document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204539>Question 1204539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the Rational Function f(x)= p(x)/q(x), which of the following statements are true? Check all that apply.\r<BR>\n" );
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document.write( "a. If q(x) has a higher degree term than p(x) then the Horizontal Asymptote is y=0\r<BR>\n" );
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document.write( "b. If the highest degree term in p(x) is greater than the highest degree term in q(x) there will be more than one Horizontal Asymptote\r<BR>\n" );
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document.write( "c. If the highest degree term of p(x) is the same as the highest degree term of q(x) then the Horizontal Asymptote is x=0\r<BR>\n" );
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document.write( "d. If f(x) has a HORIZONTAL ASYMPTOTE y=a, then as the input values increase or decrease without bound, the output values will approach a\r<BR>\n" );
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document.write( "e. The Horizontal Asymptote is a guiding line for the function as the input values increase or decrease without bound </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840852 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=840852\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "(a) <font color=red>True</font><BR>\n" );
document.write( "(b) <font color=red>False</font><BR>\n" );
document.write( "(c) <font color=red>False</font><BR>\n" );
document.write( "(d) <font color=red>True</font><BR>\n" );
document.write( "(e) <font color=red>True</font>\r<BR>\n" );
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document.write( "Explanation for part (a)\r<BR>\n" );
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document.write( "Let,<BR>\n" );
document.write( "n = degree of the numerator<BR>\n" );
document.write( "d = degree of the denominator<BR>\n" );
document.write( "where degree = largest exponent of a single variable polynomial\r<BR>\n" );
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document.write( "If n < d, then this will mean the denominator polynomial races faster to infinity compared to the numerator polynomial. \r<BR>\n" );
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document.write( "As such, the denominator gets much larger and the overall fraction approaches 0. <BR>\n" );
document.write( "An example would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x%2B5%29%2F%28x%5E2-7x%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%2B5%29%2F%28x%5E2-7x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use a graphing tool such as a TI83, Desmos, or GeoGebra to plot out that equation. You should notice the curve steadily approaches the x axis. It will never actually arrive there.\r<BR>\n" );
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document.write( "Or you can use technology to create a table of values to see what happens to the value of y when x gets really large.<BR>\n" );
document.write( "For very large values of x, the y values should be really small or 0 depending on how things are rounded. As mentioned, the y value never actually will equal 0 itself.<BR>\n" );
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document.write( "Explanation for part (b)\r<BR>\n" );
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document.write( "Again we define the following<BR>\n" );
document.write( "n = degree of the numerator<BR>\n" );
document.write( "d = degree of the denominator\r<BR>\n" );
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document.write( "If n = d+1, then we'll have an oblique asymptote or slant asymptote. \r<BR>\n" );
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document.write( "An example would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%285x%5E3-2%29%2F%28x%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"y+=+%285x%5E3-2%29%2F%28x%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, in which case the slant asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+5x ALIGN=MIDDLE  ALT=\"y+=+5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For other cases when n > d+1, the curve's endpoints will approach some other curve. <BR>\n" );
document.write( "Meaning the asymptote is no longer a straight line.<BR>\n" );
document.write( "Some textbooks would consider the asymptote to be \"none\", but I think a curve can be an asymptote.\r<BR>\n" );
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document.write( "For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28x%5E4-7x%5E2%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"y+=+%28x%5E4-7x%5E2%29%2F%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has its the endpoints approach the cubic curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E3-5x%5E2%2B18x-90 ALIGN=MIDDLE  ALT=\"y+=+x%5E3-5x%5E2%2B18x-90\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "This can be determined using polynomial long division or synthetic division. <BR>\n" );
document.write( "Synthetic division can be used only when the degree of the denominator is 1.\r<BR>\n" );
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document.write( "WolframAlpha can be used to determine the asymptotes.<BR>\n" );
document.write( "For example<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=asymptotes+%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29\">https://www.wolframalpha.com/input?i=asymptotes+%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29</a><BR>\n" );
document.write( "The search input was \"<font color=red>asymptotes (2x^3 + 4x^2 - 9)/(3 - x)</font>\" without quotes.<BR>\n" );
document.write( "The \"vertical asymptote\" section is basically saying the vertical asymptote is x = 3.<BR>\n" );
document.write( "The \"parabolic asymptote\" section states that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+4x%5E2+-+9%29%2F%283+-+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is asymptotic to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2+-+10x+-+30 ALIGN=MIDDLE  ALT=\"-2x%5E2+-+10x+-+30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "WolframAlpha also provides a graph along with the asymptotes as dashed lines/curves.\r<BR>\n" );
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document.write( "Explanation for part (c)\r<BR>\n" );
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document.write( "n = degree of the numerator<BR>\n" );
document.write( "d = degree of the denominator\r<BR>\n" );
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document.write( "If n = d, then the horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Fb ALIGN=MIDDLE  ALT=\"y+=+a%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'a' is the leading coefficient of the numerator, while b is the leading coefficient of the denominator.\r<BR>\n" );
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document.write( "For example, the horizontal asymptote of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%283x%2B26%29%2F%287x%2B4%29 ALIGN=MIDDLE  ALT=\"y+=+%283x%2B26%29%2F%287x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3%2F7 ALIGN=MIDDLE  ALT=\"y+=+3%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Explanation for part (d)\r<BR>\n" );
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document.write( "This statement is true. <BR>\n" );
document.write( "It can be rephrased to \"as x goes to positive or negative infinity, the value of y gets closer to 'a'\"\r<BR>\n" );
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document.write( "Explanation for part (e)\r<BR>\n" );
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document.write( "The wording \"guiding line\" is a bit strange, but I'm assuming it refers to the idea the curve steadily approaches this line as x gets really big or really small.\r<BR>\n" );
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document.write( "For some rational functions, the curve will cross over the horizontal asymptote. Perhaps more than once. Other rational functions will not have the curve ever touch the horizontal asymptote at all.\r<BR>\n" );
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document.write( "For either case, as x goes off to infinity, the curve gets closer to this line. \r<BR>\n" );
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document.write( "Further Reading<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mathsisfun.com/algebra/asymptote.html\">https://www.mathsisfun.com/algebra/asymptote.html</a><BR>\n" );
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