document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77604>Question 77604</A>: <I><SPAN STYLE=\"word-break: break-all;\"> will someone help me on how to get the particular equation for the quadratic function and will you please show specific step.<BR>\n" );
document.write( "The sluggers were up to bat in the bottom of the ninth inning in a game against <BR>\n" );
document.write( "the ditchers. Theo Slam was up to bat. As he plugged the ball toward left <BR>\n" );
document.write( "field, it rose from 3 to 10 to 50 feet as the horizontal distance from home <BR>\n" );
document.write( "plate increased from 0 to 2 to 14 feet. Assume that the vertical distance <BR>\n" );
document.write( "varies quadratically with the horizontal distance.\r<BR>\n" );
document.write( "\n" );
document.write( "Can someone please help me!!!!!!!!!!!!!!''.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55671 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=55671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's a way to get the equation.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Presume the equation you are looking for is of the quadratic form:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+Ax%5E2+%2B+Bx+%2B+C ALIGN=MIDDLE  ALT=\"H+=+Ax%5E2+%2B+Bx+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "where H = the height of the ball and x the distance the ball has traveled horizontally<BR>\n" );
document.write( "after contact with the bat. A, B, and C are constants that we will find by using the data<BR>\n" );
document.write( "that are given in the problem.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "For example, we are told that at the point of contact with the bat, x is zero and the ball <BR>\n" );
document.write( "is 3 feet above the ground. Put these data into the equation we are developing and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+A%2A%280%29%5E2+%2B+B%2A%280%29+%2B+C ALIGN=MIDDLE  ALT=\"3+=+A%2A%280%29%5E2+%2B+B%2A%280%29+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice that the terms containing A and B are both zero and therefore the equation<BR>\n" );
document.write( "reduces to 3 = C.  Substitute this into the equation and the result is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+A%2Ax%5E2+%2B+B%2Ax+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+A%2Ax%5E2+%2B+B%2Ax+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract 3 from both sides so that it becomes a part of the left side:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+-+3+=+A%2Ax%5E2+%2B+B%2Ax ALIGN=MIDDLE  ALT=\"H+-+3+=+A%2Ax%5E2+%2B+B%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We have two more data points.  One of them is that when x = 2 ft the height of the ball<BR>\n" );
document.write( "is 10 ft.  Substitute 10 for H and 2 for x to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10++-+3+=+A%2A%282%29%5E2+%2B+B%2A%282%29 ALIGN=MIDDLE  ALT=\"10++-+3+=+A%2A%282%29%5E2+%2B+B%2A%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The left side terms can be combined and the right side simplified to give:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+=+4%2AA+%2B+2%2AB ALIGN=MIDDLE  ALT=\"7+=+4%2AA+%2B+2%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Remember this equation.  Now let's do the same thing with the final data point.  That point<BR>\n" );
document.write( "is that when x = 14 ft the height of the ball is 50 ft.  Substitute these two values into<BR>\n" );
document.write( "the equation to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50+-+3+=+A%2A%2814%29%5E2+%2B+B%2A14 ALIGN=MIDDLE  ALT=\"50+-+3+=+A%2A%2814%29%5E2+%2B+B%2A14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This simplifies to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=47+=+196%2AA+%2B+14%2AB ALIGN=MIDDLE  ALT=\"47+=+196%2AA+%2B+14%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So now we have two equations involving the two unknowns A and B.  They can be solved as<BR>\n" );
document.write( "a set.  These equations are:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+=+4%2AA+%2B+2%2AB ALIGN=MIDDLE  ALT=\"7+=+4%2AA+%2B+2%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=47+=+196%2AA+%2B+14%2AB ALIGN=MIDDLE  ALT=\"47+=+196%2AA+%2B+14%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's solve it by variable elimination.  Let's eliminate the terms containing B.  Make the<BR>\n" );
document.write( "top equation have a B term equal to the corresponding B term in the bottom equation.<BR>\n" );
document.write( "Do that by multiplying all the terms in the top equation by 7.  When you do the set of<BR>\n" );
document.write( "equations becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49+=+28%2AA+%2B+14%2AB ALIGN=MIDDLE  ALT=\"49+=+28%2AA+%2B+14%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=47+=+196%2AA+%2B+14%2AB ALIGN=MIDDLE  ALT=\"47+=+196%2AA+%2B+14%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you subtract the bottom equation from the top one, the B terms cancel each other and<BR>\n" );
document.write( "you are left with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+-168%2AA ALIGN=MIDDLE  ALT=\"2+=+-168%2AA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve for A by dividing both sides by -168 and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+2%2F%28-168%29+=+-1%2F84 ALIGN=MIDDLE  ALT=\"A+=+2%2F%28-168%29+=+-1%2F84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "You can now solve for B by returning to an equation containing A and B as the two<BR>\n" );
document.write( "variables, and substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F84 ALIGN=MIDDLE  ALT=\"-1%2F84\"   DISABLED_style= \"max-width:90%; height:auto;\" > for A.  Just to help us, let's go all the way <BR>\n" );
