document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027807>Question 1027807</A>: <I><SPAN STYLE=\"word-break: break-all;\">   Please help me with these problem did any of this before. <BR>\n" );
document.write( " Give an example of each of the following: \r<BR>\n" );
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document.write( " a. A function with vertical asymptotes at x=-3 and x=1, and a horizontal asymptote at y=2. \r<BR>\n" );
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document.write( " b. A word problem whose solution is 12!/9!3! \r<BR>\n" );
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document.write( " c. The equation of an ellipse with vertices at (0,0) and (-8,0). \r<BR>\n" );
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document.write( " d. A logarithmic expression equivalent to 5.  <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #643091 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=643091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. To have those vertical asymptotes, a rational functions should have in the denominator the factors<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%28-3%29%29=%28x%2B3%29 ALIGN=MIDDLE  ALT=\"%28x-%28-3%29%29=%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29 ALIGN=MIDDLE  ALT=\"%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x-1%29=x%5E2%2B2x-3 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x-1%29=x%5E2%2B2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a denominator<BR>\n" );
document.write( "a rational function with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a horizontal asymptote<BR>\n" );
document.write( "must have for a numerator a quadratic polynomial with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a leading coefficient.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2 ALIGN=MIDDLE  ALT=\"2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a suitable numerator for that, and its value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not zero.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28f%28x%29=2x%5E2%2F%28x%5E2%2B2x-3%29%29 ALIGN=MIDDLE  ALT=\"highlight%28f%28x%29=2x%5E2%2F%28x%5E2%2B2x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has all the required asymptotes.\r<BR>\n" );
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document.write( "FURTHER EXPLANATION:<BR>\n" );
document.write( "A rational function may have vertical asymptotes at the values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that make<BR>\n" );
document.write( "the denominator zero.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%2B2%29%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%2B2%29%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not exist for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a vertical asymptote,<BR>\n" );
document.write( "because at valued close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > the numerator is close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2=3 ALIGN=MIDDLE  ALT=\"1%2B2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the denominator approaches zero,<BR>\n" );
document.write( "making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28f%28x%29%29 ALIGN=MIDDLE  ALT=\"abs%28f%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > increase without bounds.<BR>\n" );
document.write( "However, if a value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes both, numerator and denominator zero,<BR>\n" );
document.write( "there may not be a vertical asymptote at that value.<BR>\n" );
document.write( "For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-1%29%28x%2B2%29%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-1%29%28x%2B2%29%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not exist for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E1 ALIGN=MIDDLE  ALT=\"x%3C%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so within its domain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is just a linear function,<BR>\n" );
document.write( "and graphs as a slanted line, with a hole at the point (1,3), and no vertical asymptote.<BR>\n" );
document.write( "On the other hand, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28x-1%29%28x%2B2%29%2F%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"g%28x%29=%28x-1%29%28x%2B2%29%2F%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is {{g(x)=(x+2)/(x-1)}}} for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E1 ALIGN=MIDDLE  ALT=\"x%3C%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a vertical asymptote.<BR>\n" );
document.write( "A rational function has a horizontal asymptote <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k ALIGN=MIDDLE  ALT=\"y=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when numerator and denominator are polynomials of the same degree, and the ratio of the leading coefficients is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It is obvious that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E2%2B1%29%2Fx%5E2=2%2B1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E2%2B1%29%2Fx%5E2=2%2B1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a vertical asymptote,<BR>\n" );
document.write( "but so does <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%282x%5E5%2Bx%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1%29%2F%28x%5E5-x%5E4%2Bx%5E3-x%5E2%2Bx-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "b. How many different possibilities has an academic coach trying to choose a team of 3 students to represent the school in an academic contest? (The order the team members are chosen or listed does not matter).<BR>\n" );
document.write( "EXPLANATION:<BR>\n" );
document.write( "Choosing a set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > from a larger set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is a question of combinations, because the choosing order, position, or role of each team member is not important.<BR>\n" );
document.write( "When choosing, there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > possibilities for the first choice, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-1=11 ALIGN=MIDDLE  ALT=\"12-1=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second choice, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2=10 ALIGN=MIDDLE  ALT=\"1-2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the third choice.<BR>\n" );
document.write( "That is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A11%2A10=12%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%2F%289%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29=12%21%2F9%21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "But there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21=3%2A2%2A1 ALIGN=MIDDLE  ALT=\"3%21=3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to choose the same team, because any of the {{[3}}} members could have been chosen first, followed in each case, by any of the {{[3-1=2}}} other members, leaving just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1=1 ALIGN=MIDDLE  ALT=\"2-1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > choice.<BR>\n" );
document.write( "So there may be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%21%2F9%21 ALIGN=MIDDLE  ALT=\"12%21%2F9%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > lists of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > members, but since each set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be listed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%21 ALIGN=MIDDLE  ALT=\"3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways, there are really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%21%2F%289%21%2A3%21%29 ALIGN=MIDDLE  ALT=\"12%21%2F%289%21%2A3%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > teams.<BR>\n" );
document.write( "NOTE: An alternate word problem is<BR>\n" );
document.write( "How many different cheerleading team of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > members can be chosen from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > students showing up at the tryouts?<BR>\n" );
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document.write( "c. The equation of an ellipse with vertices at (0,0) and (-8,0) could be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2%2F16%2By%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2%2F16%2By%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "EXPLANATION:<BR>\n" );
document.write( "The points called vertices are the ends of the major axis, since the ends of the minor axis are usually called co-vertices.<BR>\n" );
document.write( "The given points are on a horizontal line (axis), so the segments between them has to be the horizontal major axis.<BR>\n" );
document.write( "Since the distance between the vertices is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0-%28-8%29=8 ALIGN=MIDDLE  ALT=\"0-%28-8%29=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the semi-major axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F2=4 ALIGN=MIDDLE  ALT=\"8%2F2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the equation of the ellipse must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2%2F16%2By%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2%2F16%2By%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > with any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the semi-minor axis) such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cb%3C4 ALIGN=MIDDLE  ALT=\"0%3Cb%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "d. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28100000%29 ALIGN=MIDDLE  ALT=\"log%28100000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a logarithmic expression equivalent to 5. So is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C32%29 ALIGN=MIDDLE  ALT=\"log%282%2C32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "EXPLANATION:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100000=10%5E5 ALIGN=MIDDLE  ALT=\"100000=10%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28100000%29=log%2810%2C100000%29=log%2810%2C10%5E5%29=5 ALIGN=MIDDLE  ALT=\"log%28100000%29=log%2810%2C100000%29=log%2810%2C10%5E5%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E5=32 ALIGN=MIDDLE  ALT=\"2%5E5=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C32%29=log%282%2C2%5E5%29=5 ALIGN=MIDDLE  ALT=\"log%282%2C32%29=log%282%2C2%5E5%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );