document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116017>Question 116017</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone help me with this graphing?<BR>\n" );
document.write( "  Describe the transformations on the following graph of f(x)=log(x). State the replacement of the verticle asymptote and x-intercept after the transformation. For example, \"left 1\" or stretched vertically by a factor of 2\" or descriptions.On my graph i see a line from -2 on the y axis and 10 on the x-axis<BR>\n" );
document.write( "a)g(x)=log(x+2)<BR>\n" );
document.write( "  Description of transformation:<BR>\n" );
document.write( "  Vertical asymptote:<BR>\n" );
document.write( "  x-intercept in (x,y) form:\r<BR>\n" );
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document.write( "b) g(x)= -log(x)<BR>\n" );
document.write( "   Description of transformation:<BR>\n" );
document.write( "   Vertical asymptote:<BR>\n" );
document.write( "   x-intercept in (x,y)form:<BR>\n" );
document.write( "Need help asap<BR>\n" );
document.write( "Thankyou </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84399 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=84399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "From the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can see that only positive x values will work. In other words, the domain of f is  (0,<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \infty\">). Now let's find the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=log%2810%2C%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"g%28x%29=log%2810%2C%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2%3E0 ALIGN=MIDDLE  ALT=\"x%2B2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the inner expression greater than zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0-2 ALIGN=MIDDLE  ALT=\"x%3E0-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 2 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E-2 ALIGN=MIDDLE  ALT=\"x%3E-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "So that means x must be greater than -2\r<BR>\n" );
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document.write( "So here is the domain in interval notation: (-2,<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \infty\">)\r<BR>\n" );
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document.write( "Notice how the endpoint of the domain has been shifted to the left two units. So what this did was simply shift every x value 2 units to the left\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the transformation  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29%29 ALIGN=MIDDLE  ALT=\"g%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > shifts the entire graph of   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  two units to the left\r<BR>\n" );
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document.write( "Notice if we graph <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+log%2810%2C%28x%29%29%2Clog%2810%2C%28x%2B2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=log%2810%2C%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"g%28x%29=log%2810%2C%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "and we can visually verify the transformation\r<BR>\n" );
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document.write( "<b>Vertical Asymptote</b>:\r<BR>\n" );
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document.write( "From the graph, we can see that the vertical asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <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;\" >. Since we've shifted the graph 2 units to the left, we've also shifted the vertical asymptote 2 units to the left. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the vertical asymptote for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can visually verify this if we look at the graph above\r<BR>\n" );
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document.write( "<b>x-intercept in (x, y) form</b>:\r<BR>\n" );
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document.write( "From the graph, we can see that the x-intercept of <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 (1,0). Since we've shifted everything two units to the left, the x-intercept shifts  two units to the left also.\r<BR>\n" );
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document.write( "So subtract 2 from 1 to get \r<BR>\n" );
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document.write( "(1-2,0)---->(-1,0)\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the x-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (-1,0)\r<BR>\n" );
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document.write( "Once again, we can visually verify this if we look at the graph above\r<BR>\n" );
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document.write( "b) \r<BR>\n" );
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document.write( "<b>Description of transformation</b>:\r<BR>\n" );
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document.write( "Remember, <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 the same as y. So this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=log%28x%29 ALIGN=MIDDLE  ALT=\"y=log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now if we negate both sides to get  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y=-log%28x%29 ALIGN=MIDDLE  ALT=\"-y=-log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is simply making each y coordinate becomes it's opposite. So something like (0,2) becomes (0,-2) and (3,-2) becomes (3,2), etc\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So what's happening is that the graph is being reflected over the x-axis\r<BR>\n" );
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document.write( "Notice if we graph <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+log%2810%2C%28x%29%29%2C+-log%2810%2C%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=-log%2810%2C%28x%29%29 ALIGN=MIDDLE  ALT=\"g%28x%29=-log%2810%2C%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "and we can visually verify the transformation\r<BR>\n" );
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document.write( "<b>Vertical Asymptote</b>:\r<BR>\n" );
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document.write( "From the graph, we can see that the vertical asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the transformation reflected the graph across the x-axis, the vertical asymptote of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as the vertical asymptote of <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;\" > \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the vertical asymptote of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can verify this by looking at the graph above\r<BR>\n" );
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document.write( "<b>x-intercept in (x, y) form</b>:\r<BR>\n" );
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document.write( "From the graph, we can see that the x-intercept of <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 (1,0). Since we've reflected everything with respect to the x-axis, the point on the x-axis is not affected. In other words the x-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as the x-intercept of <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;\" > \r<BR>\n" );
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document.write( "So the x-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is (1,0)\r<BR>\n" );
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document.write( "Once again, we can visually verify this if we look at the graph above\r<BR>\n" );
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