course Mth 163 assignment #003003.
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17:21:36 `questionNumber 30000 `q001. Note that this assignment has 6 questions The function y = a x^2 + b x + c takes the value y = 0 when x = [ -b + `sqrt(b^2 - 4 a c ] / (2 a) or when x = [ -b - `sqrt(b^2 - 4 a c ] / (2 a). For the function y = - 0.45833 x^2 + 5.33333 x - 6.875, which you obtained as a quadratic model of the points (1, -2), (3, 5) and (7, 8) in the preceding assignment, find the values of x for which y = 0. Compare to the estimates you made from the graph through (1,-2), (3, 5) and (7, 8) in Assignment 1.
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RESPONSE --> For the given function y = - 0.45833 x^2 + 5.33333 x - 6.875, we know that a = -0.45833, b = 5.33333, and c = -6.875. So, from this information the quadratic formula shows up that for the function y = 0 when x = [-5.33333 + `sqrt(5.33333^2 - 4 * (-0.45833 ) * (-6.875)) ] / ( 2 * (-0.45833)) = 1.47638 and also when x = [-5.33333 - `sqrt(5.33333^2 - 4 * (-0.45833 ) * (-6.875)) ] / ( 2 * (-0.45833)) = 10.16006 confidence assessment: 2
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17:24:13 `questionNumber 30000 `q002. Extend the smooth curve in your sketch to include both points at which y = 0. Estimate the x value at which y takes its maximum value.
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RESPONSE --> My graph shows how the parabola passes through the x-axis at 1.5 and where it takes is appoximately at 10. The parabola peaks at about x = 6 and y = 8.5 confidence assessment: 2
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17:26:42 `questionNumber 30000 `q003. For the function of the preceding two questions, y will take its maximum value when x is halfway between the two values at which y = 0. Recall that these two values are approximately x = 1.48 and x = 10.16. At what x value will the function take its maximum value? What will be this value? What are the coordinates of the highest point on the graph?
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RESPONSE --> The function will take its maximum value at the vertex which is points (5.82, 8.64), where x = 5.82. confidence assessment: 2
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17:31:12 `questionNumber 30000 `q004. The function y = a x^2 + b x + c has a graph which is a parabola. This parabola will have either a highest point or a lowest point, depending upon whether it opens upward or downward. In either case this highest or lowest point is called the vertex of the parabola. The vertex of a parabola will occur when x = -b / (2a). At what x value, accurate to five significant figures, will the function y = - 0.458333 x^2 + 5.33333 x - 6.875 take its maximum value? Accurate to five significant figures, what is the corresponding y value?
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RESPONSE --> First to solve this problem you use the formula x = - b / ( 2 a ) to find the value of x when the function is maximized. The value of x is 5.81818, you take this value and substitute it into y = - 0.458333 x^2 + 5.33333 x - 6.875 to obtain the y value which comes out to be y = 8.64024. Therefore, the vertex of the parabola lies on the points (5.81818, 8.64024). confidence assessment: 3
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17:41:45 `questionNumber 30000 `q005. As we just saw the vertex of the parabola defined by the function y = - 0.45833 x^2 + 5.33333 x - 6.875 lies at (5.8182, 8.6402). What is the value of x at a point on the parabola which lies 1 unit to the right of the vertex, and what is the value of x at a point on the parabola which lies one unit to the left of the vertex? What is the value of y corresponding to each of these x values? By how much does each of these y values differ from the y value at the vertex, and how could you have determined this number by the equation of the function?
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RESPONSE --> The x values to the left and right of the vertex on the parabola are x = 6.8182 and x = 4.8182. The y values to both the left and the right sides of the vertex come out to be the same thing which is y = 8.1818. We find these y values by substituting the x values into the function y = - 0.458333 x^2 + 5.33333 x - 6.875. Both if these values are different from the y value at the vertex because anytime you it to the left or right of the vertex of the parsbola, the y value always differs from the y value at the vertex. confidence assessment: 2
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17:53:29 `questionNumber 30000 `q006. In the preceding problem we saw an instance of the following rule: The function y = a x^2 + b x + c has a graph which is a parabola. This parabola has a vertex. If we move 1 unit in the x direction from the vertex, moving either 1 unit to the right or to the left, then move vertically a units, we end up at another point on the graph of the parabola. In assignment 2 we obtained the solution a = -1, b = 10, c = 100 for a system of three simultaneous linear equations. If these linear equations had been obtained from 3 points on a graph, we would then have the quadratic model y = -1 x^2 + 10 x + 100 for those points. What would be the coordinates of the vertex of this parabola? What would be the coordinates of the points on the parabola which lie 1 unit to the right and one unit to the left of the vertex? Sketch a graph with these three points, and sketch a parabola through these points. Will this parabola ever touch the x axis?
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RESPONSE --> The coordinates for the vertex of this parabola are ( 5, 125 ). The points to the right and the left of the parabola are ( 4, 124 ) and ( 6, 124 ). The parabola will everntually touch the x- axis on both the left and right sides of the vertex. confidence assessment: 2
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