document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111985>Question 1111985</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The temperature of a patient during an illness is given by the function&#8203; below, where T is the&#8203; temperature, in degrees&#8203; Fahrenheit, at time&#8203; t, in&#8203; days, after the onset of the illness. Graph the function using a graphing calculator and use the MAXIMUM feature to find at what time the&#8203; patient's temperature was highest. What was the highest&#8203; temperature?<BR>\n" );
document.write( "T(t)=0.7tsquared +4.2t+98.6, 0 less than or equal too t less than or equal to 6  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #727141 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=727141\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the function you provided is:\r<BR>\n" );
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document.write( "T(t) = .7 * t^2 + 4.2 * t + 98.6\r<BR>\n" );
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document.write( "it will not give you a maximum temperature.\r<BR>\n" );
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document.write( "this is because the coefficient of the t^2 term is positive.\r<BR>\n" );
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document.write( "in order for the function to give you a maximum temperature, the coefficint of the t^2 term has to be negative.\r<BR>\n" );
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document.write( "to graph this function:\r<BR>\n" );
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document.write( "let x = t and let y = T(t).\r<BR>\n" );
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document.write( "the function becomes y = .7 * x^2 + 4.2 * x + 98.6\r<BR>\n" );
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document.write( "the domain is 0 <= x <= 6.\r<BR>\n" );
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document.write( "here's the graph of the function as given.\r<BR>\n" );
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document.write( "you can see that there is no maximum.\r<BR>\n" );
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document.write( "there is only a minimum.\r<BR>\n" );
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document.write( "here is the graph of the function when the coefficient of the x^2 term is negative.\r<BR>\n" );
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document.write( "now you have a maximum.\r<BR>\n" );
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document.write( "from the graph, you can see that, when x = 0, the temperature is 98.6, and when x = 3, the temperature is 104.9, and when x = 6, the temperature is 98.6 again.\r<BR>\n" );
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document.write( "the maximum temperature is when x = 3.\r<BR>\n" );
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document.write( "the maximum temperature is 104.9 degrees.\r<BR>\n" );
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document.write( "we allowed y to be equal to T(t) and we allowed x to be equal to t in order for the graphing software to be able to handle the function.\r<BR>\n" );
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document.write( "some graphing functions can handle it the way it was originally written, but a lot can't, which is why we made the equivalency.\r<BR>\n" );
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document.write( "bottom line:\r<BR>\n" );
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document.write( "coefficient of the x^2 term, or the t^2 term, needs to be negative in order to get a maximum out of this quadratic function.\r<BR>\n" );
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document.write( "since the quadratic equation is in standard form, you could also find the maximum vaalue by using the formula:\r<BR>\n" );
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document.write( "x = -b/2a\r<BR>\n" );
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document.write( "the standard form of a quadratic equation is y = ax^2 + bx+ c, when y is equal to 0.\r<BR>\n" );
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document.write( "in this form:\r<BR>\n" );
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document.write( "a = the coefficient of the x^2 term.<BR>\n" );
document.write( "b = the coefficient of the x term.<BR>\n" );
document.write( "c = the constant term.\r<BR>\n" );
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document.write( "your equation of y = -.7 * x^2 + 4.2 * x + 98.6 gets you:\r<BR>\n" );
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document.write( "a = -.7<BR>\n" );
document.write( "b = 4.2<BR>\n" );
document.write( "c = 98.6\r<BR>\n" );
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document.write( "x = -b/2a becomes x = -4.2 / -1.4 which gets you x = 3.\r<BR>\n" );
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document.write( "when x = 3, y = -.7 * 3^2 + 4.2 * 3 + 98.6.\r<BR>\n" );
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document.write( "this results in y = 104.9\r<BR>\n" );
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document.write( "this agrees with what the graph is telling you, as it should.\r<BR>\n" );
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