document.write( "Question 87833: I need help with this: graph the following functions separately:
\n" ); document.write( " f(x)=5^x
\n" ); document.write( " f(x)=-2+e^x \n" ); document.write( "

Algebra.Com's Answer #63739 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
#1 \r
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=5%5Ex\"$ Start with the given expression\r
\n" ); document.write( "\n" ); document.write( "In order to plot an exponential function, we need to plot some points\r
\n" ); document.write( "\n" ); document.write( "We can start at any x value, so lets start at x=-5\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-5%29=5%5E-5\"$ Plug in $\"x=-5\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-5%29=0.00032\"$ Raise 5 to the negative fifth power to get 0.00032\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=-5\"$ , $\"y=0.00032\"$ . So our first point is (-5,0.00032)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-4%29=5%5E-4\"$ Plug in $\"x=-4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=0.0016\"$ Raise 5 to the negative fourth power to get 0.0016\r
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\n" ); document.write( "\n" ); document.write( "So when $\"x=-4\"$ , $\"y=0.0016\"$ . So our second point is (-4,0.0016)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-3%29=5%5E-3\"$ Plug in $\"x=-3\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-3%29=0.008\"$ Raise 5 to the negative third power to get 0.008\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-3\"$ , $\"y=0.008\"$ . So our third point is (-3,0.008)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-2%29=5%5E-2\"$ Plug in $\"x=-2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-2%29=0.04\"$ Raise 5 to the negative second power to get 0.04\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-2\"$ , $\"y=0.04\"$ . So our fourth point is (-2,0.04)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-1%29=5%5E-1\"$ Plug in $\"x=-1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=0.2\"$ Raise 5 to the negative first power to get 0.2\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-1\"$ , $\"y=0.2\"$ . So our fifth point is (-1,0.2)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%280%29=5%5E0\"$ Plug in $\"x=0\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%280%29=1\"$ Raise 5 to the zeroth power to get 1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=0\"$ , $\"y=1\"$ . So our sixth point is (0,1)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%281%29=5%5E1\"$ Plug in $\"x=1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%281%29=5\"$ Raise 5 to the first power to get 5\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=1\"$ , $\"y=5\"$ . So our seventh point is (1,5)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%282%29=5%5E2\"$ Plug in $\"x=2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%282%29=25\"$ Raise 5 to the second power to get 25\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=2\"$ , $\"y=25\"$ . So our eighth point is (2,25)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%283%29=5%5E3\"$ Plug in $\"x=3\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%283%29=125\"$ Raise 5 to the third power to get 125\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=3\"$ , $\"y=125\"$ . So our ninth point is (3,125)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%284%29=5%5E4\"$ Plug in $\"x=4\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%284%29=625\"$ Raise 5 to the fourth power to get 625\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=4\"$ , $\"y=625\"$ . So our tenth point is (4,625)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%285%29=5%5E5\"$ Plug in $\"x=5\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%285%29=3125\"$ Raise 5 to the fifth power to get 3125\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=5\"$ , $\"y=3125\"$ . So our eleventh point is (5,3125)\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================================\r
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\n" ); document.write( "\n" ); document.write( "Now lets make a table of the values we have calculated\r
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document.write( " x y
 -5 0.00032
 -4 0.0016
 -3 0.008
 -2 0.04
 -1 0.2
 0 1
 1 5
 2 25
 3 125
 4 625
 5 3125\r\n" );
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Now plot the points\r
\n" ); document.write( "\n" ); document.write( "

\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Connect the points to form the graph $\"y=5%5Ex\"$ \r
\n" ); document.write( "\n" ); document.write( "

