document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243399>Question 243399</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PLEASE HELP ME SOLVE THIS: \r<BR>\n" );
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document.write( "Determine whether the quadratic function f(x)=3x^+18x-23 has a maximun value or minimum value. Find the range of the function f(x)expressed using interval notation.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178344 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=178344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> equation is:\r<BR>\n" );
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document.write( "3x^2 + 18x - 23\r<BR>\n" );
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document.write( "It doesn't look it will factor easily, so use the quadratic formula.\r<BR>\n" );
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document.write( "quadratic equation general form is ax^2 + bx + c\r<BR>\n" );
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document.write( "In this equation:\r<BR>\n" );
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document.write( "a = 3<BR>\n" );
document.write( "b = 18<BR>\n" );
document.write( "c = -23\r<BR>\n" );
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document.write( "quadratic formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-b%29+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28%28-b%29+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b^2 - 4ac is the discriminant.\r<BR>\n" );
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document.write( "formula becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-18%29+%2B-+sqrt%2818%5E2-4%2A3%2A%28-23%29%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28%28-18%29+%2B-+sqrt%2818%5E2-4%2A3%2A%28-23%29%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "{{((-18) +- sqrt(600))/(6)}}} which becomes:\r<BR>\n" );
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document.write( "x = 1.082482905 or x = -7.082482905\r<BR>\n" );
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document.write( "The roots are real and those are the points where the equation crosses the x-axis.\r<BR>\n" );
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document.write( "The minimum / maximum point of this equation is given by the formula:\r<BR>\n" );
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document.write( "x = -b/2a and y = f(-b/2a)\r<BR>\n" );
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document.write( "the value for x = -(18) / (2*3) = -18/6 = -3\r<BR>\n" );
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document.write( "the value for y = f(-b/2a) = f(-3) = 3*(-3)^2 + 18*(-3) - 23 = 27 - 54 - 23 = 27 - 77 = -50\r<BR>\n" );
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document.write( "The max / min point is equal to (-2,-50)\r<BR>\n" );
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document.write( "The range of this function is dependent on the domain.\r<BR>\n" );
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document.write( "The domain of this function looks like it is all real values of x because x can be positive and negative and is all real (no negative square roots and no divisions by 0 to restrict the domain).\r<BR>\n" );
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document.write( "The range is all real values of y but the minimum / maximum value of y is determined by the minimimum / maximum point.\r<BR>\n" );
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document.write( "In this case the min / max point is a minimum because the x^2 term is positive.\r<BR>\n" );
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document.write( "This means the range of the function will be all real values of y greater than or equal to the minimum point which is -50.\r<BR>\n" );
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document.write( "In interval notation this would be x >= -50\r<BR>\n" );
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document.write( "It can also be written as -50 <= x < infinity which in symbol form looks like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-50+%3C=+x+%3C+infinity ALIGN=MIDDLE  ALT=\"-50+%3C=+x+%3C+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "a graph of your equation looks like this:\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-10%2C10%2C-100%2C100%2C3x%5E2%2B18x-23%2C-50%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-100%2C100%2C3x%5E2%2B18x-23%2C-50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I placed a horizontal line at y = -50 to show you that the minimum point was there.\r<BR>\n" );
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document.write( "Since I'm not sure which form of interval notation you are looking for, the only other form interval notation I know of would be:\r<BR>\n" );
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document.write( "y = [ -50, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "This means the value of y is greater than or equal to 50 and smaller than infinity.\r<BR>\n" );
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