Question 87833
#1 

{{{f(x)=5^x}}} Start with the given expression

In order to plot an exponential function, we need to plot some points

We can start at any x value, so lets start at x=-5


{{{f(-5)=5^-5}}}  Plug in {{{x=-5}}}


{{{f(-5)=0.00032}}}  Raise 5  to the negative fifth power to get 0.00032


So when {{{x=-5}}}, {{{y=0.00032}}}. So our  first point is (-5,0.00032)



----Now lets find another point----
{{{f(-4)=5^-4}}}  Plug in {{{x=-4}}}


{{{f(-4)=0.0016}}}  Raise 5  to the negative fourth power to get 0.0016


So when {{{x=-4}}}, {{{y=0.0016}}}. So our  second point is (-4,0.0016)



----Now lets find another point----
{{{f(-3)=5^-3}}}  Plug in {{{x=-3}}}


{{{f(-3)=0.008}}}  Raise 5  to the negative third power to get 0.008


So when {{{x=-3}}}, {{{y=0.008}}}. So our  third point is (-3,0.008)



----Now lets find another point----
{{{f(-2)=5^-2}}}  Plug in {{{x=-2}}}


{{{f(-2)=0.04}}}  Raise 5  to the negative second power to get 0.04


So when {{{x=-2}}}, {{{y=0.04}}}. So our  fourth point is (-2,0.04)



----Now lets find another point----
{{{f(-1)=5^-1}}}  Plug in {{{x=-1}}}


{{{f(-1)=0.2}}}  Raise 5  to the negative first power to get 0.2


So when {{{x=-1}}}, {{{y=0.2}}}. So our  fifth point is (-1,0.2)



----Now lets find another point----
{{{f(0)=5^0}}}  Plug in {{{x=0}}}


{{{f(0)=1}}}  Raise 5  to the zeroth power to get 1


So when {{{x=0}}}, {{{y=1}}}. So our  sixth point is (0,1)



----Now lets find another point----
{{{f(1)=5^1}}}  Plug in {{{x=1}}}


{{{f(1)=5}}}  Raise 5  to the first power to get 5


So when {{{x=1}}}, {{{y=5}}}. So our  seventh point is (1,5)



----Now lets find another point----
{{{f(2)=5^2}}}  Plug in {{{x=2}}}


{{{f(2)=25}}}  Raise 5  to the second power to get 25


So when {{{x=2}}}, {{{y=25}}}. So our  eighth point is (2,25)



----Now lets find another point----
{{{f(3)=5^3}}}  Plug in {{{x=3}}}


{{{f(3)=125}}}  Raise 5  to the third power to get 125


So when {{{x=3}}}, {{{y=125}}}. So our  ninth point is (3,125)



----Now lets find another point----
{{{f(4)=5^4}}}  Plug in {{{x=4}}}


{{{f(4)=625}}}  Raise 5  to the fourth power to get 625


So when {{{x=4}}}, {{{y=625}}}. So our  tenth point is (4,625)



----Now lets find another point----
{{{f(5)=5^5}}}  Plug in {{{x=5}}}


{{{f(5)=3125}}}  Raise 5  to the fifth power to get 3125


So when {{{x=5}}}, {{{y=3125}}}. So our  eleventh point is (5,3125)




===============================================================================


Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> -5</TD><TD>0.00032</TD></TR> 
<TR><TD> -4</TD><TD>0.0016</TD></TR> 
<TR><TD> -3</TD><TD>0.008</TD></TR> 
<TR><TD> -2</TD><TD>0.04</TD></TR> 
<TR><TD> -1</TD><TD>0.2</TD></TR> 
<TR><TD> 0</TD><TD>1</TD></TR> 
<TR><TD> 1</TD><TD>5</TD></TR> 
<TR><TD> 2</TD><TD>25</TD></TR> 
<TR><TD> 3</TD><TD>125</TD></TR> 
<TR><TD> 4</TD><TD>625</TD></TR> 
<TR><TD> 5</TD><TD>3125</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing( 900, 900, -10, 10, -10, 10,
  grid( 1 ),circle(-5,0.00032,0.05),
circle(-5,0.00032,0.08),
circle(-4,0.0016,0.05),
circle(-4,0.0016,0.08),
circle(-3,0.008,0.05),
circle(-3,0.008,0.08),
circle(-2,0.04,0.05),
circle(-2,0.04,0.08),
circle(-1,0.2,0.05),
circle(-1,0.2,0.08),
circle(0,1,0.05),
circle(0,1,0.08),
circle(1,5,0.05),
circle(1,5,0.08),
circle(2,25,0.05),
circle(2,25,0.08),
circle(3,125,0.05),
circle(3,125,0.08),
circle(4,625,0.05),
circle(4,625,0.08),
circle(5,3125,0.05),
circle(5,3125,0.08))}}}



Connect the points to form the graph {{{y=5^x}}}

{{{drawing( 900, 900, -10, 10, -10, 10,
grid( 1 ),
graph( 900, 900, -10, 10, -10, 10, 5^x),
circle(-5,0.00032,0.05),
circle(-5,0.00032,0.08),
circle(-4,0.0016,0.05),
circle(-4,0.0016,0.08),
circle(-3,0.008,0.05),
circle(-3,0.008,0.08),
circle(-2,0.04,0.05),
circle(-2,0.04,0.08),
circle(-1,0.2,0.05),
circle(-1,0.2,0.08),
circle(0,1,0.05),
circle(0,1,0.08),
circle(1,5,0.05),
circle(1,5,0.08),
circle(2,25,0.05),
circle(2,25,0.08),
circle(3,125,0.05),
circle(3,125,0.08),
circle(4,625,0.05),
circle(4,625,0.08),
circle(5,3125,0.05),
circle(5,3125,0.08))}}} Graph of {{{y=5^x}}} through the points solved for earlier