Graph of $\"y=5%5Ex\"$ through the points solved for earlier\r
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\n" ); document.write( "\n" ); document.write( "-----------------------------------------------------------------------------\r
\n" ); document.write( "\n" ); document.write( "#2\r
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=-2%2Be%5Ex\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=e%5Ex-2\"$ Rearrange\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In order to plot an exponential function, we need to plot some points\r
\n" ); document.write( "\n" ); document.write( "We can start at any x value, so lets start at x=-5\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-5%29=e%5E-5-2\"$ Plug in $\"x=-5\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-5%29=0.007-2\"$ Raise e (note: e=2.71828182846) to the negative fifth power to get 0.007\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-5%29=-1.993\"$ Combine like terms 0.007 and -2 to get -1.993\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-5\"$ , $\"y=-1.993\"$ . So our first point is (-5,-1.993)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-4%29=e%5E-4-2\"$ Plug in $\"x=-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=0.018-2\"$ Raise e to the negative fourth power to get 0.018\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=-1.982\"$ Combine like terms 0.018 and -2 to get -1.982\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-4\"$ , $\"y=-1.982\"$ . So our second point is (-4,-1.982)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-3%29=e%5E-3-2\"$ Plug in $\"x=-3\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-3%29=0.05-2\"$ Raise e to the negative third power to get 0.05\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-3%29=-1.95\"$ Combine like terms 0.05 and -2 to get -1.95\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-3\"$ , $\"y=-1.95\"$ . So our third point is (-3,-1.95)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-2%29=e%5E-2-2\"$ Plug in $\"x=-2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-2%29=0.135-2\"$ Raise e to the negative second power to get 0.135\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-2%29=-1.865\"$ Combine like terms 0.135 and -2 to get -1.865\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-2\"$ , $\"y=-1.865\"$ . So our fourth point is (-2,-1.865)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%28-1%29=e%5E-1-2\"$ Plug in $\"x=-1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=0.368-2\"$ Raise e to the negative first power to get 0.368\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=-1.632\"$ Combine like terms 0.368 and -2 to get -1.632\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=-1\"$ , $\"y=-1.632\"$ . So our fifth point is (-1,-1.632)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%280%29=e%5E0-2\"$ Plug in $\"x=0\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%280%29=1-2\"$ Raise e to the zeroth power to get 1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%280%29=-1\"$ Combine like terms 1 and -2 to get -1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=0\"$ , $\"y=-1\"$ . So our sixth point is (0,-1)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%281%29=e%5E1-2\"$ Plug in $\"x=1\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%281%29=2.718-2\"$ Raise e to the first power to get 2.718\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%281%29=0.718\"$ Combine like terms 2.718 and -2 to get 0.718\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=1\"$ , $\"y=0.718\"$ . So our seventh point is (1,0.718)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%282%29=e%5E2-2\"$ Plug in $\"x=2\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%282%29=7.389-2\"$ Raise e to the second power to get 7.389\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%282%29=5.389\"$ Combine like terms 7.389 and -2 to get 5.389\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=2\"$ , $\"y=5.389\"$ . So our eighth point is (2,5.389)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%283%29=e%5E3-2\"$ Plug in $\"x=3\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%283%29=20.086-2\"$ Raise e to the third power to get 20.086\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%283%29=18.086\"$ Combine like terms 20.086 and -2 to get 18.086\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=3\"$ , $\"y=18.086\"$ . So our ninth point is (3,18.086)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%284%29=e%5E4-2\"$ Plug in $\"x=4\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%284%29=54.598-2\"$ Raise e to the fourth power to get 54.598\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%284%29=52.598\"$ Combine like terms 54.598 and -2 to get 52.598\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=4\"$ , $\"y=52.598\"$ . So our tenth point is (4,52.598)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "----Now lets find another point----
\n" ); document.write( " $\"f%285%29=e%5E5-2\"$ Plug in $\"x=5\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%285%29=148.413-2\"$ Raise e to the fifth power to get 148.413\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"f%285%29=146.413\"$ Combine like terms 148.413 and -2 to get 146.413\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So when $\"x=5\"$ , $\"y=146.413\"$ . So our eleventh point is (5,146.413)\r
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\n" ); document.write( "\n" ); document.write( "===============================================================================\r
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\n" ); document.write( "\n" ); document.write( "Now lets make a table of the values we have calculated\r
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document.write( " x y
 -5 -1.993
 -4 -1.982
 -3 -1.95
 -2 -1.865
 -1 -1.632
 0 -1
 1 0.718
 2 5.389
 3 18.086
 4 52.598
 5 146.413\r\n" );
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Now plot the points\r
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\r
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\n" ); document.write( "\n" ); document.write( "Connect the points to form the graph $\"y=e%5Ex-2\"$ \r
\n" ); document.write( "\n" ); document.write( "

Graph of $\"y=e%5Ex-2\"$ through the points solved for earlier\r
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\n" ); document.write( "\n" ); document.write( "If this seems like a lot of work, you can reduce the number of points to work with \n" ); document.write( "

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