document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=599183>Question 599183</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ernie Thayer1 hits a baseball and it travels for 409.6 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:<BR>\n" );
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document.write( "y=ax2 +bx+c, where x is the horizontal distance that the ball has travelled, and y is the height<BR>\n" );
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document.write( "of the ball at the given distance. Background: To determine the coefficients in a polynomial equation of degree<BR>\n" );
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document.write( "y=a0xn +a1xn-1 +a2xn-2 +···+an-2x2 +an-1x+an, \r<BR>\n" );
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document.write( "one generally to know n + 1 points. With additional information about the poly- nomial, the number of points that are needed can sometimes be reduced. (This additional information is sometimes called symmetry.)<BR>\n" );
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document.write( "In addition, if we know certain kinds of points, then there may be special forms of the polynomial equation that are easier to work with. For example, if we know that a cubic has its 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:<BR>\n" );
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document.write( "y = a(x - 1)(x - 2)(x - 3). \r<BR>\n" );
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document.write( "We still need a fourth point on the cubic to determine the value of the constant a.<BR>\n" );
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document.write( "In this lab, we will focus on 2nd degree or quadratic polynomial equations: \r<BR>\n" );
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document.write( "y=ax2 +bx+c. \r<BR>\n" );
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document.write( "1. There are many quadratic equations that you could use to model the dis- tance x and the height y, but we want to find one that is realistic. Because of this real-life interaction, not every quadratic equation will be acceptable. Let’s consider some of the properties that an appropriate quadratic equation will have. Answer the following questions:<BR>\n" );
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document.write( "(a) The following questions deal with the initial height of the ball: \r<BR>\n" );
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document.write( "With respect to the real life situation, what is happening when x = 0? \r<BR>\n" );
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document.write( "What are the possible values for y when x = 0? \r<BR>\n" );
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document.write( "With respect to the graph of the quadratic equation, what is this \r<BR>\n" );
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document.write( "point called? <BR>\n" );
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document.write( "(b) The following questions deal with the maximum height of the ball: \r<BR>\n" );
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document.write( "What is the name of the point on the graph (a parabola) where the maximum height is attained?<BR>\n" );
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document.write( "Are some values of y unreasonable? \r<BR>\n" );
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document.write( "How is this information important for choosing a window for the \r<BR>\n" );
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document.write( "graph? <BR>\n" );
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document.write( "(c) The following questions deal with the ending values for the flight of the ball:<BR>\n" );
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document.write( "What would be the height of the ball when x = 409.6 feet? \r<BR>\n" );
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document.write( "This point on the graph has a name. What is it called? \r<BR>\n" );
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document.write( "With respect to the quadratic polynomial, this value of x has a name. What is it called?<BR>\n" );
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document.write( "(d) Using the information collected above, explain why the following equa- tions are poor models of the situation. Give coordinates of points on the graph that support your claim.<BR>\n" );
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document.write( "i. y = -0.002x(x - 409.6). ii. y = -0.5x2 +216x+3. \r<BR>\n" );
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document.write( "iii. y = -0.002x2 + 0.879x + 3.981. iv. y = -0.002x2 + 0.8732x - 3.981. \r<BR>\n" );
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document.write( "(e) Why is the constant a in the equation y = ax2 +bx+c negative in a reasonable model?<BR>\n" );
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document.write( "Part II. Using given information about the graph to find the quadratic equation.<BR>\n" );
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document.write( "2. There are different algebraic ways to find a reasonable quadratic model of the situation. We have some information about the path of the ball, giving some information about points of its graph (a parabola).<BR>\n" );
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document.write( "Suppose the ball was 4.56 feet above ground when Ernie Thayer hit it, and that it reached a maximum height of approximately 108.42 feet when it was approximately 202.6 feet away from where he hit the ball. The ball lands after travelling a ground distance of approximately 409.6 feet.<BR>\n" );
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document.write( "We will find equations to model the situation by using two algebraic meth- ods. (Show all your work for each part.)<BR>\n" );
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document.write( "(a) Find an equation of the form y = C(x - z1)(x - z2) where z1 and z2 are the zeros (or roots) of the quadratic polynomial (or x-intercepts of the graph) and C is a scaling constant that needs to be determined.<BR>\n" );
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document.write( "Find the other root. (Hint: Use the known root, the vertex, and a symmetry property of the graph.)<BR>\n" );
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document.write( "Find the constant C. (Round this value to three significant posi- tions. Leading zeros are not significant, but trailing zeros are signif- icant. For example, 0.0003478 would be rounded to 0.000348 and 0.0003501 would be rounded to 0.000350. Hint: To find C, you can use the initial height.)<BR>\n" );
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document.write( "Write out the equation that you found and algebraically check that it satisfies all the necessary conditions. (Note: Show your work! Expect small errors because of rounding.)<BR>\n" );
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document.write( "(b) Find an equation of the form y = A(x - h)2 + k where the vertex is at (h, k) and the constant A is a scaling factor.<BR>\n" );
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document.write( "Based on the information that you were given, what are the coordi- nates of the vertex.<BR>\n" );
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document.write( "Find the constant A. (Round this value to three significant positions.) \r<BR>\n" );
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document.write( "Write out the equation that you found and algebraically check that it satisfies all the necessary conditions. (Note: Show your work! Expect small errors because of rounding.)<BR>\n" );
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document.write( "Part III. Discussion of when algebra can be used to find a quadratic equation. \r<BR>\n" );
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document.write( "3. In part II, we used two different methods to find a quadratic equation to model the situation. Each required different information about the graph.<BR>\n" );
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document.write( "(a) Can you use algebra to find a quadratic equation if you know the coor- dinates of just one point on its graph?<BR>\n" );
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document.write( "(b) Can you use algebra to find a quadratic equation if you know the co- ordinates of just two points on its graph? If so, what information is needed?<BR>\n" );
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document.write( "3.<BR>\n" );
document.write( "(c) Rounding errors in the data might result in your finding different cor- rect equations for your answers in parts (a) and (b) above for Problem 2. What characteristic of the data causes this to happen?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #378950 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=378950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You shouldn't post giant homework assignments here. We're here for homework help, not just to do your entire assignment.\r<BR>\n" );
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document.write( "Do as much as you can first, then post specific questions on the ones you don't understand. </SPAN>\n" );
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