document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200454>Question 200454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "Find an exponential function passing through (1,3) and (3,4.32)\r<BR>\n" );
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document.write( "thanks for the study help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153003 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=153003\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Exponential functions are in the form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k%2Aa%5Ex ALIGN=MIDDLE  ALT=\"y+=+k%2Aa%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k and a are constants. So our task is to find the values of k and a.<br><BR>\n" );
document.write( "We can do this by using the points we know will fit the equation: (1, 3) and (3, 4.32). Let's start with the first point. Substituting the x and y coordinates from (1, 3) for the x and y in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k%2Aa%5Ex ALIGN=MIDDLE  ALT=\"y+=+k%2Aa%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29+=+k%2Aa%5E%281%29 ALIGN=MIDDLE  ALT=\"%283%29+=+k%2Aa%5E%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+k%2Aa ALIGN=MIDDLE  ALT=\"3+=+k%2Aa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides by a we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Fa+=+k ALIGN=MIDDLE  ALT=\"3%2Fa+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now let's do it with the second point. We will use the x and y from (3, 4.32) <b>and</b> the \"value\" we just got for k: 3/a. Substituting these 3 values into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k%2Aa%5Ex ALIGN=MIDDLE  ALT=\"y+=+k%2Aa%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284.32%29+=+%283%2Fa%29%2Aa%5E3 ALIGN=MIDDLE  ALT=\"%284.32%29+=+%283%2Fa%29%2Aa%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.32+=+3a%5E2 ALIGN=MIDDLE  ALT=\"4.32+=+3a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides by 3 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.44+=+a%5E2 ALIGN=MIDDLE  ALT=\"1.44+=+a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finding the square root of both sides we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.2+=+a ALIGN=MIDDLE  ALT=\"1.2+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.2+=+a ALIGN=MIDDLE  ALT=\"-1.2+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we usually want a > 0 then we <i>may</i> reject the a = -1.2.<BR>\n" );
document.write( "As a fraction 1.2 is 6/5. If a = 6/5 then k = 3/(6/5) = 5/2. So our exponential equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%285%2F2%29%286%2F5%29%5Ex ALIGN=MIDDLE  ALT=\"y+=+%285%2F2%29%286%2F5%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > or ,as decimals, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2.5%281.2%29%5Ex ALIGN=MIDDLE  ALT=\"y+=+2.5%281.2%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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