document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203271>Question 1203271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Appreciate if you could explain a step by step in detail how to get the answer (C).\r<BR>\n" );
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document.write( "Tried to use these Y = a(x-h)² + k and -b/2a but not sure this is the way to do so.<BR>\n" );
document.write( "Thank you!\r<BR>\n" );
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document.write( "The function f is defined for all real numbers x by f(x)=ax^2+bx+c where a, b, c are constants and a is negative. In the xy-plane, the x-coordinate of the vertex of the parabola y=f(x) is -1. If t is a number for which f(t)>f(0), which of the following must be true? \r<BR>\n" );
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document.write( "I. -2 < t < 0<BR>\n" );
document.write( "II. f(t) < f (-2)<BR>\n" );
document.write( "III.f(t) > f (1)\r<BR>\n" );
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document.write( "(A) I only<BR>\n" );
document.write( "(B) II only<BR>\n" );
document.write( "(C) I and III only<BR>\n" );
document.write( "(D) II and III only<BR>\n" );
document.write( "(E) I, II, and III  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838698 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838698\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "The function f is defined for all real numbers x by f(x)=ax^2+bx+c where a, b, c are constants and a is negative.<BR>\n" );
document.write( "In the xy-plane, the x-coordinate of the vertex of the parabola y=f(x) is -1. <BR>\n" );
document.write( "If t is a number for which f(t) > f(0), which of the following must be true?<BR>\n" );
document.write( "I. -2 < t < 0<BR>\n" );
document.write( "II.   f(t) < f (-2)<BR>\n" );
document.write( "III. f(t) > f (1)<BR>\n" );
document.write( "(A) I only<BR>\n" );
document.write( "(B) II only<BR>\n" );
document.write( "(C) I and III only<BR>\n" );
document.write( "(D) II and III only<BR>\n" );
document.write( "(E) I, II, and III<BR>\n" );
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document.write( "This problem is not to solve it by applying formal algebraic transformations/inequalities.<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is for MENTAL solution, using reasoning.<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;You should apply your geometric imagination.\r<BR>\n" );
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document.write( "              The preliminary analysis\r\n" );
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document.write( "The given parabola is opened downward and has the vertex at x= -1 (given).\r\n" );
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document.write( "It means that the parabola has the maximum at x= -1.\r\n" );
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document.write( "On the coordinate plane, draw (mentally) such a parabola and horizontal line y = f(0).\r\n" );
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document.write( "It is clear, that this line is BELOW the vertex.\r\n" );
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document.write( "The point (0,f(0)) lies on the parabola and on the horizontal line.  \r\n" );
document.write( "Hence, the symmetric point (-2, f(-2)) ALSO lies on the parabola and on the same horizontal line.\r\n" );
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document.write( "So, you conclude that at -2 < t < 0,  the points of parabola are ABOVE the points of the line.\r\n" );
document.write( "Geometrically, the opposite is also OBVIOUS: if the point of the line is under the parabola, then  -2 < t < 0.\r\n" );
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document.write( "Also, notice that f(0) = f(-2) due to symmetry relative the axis x= -1.\r\n" );
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document.write( "      Now we are ready to analyze cases (I), (II) and (III).\r\n" );
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document.write( "So, if t is a number for which f(t) > f(0), it means that the parabola is above the line,\r\n" );
document.write( "and hence, (I) is TRUE:  -2 < t < 0.  \r\n" );
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document.write( "Next, if t is a number for which f(t) > f(0), it means that the parabola is above the line,\r\n" );
document.write( "and hence, (II) is NOT NOT.\r\n" );
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document.write( "And finally, if t is a number for which f(t) > f(0), it means that f(t) > f(0) > f(1),\r\n" );
document.write( "i.e. (III) is TRUE.\r\n" );
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document.write( "    +--------------------------------------------------+\r\n" );
document.write( "    |   At this point, the major part of the problem   |\r\n" );
document.write( "    |                 is just done.                    |\r\n" );
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document.write( "Having it, the rest is just  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28OBVIOUS%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28OBVIOUS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > : \r\n" );
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document.write( "       of 5 options (A), (B), (C), (D) and (E), only (C) is true.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "This problem seems to be complicated ONLY BECAUSE<BR>\n" );
document.write( "they suppressed you by great number of words and options.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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