document.write( "back to our original equation of:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+A%2Ax%5E2+%2B+B%2Ax+%2B+C ALIGN=MIDDLE  ALT=\"H+=+A%2Ax%5E2+%2B+B%2Ax+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=-1%2F84 ALIGN=MIDDLE  ALT=\"A=-1%2F84\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+3 ALIGN=MIDDLE  ALT=\"C+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Substitute those into the equation and you<BR>\n" );
document.write( "then have:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-1%2F84%29%2Ax%5E2+%2B+B%2Ax+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+%28-1%2F84%29%2Ax%5E2+%2B+B%2Ax+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let's take one of the data points that has an x value other than zero.  It's probably<BR>\n" );
document.write( "easier to use x = 2 for a height of 10 ft. When you substitute those values you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+%28-1%2F84%29%2A2%5E2+%2B+B%2A2+%2B+3 ALIGN=MIDDLE  ALT=\"10+=+%28-1%2F84%29%2A2%5E2+%2B+B%2A2+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Multiply the first term on the right side and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+-4%2F84+%2B+B%2A2+%2B+3 ALIGN=MIDDLE  ALT=\"10+=+-4%2F84+%2B+B%2A2+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract 3 from both sides to eliminate the 3 on the right side.  This results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+=+-4%2F84+%2B+B%2A2 ALIGN=MIDDLE  ALT=\"7+=+-4%2F84+%2B+B%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "By dividing the numerator and denominator by 4, note that the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F84 ALIGN=MIDDLE  ALT=\"-4%2F84\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies <BR>\n" );
document.write( "to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F21 ALIGN=MIDDLE  ALT=\"-1%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+=+-1%2F21+%2B+B%2A2 ALIGN=MIDDLE  ALT=\"7+=+-1%2F21+%2B+B%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F21 ALIGN=MIDDLE  ALT=\"-1%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7+%2B+1%2F21+=+B%2A2 ALIGN=MIDDLE  ALT=\"7+%2B+1%2F21+=+B%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Putting the 7 over a denominator of 21 results in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=147%2F21 ALIGN=MIDDLE  ALT=\"147%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > and adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F21 ALIGN=MIDDLE  ALT=\"1%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "changes the left side to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=148%2F21+=+2%2AB ALIGN=MIDDLE  ALT=\"148%2F21+=+2%2AB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Finally, solve for B by dividing both sides of this equation by 2 to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=74%2F21+=+B ALIGN=MIDDLE  ALT=\"74%2F21+=+B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now putting this into the equation along with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-1%2F84 ALIGN=MIDDLE  ALT=\"A+=+-1%2F84\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+3 ALIGN=MIDDLE  ALT=\"C+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-1%2F84%29%2Ax%5E2+%2B+%2874%2F21%29%2Ax+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+%28-1%2F84%29%2Ax%5E2+%2B+%2874%2F21%29%2Ax+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That's the equation that will model the path of the ball.  Let's try it to see how close<BR>\n" );
document.write( "it comes to the actual height of the ball for the three given values of x. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "When x = 0, as we have seen previously, the two terms containing x go to zero and the<BR>\n" );
document.write( "equation reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+3 ALIGN=MIDDLE  ALT=\"H+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "When x = 2, the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-1%2F84%29%2A2%5E2+%2B+%2874%2F21%29%2A2+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+%28-1%2F84%29%2A2%5E2+%2B+%2874%2F21%29%2A2+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you run this on a calculator you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-0.047619047619+%2B+7.047619047619+%2B+3+=+10 ALIGN=MIDDLE  ALT=\"H+=+-0.047619047619+%2B+7.047619047619+%2B+3+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That works.  As a further check let's try the point where x is 14 ft and see what height<BR>\n" );
document.write( "the ball is at at that distance.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-1%2F84%29%2A14%5E2+%2B+%2874%2F21%29%2A14+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+%28-1%2F84%29%2A14%5E2+%2B+%2874%2F21%29%2A14+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "again, using a calculator you find that the individual terms are:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+-2.333333333333+%2B+49.333333333333+%2B+3+=+47+%2B+3+=+50 ALIGN=MIDDLE  ALT=\"H+=+-2.333333333333+%2B+49.333333333333+%2B+3+=+47+%2B+3+=+50\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That works also.  The graph of the equation:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H+=+%28-1%2F84%29%2Ax%5E2+%2B+%2874%2F21%29%2Ax+%2B+3 ALIGN=MIDDLE  ALT=\"H+=+%28-1%2F84%29%2Ax%5E2+%2B+%2874%2F21%29%2Ax+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "models the path of the ball after the batter hits it. And the graph looks like this:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C300%2C-40%2C300%2C-20%2C300%2C%28-1%2F84%29%2Ax%5E2+%2B+%2874%2F21%29%2Ax+%2B+3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "From the graph you can see that the top part of the graph is about 254 feet up in the<BR>\n" );
document.write( "air, probably higher than a batter can hit it, and the ball goes horizontally about 286 <BR>\n" );
document.write( "feet before it hits the ground ... not likely to be far enough to be a home run unless<BR>\n" );
document.write( "you're talking Little League.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you understand the problem and how to develop the equation. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );