document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=159065>Question 159065</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the number of new homes built, H, in a city over a period of time, t, is graphed on a rectangular coordinate system where time is on the horizontal axis. Suppose that the number of homes built can be modeled by an exponential function, H= p * at , where p is the number of new homes built in the first year recorded. If you were a homebuilder looking for work, would you prefer that the value of a to be between 0 and 1 or larger than 1? Explain your reasoning.<BR>\n" );
document.write( "Thank you for helping me out on this =]  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #117298 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=117298\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the answer is if it's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax ALIGN=MIDDLE  ALT=\"a%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >, any a value is ok but the bigger the x value the better.  if it's <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Ex ALIGN=MIDDLE  ALT=\"a%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >, any value > 1 is good because there will be growth, and any value < 1 is not good because there will be no growth.<BR>\n" );
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document.write( "to show you, i'll graph 5 equations with different (a) values used in each equation starting with a value of <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;\" > and ending with a value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3E+1 ALIGN=MIDDLE  ALT=\"a+%3E+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "so that the graphs will show the proper x-axis values and the proper y-axis values, i had to cross the origin with both x and y values.<BR>\n" );
document.write( "for your purposes, you just need to look at the values where x >= 0.<BR>\n" );
document.write( "disregard the values where x < 0 since they are there only to force the x-axis and y-axis to show up properly and have no other meaning for the discussion.<BR>\n" );
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document.write( "in both of these graphs, i am showing the total number of houses projected each year.  any growth will be the number of houses in that year minus the number of houses in the previous year.<BR>\n" );
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document.write( "for linear growth, the general equation will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28a%2Ax%29%2A%28h%29%2B%28h%29 ALIGN=MIDDLE  ALT=\"y+=+%28a%2Ax%29%2A%28h%29%2B%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <BR>\n" );
document.write( "y is the total number of homes in each year (thousands).<BR>\n" );
document.write( "x is the number of years.<BR>\n" );
document.write( "a is the multiplication factor for each year.<BR>\n" );
document.write( "h is the starting number of houses in thousands.<BR>\n" );
document.write( "h will be set to 100 (thousand).<BR>\n" );
document.write( "the additional h is required to make the number of houses equal to 100 (thousand) when x = 0.<BR>\n" );
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document.write( "for the first equation, a is set to .25<BR>\n" );
document.write( "for the second equation, a is set to .5<BR>\n" );
document.write( "for the third equation, a is set to .75<BR>\n" );
document.write( "for the 4th equation, a is set to 1.00<BR>\n" );
document.write( "for the 5th equation, a is set to 1.25<BR>\n" );
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document.write( "scan down below the graph for more information.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281000%2C800%2C-1%2C20%2C-200%2C2000%2C.25%2Ax%2A100%2B100%2C.5%2Ax%2A100%2B100%2C.75%2Ax%2A100%2B100%2C1.0%2Ax%2A100%2B100%2C1.25%2Ax%2A100%2B100%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "the growth of homes in each year is the number of houses in that year minus the number of houses in the previous year.<BR>\n" );
document.write( "as a gets larger, the number of houses required in a year goes higher.<BR>\n" );
document.write( "for example, <BR>\n" );
document.write( "in year 10 with a = .25, the number of houses is 350.<BR>\n" );
document.write( "in year 10 with a = .75, the number of houses is 850.<BR>\n" );
document.write( "in year 10 with a = 1.25, the number of houses is 1350. <BR>\n" );
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document.write( "for exponential growth, the general equation will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28a%5Ex%29%2A%28h%29%29 ALIGN=MIDDLE  ALT=\"y+=+%28a%5Ex%29%2A%28h%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <BR>\n" );
document.write( "y is the total number of homes in each year.<BR>\n" );
document.write( "x is the number of years.<BR>\n" );
document.write( "a is the base factor that will be raised to the x power for each year.<BR>\n" );
document.write( "h is the starting number of houses in thousands.<BR>\n" );
document.write( "h will be set to 100 (thousand).<BR>\n" );
document.write( "the additional h is not required here because a^0 = 1 for all values of a.<BR>\n" );
document.write( "-------------------------------------------<BR>\n" );
document.write( "for the first equation, a is set to .25<BR>\n" );
document.write( "for the second equation, a is set to .5<BR>\n" );
document.write( "for the third equation, a is set to .75<BR>\n" );
document.write( "for the 4th equation, a is set to 1.00<BR>\n" );
document.write( "for the 5th equation, a is set to 1.25<BR>\n" );
document.write( "---------------------------------------------<BR>\n" );
document.write( "scan down below the graph for more information.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%281000%2C800%2C-1%2C10%2C-200%2C500%2C.25%5Ex%2A100%2C.5%5Ex%2A100%2C.75%5Ex%2A100%2C1.0%5Ex%2A100%2C1.25%5Ex%2A100%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "with an exponential factor, interesting things happen.<BR>\n" );
document.write( "when a is > 1, the growth takes off.<BR>\n" );
document.write( "when a = 1 there is no growth.<BR>\n" );
document.write( "when a < 1 there is actually a decrease in the number of houses required each year.<BR>\n" );
document.write( "if we take year 6 as an example, we get the following values:<BR>\n" );
document.write( "when a = .25, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=.25%5E6%2A100 ALIGN=MIDDLE  ALT=\"y=.25%5E6%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals .0244....., a number very close to 0.  <BR>\n" );
document.write( "when a = .50, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+.5%5E6%2A100 ALIGN=MIDDLE  ALT=\"y+=+.5%5E6%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals 1.5625, a number still very close to 0.<BR>\n" );
document.write( "when a = .75 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+.75%5E6%2A100 ALIGN=MIDDLE  ALT=\"y+=+.75%5E6%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals 17.7978..., a number that is further away from 0 but still less then the starting value of 100.<BR>\n" );
document.write( "when a = 1.00 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+1.0%5E6%2A100 ALIGN=MIDDLE  ALT=\"y+=+1.0%5E6%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals 100.  the number of houses required each year remains the same.<BR>\n" );
document.write( "when a = 1.25 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+1.25%5E6%2A100 ALIGN=MIDDLE  ALT=\"y+=+1.25%5E6%2A100\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals 381.469....<BR>\n" );
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document.write( "as x gets larger, any value of a less than 1 will tend towards 0.<BR>\n" );
document.write( "take x = 100<BR>\n" );
document.write( ".25^100 = 6.223.....^-61 which is a very small number and will keep getting smaller as x gets larger.<BR>\n" );
document.write( ".75^100 = 3.207.....^-13 which is still avery small number and will keep getting smaller as x gets larger.<BR>\n" );
document.write( "1.0^100 = 1 and will always be 1 as x gets larger because 1 to any power = 1.<BR>\n" );
document.write( "1.25^100 = 4.909...^9 which is a very big number and will keep getting bigger as x gets large\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
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