document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204295>Question 1204295</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What can you say about the functions graphed below? Explain your reasoning: Is it a log or an exponential? For an exponential, is the exponent x or -x? What is the equation for the asymptote? Which direction and how many units is it shifted?\r<BR>\n" );
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document.write( "Graph A) https://app.gemoo.com/share/image-annotation/572889363786969088?codeId=Ml2kEYAjRgoXb&origin=imageurlgenerator\r<BR>\n" );
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document.write( "Graph B) https://app.gemoo.com/share/image-annotation/572889769845932032?codeId=v6gK8ZKQR0YZa&origin=imageurlgenerator </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840456 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Graph A represents a log function.\r<BR>\n" );
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document.write( "This particular curve passes through (3,3) and (12,6).\r<BR>\n" );
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document.write( "Consider the log equation template<BR>\n" );
document.write( "y = a + b*log(x)<BR>\n" );
document.write( "If we plugged in x = 1, then we get y = a.<BR>\n" );
document.write( "This is because log(1) = 0 regardless of the base.\r<BR>\n" );
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document.write( "Unfortunately x = 1 isn't defined on your log graph. <BR>\n" );
document.write( "But let's shift (3,3) and (12,6) two units to the left so that x = 1 would be defined.\r<BR>\n" );
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document.write( "(3,3) moves to (1,3)<BR>\n" );
document.write( "(12,6) moves to (10,6)\r<BR>\n" );
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document.write( "If we plugged the coordinates of (1,3) into y = a+b*log(x), then we'd find that a = 3.\r<BR>\n" );
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document.write( "So y = a+b*log(x) updates to y = 3+b*log(x)\r<BR>\n" );
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document.write( "Now try the coordinates of (10,6) to determine b.<BR>\n" );
document.write( "y = 3+b*log(x)<BR>\n" );
document.write( "6 = 3+b*log(10)<BR>\n" );
document.write( "6 = 3+b*1 ...... I'm using log base 10<BR>\n" );
document.write( "6 = 3+b<BR>\n" );
document.write( "b = 6-3<BR>\n" );
document.write( "b = 3\r<BR>\n" );
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document.write( "We have y = 3 + b*log(x) update to y = 3 + 3*log(x)\r<BR>\n" );
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document.write( "This would be the answer if (1,3) and (10,6) were points on the curve. <BR>\n" );
document.write( "However, recall we shifted things over 2 units to the left. <BR>\n" );
document.write( "Shift 2 units to the right to undo that previous shift.\r<BR>\n" );
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document.write( "This means we'll replace x with x-2 to apply this shift we need.\r<BR>\n" );
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document.write( "The function that represents graph A is <font color=red>y = 3 + 3*log(x-2)</font> where the log is base 10.\r<BR>\n" );
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document.write( "The vertical asymptote is x = 2 because this x value makes the stuff inside the log, the x-2 portion, equal to zero.<BR>\n" );
document.write( "Recall that log(x) has the domain x > 0\r<BR>\n" );
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document.write( "Therefore the domain of y = 3 + 3*log(x-2) is x > 2.\r<BR>\n" );
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document.write( "The curve looks like it touches x = 2, but it never actually gets there. Think of it like an electric fence.\r<BR>\n" );
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document.write( "Now onto graph B. It appears this curve goes through (1,8) and (3,2).\r<BR>\n" );
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document.write( "This is an exponential decay curve. <BR>\n" );
document.write( "This is due to the fact it decreases as we move to the right.<BR>\n" );
document.write( "A real world example would be a half-life function.<BR>\n" );
document.write( "The exponent is -x to indicate this decay.\r<BR>\n" );
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document.write( "The horizontal asymptote of the function y = a*b^(-x) is y = 0.\r<BR>\n" );
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document.write( "This curve shows the horizontal asymptote is y = 1.\r<BR>\n" );
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document.write( "Let's shift everything down by 1 so the horizontal asymptote y = 1 overlaps with the x axis (aka y = 0).\r<BR>\n" );
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document.write( "(1,8) moves to (1,7)<BR>\n" );
document.write( "(3,2) moves to (3,1)\r<BR>\n" );
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document.write( "Let's look at the template<BR>\n" );
document.write( "y = a*b^(-x)\r<BR>\n" );
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document.write( "To make the b term go away, we would plug in x = 0. But it's not clear what x = 0 leads to based on the graph you've provided (even after the vertical shift downward).\r<BR>\n" );
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document.write( "Let's move those points over 1 spot to the left.<BR>\n" );
document.write( "(1,7) moves to (0,7)<BR>\n" );
document.write( "(3,1) moves to (2,1)\r<BR>\n" );
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document.write( "Plug in the coordinates of (0,7) to find y = a*b^(-x) leads to a = 7.<BR>\n" );
document.write( "So we now have y = 7*b^(-x)\r<BR>\n" );
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document.write( "Plug in (2,1) and solve for b<BR>\n" );
document.write( "y = 7*b^(-x)<BR>\n" );
document.write( "1 = 7*b^(-2)<BR>\n" );
document.write( "1 = 7/(b^2)<BR>\n" );
document.write( "1*b^2 = 7<BR>\n" );
document.write( "b^2 = 7<BR>\n" );
document.write( "b = sqrt(7)<BR>\n" );
document.write( "b = 7^(1/2)<BR>\n" );
document.write( "b = 7^(0.5)\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "y = 7*b^(-x)<BR>\n" );
document.write( "y = 7*(7^(0.5))^(-x)<BR>\n" );
document.write( "y = 7*7^(-0.5x)\r<BR>\n" );
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document.write( "Next we'll shift 1 unit up and 1 unit right to undo those previous two shifts we did earlier.<BR>\n" );
document.write( "y = 7*7^(-0.5x)<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "<font color=red>y = 7*7^(-0.5(x-1))+1</font><BR>\n" );
document.write( "I'll let you simplify if needed.\r<BR>\n" );
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document.write( "There is probably a much more elegant, clever, and efficient way to determine this function. <BR>\n" );
document.write( "I'll let another tutor provide that route. Feel free to explore other methods.\r<BR>\n" );
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document.write( "I recommend using a tool like GeoGebra or Desmos to confirm we have the correct functions.\r<BR>\n" );
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document.write( "Here is a link to the interactive Desmos graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/wc4mvapjx7\">https://www.desmos.com/calculator/wc4mvapjx7</a><BR>\n" );
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