document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191207>Question 1191207</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is of the form f(x) = (ax + b) / (2x + c), for x ≠ -(c/2), where a, b, and c ∈ ℤ. Given that the graph of y = f(x) has asymptotes x = -5 and y = 2, and that the point P(1, -1/12) lies on the graph, what are the values of a, b, and c? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823004 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=823004\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The vertical asymptote is x = -5, which occurs because of a division by zero error.<BR>\n" );
document.write( "Plug x = -5 into the denominator to get<BR>\n" );
document.write( "2x+c = 2(-5)+c = -10+c\r<BR>\n" );
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document.write( "Set that result equal to 0 and isolate c<BR>\n" );
document.write( "-10+c = 0<BR>\n" );
document.write( "c = 0+10<BR>\n" );
document.write( "c = 10\r<BR>\n" );
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document.write( "Having 2x+10 in the denominator will lead to a denominator of zero when x = -5.\r<BR>\n" );
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document.write( "The horizontal asymptote y = 2 is the result of dividing the leading terms. This applies because the degree of the numerator and denominator are the same (both are of degree 1).<BR>\n" );
document.write( "This works because the leading terms are the largest when x gets really big (meaning that the smaller terms b and c effectively get so small we can ignore them).\r<BR>\n" );
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document.write( "Anyways, dividing the leading terms gets us this<BR>\n" );
document.write( "(ax)/(2x) = a/2<BR>\n" );
document.write( "where x is nonzero\r<BR>\n" );
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document.write( "Setting that equal to the horizontal asymptote value and solving for 'a' yields the following<BR>\n" );
document.write( "a/2 = 2<BR>\n" );
document.write( "a = 2*2<BR>\n" );
document.write( "a = 4\r<BR>\n" );
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document.write( "We found that<BR>\n" );
document.write( "a = 4<BR>\n" );
document.write( "c = 10\r<BR>\n" );
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document.write( "Let's update the function from<BR>\n" );
document.write( "f(x) = (ax+b)/(2x+c)<BR>\n" );
document.write( "to<BR>\n" );
document.write( "f(x) = (4x+b)/(2x+10)\r<BR>\n" );
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document.write( "We're told that P(1,-1/12) is on the f(x) curve. <BR>\n" );
document.write( "This means that the input x = 1 pairs with the output y = f(x) = -1/12\r<BR>\n" );
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document.write( "We'll use these two new values to find b<BR>\n" );
document.write( "f(x) = (4x+b)/(2x+10)<BR>\n" );
document.write( "-1/12 = (4(1)+b)/(2(1)+10)<BR>\n" );
document.write( "-1/12 = (4+b)/(12)<BR>\n" );
document.write( "-1 = 4+b .... multiplied both sides by 12<BR>\n" );
document.write( "4+b = -1<BR>\n" );
document.write( "b = -1-4<BR>\n" );
document.write( "b = -5\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "a = <font color=red>4</font><BR>\n" );
document.write( "b = <font color=red>-5</font><BR>\n" );
document.write( "c = <font color=red>10</font><BR>\n" );
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