document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=282339>Question 282339</A>: <I><SPAN STYLE=\"word-break: break-all;\"> To describe a  trajectory of a missile or a football, we use quadratic functions like y(t)=a*t^2+b*t+c, where y is the height after time t sec.  From your experience of seeing the path of football or other projectiles, what are the limitation on the parameters a, b and c?  If the function is describing the football path  t seconds after it left the player's hand, what does c mean?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #205014 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=205014\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-25%2C25%2C-75%2C15%2C%28-.01%2Ax%5E2-0%2Ax%2B6%29%2C%28-1%2Ax%5E2-0%2Ax+%2B+6%29%2C%28-100%2Ax%5E2-0%2Ax%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In the graphs above:\r<BR>\n" );
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document.write( "The coefficient of the x term is 0.<BR>\n" );
document.write( "The coefficient of the constant term is 6.<BR>\n" );
document.write( "The coefficients of the x^2 terms are -.001, -1, -100.\r<BR>\n" );
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document.write( "The more negative the coefficient of the x^2 term is, the steeper the trajectory of the parabola and the narrower its width.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-25%2C25%2C-75%2C75%2C%28-1%2Ax%5E2-5%2Ax%2B6%29%2C%28-1%2Ax%5E2-10%2Ax%2B6%29%2C%28-1%2Ax%5E2-15%2Ax%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In the graphs above:\r<BR>\n" );
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document.write( "The coefficient of the x^2 term is 1.<BR>\n" );
document.write( "The coefficient of the constant term is 6.<BR>\n" );
document.write( "The coefficients of the x terms are -5, -10, -15.\r<BR>\n" );
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document.write( "The more negative the coefficient of the x term gets, the more to the left the maximum value of the equation gets, and the higher the maximum value of the equation gets.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-25%2C25%2C-75%2C75%2C%28-1%2Ax%5E2%2B5%2Ax%2B6%29%2C%28-1%2Ax%5E2%2B10%2Ax%2B6%29%2C%28-1%2Ax%5E2%2B15%2Ax%2B6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In the graphs above:\r<BR>\n" );
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document.write( "The coefficient of the x^2 term is 1.<BR>\n" );
document.write( "The coefficient of the constant term is 6.<BR>\n" );
document.write( "The coefficients of the x terms are 5, 10, 15.\r<BR>\n" );
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document.write( "The more positive the coefficient of the x term gets, the more to the right the maximum value of the equation gets, and the higher the maximum value of the equation gets.\r<BR>\n" );
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document.write( "Impact on using the quadratic equation to describing the trajectory of the football after x second (x is used instead of t in order to show the relationship on the graph).\r<BR>\n" );
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document.write( "The standard form of the quadratic equation is ax^2 + bx + c.\r<BR>\n" );
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document.write( "a is the coefficient of the x^2 term and has to be negative.\r<BR>\n" );
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document.write( "This allows the trajectory of the football to follow an arc to reach a maximum point and then fall back to earth.\r<BR>\n" );
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document.write( "b is the coefficient of the x^2 term and can be negative or positive.\r<BR>\n" );
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document.write( "If b is more negative, the high point of the arc of the football shifts to the left.\r<BR>\n" );
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document.write( "If b is positive, the high point of the arc of the football shifts to the right.\r<BR>\n" );
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document.write( "Even though the high point of the arc of the football travel can be to the left of the y-axis, some of the arc of the football will be to the right of the y-axis which would be the period of time of interest and is therefore valid.\r<BR>\n" );
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document.write( "This happens when c is greater than 0 as it has to be.\r<BR>\n" );
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document.write( "c is the constant term and has to be positive and should be approximately the height of the football player who is throwing the football.\r<BR>\n" );
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document.write( "c represents the height of the football when t = 0 which is the point at which the ball is thrown.\r<BR>\n" );
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document.write( "You start off with c = the height of the football player who is throwing the football.\r<BR>\n" );
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document.write( "a has to be negative.\r<BR>\n" );
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document.write( "The more negative it is, the steeper the trajectory of the football, but this has no impact on the height of the trajectory.   As long as b equals 0, the height will be the value of c.\r<BR>\n" );
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document.write( "b can be negative or positive.\r<BR>\n" );
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document.write( "The value of b affects the height of the trajectory and whether or not the maximum point of the arc of the football will be shifted to the left or the right.\r<BR>\n" );
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