quadratic Models Due: February 2, 2009 1 Background Introduction: Ernie Thayer 1 hits a baseball and it travels for 412 feet before it lands. When he hits the ball, the ball is between about 2 feet and about 5 feet high. If we ignore air resistance, then physics tells us that the flight of the baseball can be modelled using a quadratic equation of the form: y = ax 2 + bx + c , where x is the horizontal distance that the ball has travelled, and y is the height of the ball at the given distance. Background: To determine the coefficients in a polynomial equation of degree n : y = a x n + a 1 x n- 1 + a 2 x n- 2 + + a n- 2 x 2 + a n- 1 x + a n , one generally to know n + 1 points. With additional information about the polynomial, the number of points that are needed can sometimes be reduced. (This additional information is sometimes called symmetry .) In addition, if we know certain kinds of points, then there may be special forms of the poly- nomial equation that are easier to work with. For example, if we know that a cubic has as zeros at 1, 2 and 3, then we can use the three x-intercepts form of the cubic to simplify our work in particular, we know that the cubic has the following form: y = a ( x- 1 )( x- 2 )( x- 3 ) . We still need a fourth point on the cubic to determine the value of the constant a . In this lab, we will focus on 2 nd degree or quadratic polynomial equations: y = ax 2 + bx + c . To determine the coefficients, one generally needs to know three points. But the graph, a parabola with a vertical axis, does have symmetry, and the number of points that are needed to determine its equation can sometimes be reduced to just two. 1 Ernest Lawrence Thayer was the author of the poem Casey at the Bat . 1 Examples: For our examples, we will work with a vertical parabola that passes through the following points: x- 3- 2- 1 1 2 3 y- 12- 16- 12 20 48 The parabola is plotted in the following figure:- 4- 3- 2- 1 1 2 3- 10 10 20 30 40 Of course these seven points are far more than is needed to determine the parabola the equa- tion is said to be overdetermined . (When an equation is overdetermined, rounding and other errors in the data can lead to inequivalent equations that model the same data.) Example 1. General form y = ax 2 + bx + c . Any three of these points can be used to find the values of a , b and c , but some choices will be more sensible than others. One not very sensible choice would be (- 2 ,- 12 ) , (- 1 ,- 16 ) , and ( 2 , 20 ) . Putting these three points into the polynomial form leads to three equations:- 12 = 4 a- 2...
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