document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98271>Question 98271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) <BR>\n" );
document.write( "Determine the average rate of change for the function:\r<BR>\n" );
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document.write( "f(x)= x^2 + 10x +16 where -3 is less than or equal to x is less than or equal to 3\r<BR>\n" );
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document.write( "2) Find the domain of the given function<BR>\n" );
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document.write( "3) Write the domain of the rational function in interval notation as<BR>\n" );
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document.write( "(P.S - If you haven't figured it out yet, I'm the person who has been posting all the questions about functions for the past week. Thank you all so much for your help! I really appreciate it)\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71455 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=71455\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1)\r<BR>\n" );
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document.write( "To find average rate between two points on a graph, simply draw a line through those two points and find the slope \r<BR>\n" );
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document.write( "So the x values of the points are given as <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 <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;\" >\r<BR>\n" );
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document.write( "Now we need to find the y values at the given x-values:\r<BR>\n" );
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document.write( "Plug in <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;\" > to find the value of y at <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=+%28-3%29%5E2+%2B+10%28-3%29+%2B16 ALIGN=MIDDLE  ALT=\"f%28-3%29=+%28-3%29%5E2+%2B+10%28-3%29+%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=+-5 ALIGN=MIDDLE  ALT=\"f%28-3%29=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate\r<BR>\n" );
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document.write( "Now plug in <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;\" > to find the value of y at <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=+%283%29%5E2+%2B+10%283%29+%2B16 ALIGN=MIDDLE  ALT=\"f%28-3%29=+%283%29%5E2+%2B+10%283%29+%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-3%29=+55 ALIGN=MIDDLE  ALT=\"f%28-3%29=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate\r<BR>\n" );
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document.write( "Now that we have the two points (-3,-5) and (3,55) we can find the slope through these points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now start with the slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%2855--5%29%2F%283--3%29 ALIGN=MIDDLE  ALT=\"m=%2855--5%29%2F%283--3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-3 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=3 ALIGN=MIDDLE  ALT=\"x%5B2%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=-5 ALIGN=MIDDLE  ALT=\"y%5B1%5D=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=55 ALIGN=MIDDLE  ALT=\"y%5B2%5D=55\"   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=m=%2860%29%2F%286%29 ALIGN=MIDDLE  ALT=\"m=%2860%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=10 ALIGN=MIDDLE  ALT=\"m=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So the average rate of change is 10\r<BR>\n" );
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document.write( "2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K%28v%29=%282v%5E2-9%29%2F%28v%5E2-v-30%29 ALIGN=MIDDLE  ALT=\"K%28v%29=%282v%5E2-9%29%2F%28v%5E2-v-30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given function \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2-v-30=0 ALIGN=MIDDLE  ALT=\"v%5E2-v-30=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28v-6%29%28v%2B5%29=0 ALIGN=MIDDLE  ALT=\"%28v-6%29%28v%2B5%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v-6=0 ALIGN=MIDDLE  ALT=\"v-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%2B5=0 ALIGN=MIDDLE  ALT=\"v%2B5=0\"   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=v=6 ALIGN=MIDDLE  ALT=\"v=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=-5 ALIGN=MIDDLE  ALT=\"v=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now solve for v in each case\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=6 ALIGN=MIDDLE  ALT=\"v=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=-5 ALIGN=MIDDLE  ALT=\"v=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > make the denominator equal to zero, that means we must exclude these values from the domain.\r<BR>\n" );
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document.write( "So our domain is: v is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%3C%3E6 ALIGN=MIDDLE  ALT=\"v%3C%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%3C%3E-5 ALIGN=MIDDLE  ALT=\"v%3C%3E-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which looks like this in interval notation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, -5\right)\cup\left(-5, 6\right)\cup\left(6,\infty \right)\">\r<BR>\n" );
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document.write( "------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "3)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%281%29%2F%28x-20%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%281%29%2F%28x-20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given function \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-20=0 ALIGN=MIDDLE  ALT=\"x-20=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=20 ALIGN=MIDDLE  ALT=\"x=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 20 to both sides\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=20 ALIGN=MIDDLE  ALT=\"x=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes the denominator equal to zero, that means we must exclude this value from the domain.\r<BR>\n" );
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document.write( "So our domain is: x is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%3C%3E20 ALIGN=MIDDLE  ALT=\"v%3C%3E20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which looks like this in interval notation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-\infty, 20\right)\cup\left(20,\infty \right)\"> </SPAN>\n" );
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