document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107482>Question 107482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The figure shows the graphs of f(x) = x3 and g(x) = ax3.<BR>\n" );
document.write( "What can you conclude about the value of a?\r<BR>\n" );
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document.write( "a. a < –1<BR>\n" );
document.write( "b. –1 < a < 0<BR>\n" );
document.write( "c. 0 < a < 1<BR>\n" );
document.write( "d. 1 < a </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #78268 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=78268\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+ax3 ALIGN=MIDDLE  ALT=\"g%28x%29+=+ax3\"   DISABLED_style= \"max-width:90%; height:auto;\" >      <BR>\n" );
document.write( "are you sure this is what you need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x3 ALIGN=MIDDLE  ALT=\"x3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or you need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I guess that cubic function is what you need:<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+ax%5E3 ALIGN=MIDDLE  ALT=\"g%28x%29+=+ax%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see what changes we get if we chaging <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "here is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C1%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, watch the graph if\r<BR>\n" );
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document.write( "a.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C+-1 ALIGN=MIDDLE  ALT=\"a+%3C+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we can take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-3 ALIGN=MIDDLE  ALT=\"a+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and see what is different if we compare it to previous\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C-3%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, watch the graph if\r<BR>\n" );
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document.write( "b. –1 < a < 0......let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-0.7 ALIGN=MIDDLE  ALT=\"a+=+-0.7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, do you see any difference?\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C-.7%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, watch the graph if\r<BR>\n" );
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document.write( "c. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+ ALIGN=MIDDLE  ALT=\"a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >< <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >......let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%2B0.7 ALIGN=MIDDLE  ALT=\"a+=+%2B0.7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, do you see any difference?\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C.7%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, watch the graph if\r<BR>\n" );
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document.write( "d. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+5 ALIGN=MIDDLE  ALT=\"a+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C5%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If a getting smaller and smaller then<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0 ALIGN=MIDDLE  ALT=\"+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, graph is getting closer and closer to from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-axis ALIGN=MIDDLE  ALT=\"y-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > and lies in II and IV quadrant.\r<BR>\n" );
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document.write( "Let see the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-0.000001 ALIGN=MIDDLE  ALT=\"a=-0.000001\"   DISABLED_style= \"max-width:90%; height:auto;\" >which is very close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >…….it is almost straight line which tend to lie on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > but still goes through II and IV quadrant.<BR>\n" );
document.write( "	<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C-.000001%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let see the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-+44 ALIGN=MIDDLE  ALT=\"a=-+44\"   DISABLED_style= \"max-width:90%; height:auto;\" >which is very distant from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >…….it is almost straight line which tend to lie on y-axis\r<BR>\n" );
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document.write( "	<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C-44%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If a getting greater and greater then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, graph is getting closer and closer to from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-axis ALIGN=MIDDLE  ALT=\"y-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > and lies in II and IV quadrant.\r<BR>\n" );
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document.write( "Let see the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.000001 ALIGN=MIDDLE  ALT=\"a=0.000001\"   DISABLED_style= \"max-width:90%; height:auto;\" >which is very close to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >…….it is almost straight line which tend to lie on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" > but still goes through I and III quadrant.\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281500%2C1500%2C-100%2C100%2C-100%2C100%2C.000001%2A%28x%5E3%29+%2B+0%2A%28x%5E2%29+%2B+0%2A%28x%29+%2B+0%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let see the graph for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=44 ALIGN=MIDDLE  ALT=\"a=44\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is very distant from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >…….it is almost straight line which tend to lie on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-axis ALIGN=MIDDLE  ALT=\"y-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >but still goes through I and III quadrant.\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Cubic_equations.solver>Graph of a cubic equation</A></TD></TR>\n" );
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