-----------------------------------------------------------------------------

#2

{{{f(x)=-2+e^x}}}


{{{f(x)=e^x-2}}} Rearrange



In order to plot an exponential function, we need to plot some points

We can start at any x value, so lets start at x=-5


{{{f(-5)=e^-5-2}}}  Plug in {{{x=-5}}}


{{{f(-5)=0.007-2}}}  Raise e (note: e=2.71828182846)  to the negative fifth power to get 0.007


{{{f(-5)=-1.993}}}  Combine like terms 0.007 and -2 to get -1.993


So when {{{x=-5}}}, {{{y=-1.993}}}. So our  first point is (-5,-1.993)



----Now lets find another point----
{{{f(-4)=e^-4-2}}}  Plug in {{{x=-4}}}


{{{f(-4)=0.018-2}}}  Raise e  to the negative fourth power to get 0.018


{{{f(-4)=-1.982}}}  Combine like terms 0.018 and -2 to get -1.982


So when {{{x=-4}}}, {{{y=-1.982}}}. So our  second point is (-4,-1.982)



----Now lets find another point----
{{{f(-3)=e^-3-2}}}  Plug in {{{x=-3}}}


{{{f(-3)=0.05-2}}}  Raise e  to the negative third power to get 0.05


{{{f(-3)=-1.95}}}  Combine like terms 0.05 and -2 to get -1.95


So when {{{x=-3}}}, {{{y=-1.95}}}. So our  third point is (-3,-1.95)



----Now lets find another point----
{{{f(-2)=e^-2-2}}}  Plug in {{{x=-2}}}


{{{f(-2)=0.135-2}}}  Raise e  to the negative second power to get 0.135


{{{f(-2)=-1.865}}}  Combine like terms 0.135 and -2 to get -1.865


So when {{{x=-2}}}, {{{y=-1.865}}}. So our  fourth point is (-2,-1.865)



----Now lets find another point----
{{{f(-1)=e^-1-2}}}  Plug in {{{x=-1}}}


{{{f(-1)=0.368-2}}}  Raise e  to the negative first power to get 0.368


{{{f(-1)=-1.632}}}  Combine like terms 0.368 and -2 to get -1.632


So when {{{x=-1}}}, {{{y=-1.632}}}. So our  fifth point is (-1,-1.632)



----Now lets find another point----
{{{f(0)=e^0-2}}}  Plug in {{{x=0}}}


{{{f(0)=1-2}}}  Raise e  to the zeroth power to get 1


{{{f(0)=-1}}}  Combine like terms 1 and -2 to get -1


So when {{{x=0}}}, {{{y=-1}}}. So our  sixth point is (0,-1)



----Now lets find another point----
{{{f(1)=e^1-2}}}  Plug in {{{x=1}}}


{{{f(1)=2.718-2}}}  Raise e  to the first power to get 2.718


{{{f(1)=0.718}}}  Combine like terms 2.718 and -2 to get 0.718


So when {{{x=1}}}, {{{y=0.718}}}. So our  seventh point is (1,0.718)



----Now lets find another point----
{{{f(2)=e^2-2}}}  Plug in {{{x=2}}}


{{{f(2)=7.389-2}}}  Raise e  to the second power to get 7.389


{{{f(2)=5.389}}}  Combine like terms 7.389 and -2 to get 5.389


So when {{{x=2}}}, {{{y=5.389}}}. So our  eighth point is (2,5.389)



----Now lets find another point----
{{{f(3)=e^3-2}}}  Plug in {{{x=3}}}


{{{f(3)=20.086-2}}}  Raise e  to the third power to get 20.086


{{{f(3)=18.086}}}  Combine like terms 20.086 and -2 to get 18.086


So when {{{x=3}}}, {{{y=18.086}}}. So our  ninth point is (3,18.086)



----Now lets find another point----
{{{f(4)=e^4-2}}}  Plug in {{{x=4}}}


{{{f(4)=54.598-2}}}  Raise e  to the fourth power to get 54.598


{{{f(4)=52.598}}}  Combine like terms 54.598 and -2 to get 52.598


So when {{{x=4}}}, {{{y=52.598}}}. So our  tenth point is (4,52.598)



----Now lets find another point----
{{{f(5)=e^5-2}}}  Plug in {{{x=5}}}


{{{f(5)=148.413-2}}}  Raise e  to the fifth power to get 148.413


{{{f(5)=146.413}}}  Combine like terms 148.413 and -2 to get 146.413


So when {{{x=5}}}, {{{y=146.413}}}. So our  eleventh point is (5,146.413)




===============================================================================


Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> -5</TD><TD>-1.993</TD></TR> 
<TR><TD> -4</TD><TD>-1.982</TD></TR> 
<TR><TD> -3</TD><TD>-1.95</TD></TR> 
<TR><TD> -2</TD><TD>-1.865</TD></TR> 
<TR><TD> -1</TD><TD>-1.632</TD></TR> 
<TR><TD> 0</TD><TD>-1</TD></TR> 
<TR><TD> 1</TD><TD>0.718</TD></TR> 
<TR><TD> 2</TD><TD>5.389</TD></TR> 
<TR><TD> 3</TD><TD>18.086</TD></TR> 
<TR><TD> 4</TD><TD>52.598</TD></TR> 
<TR><TD> 5</TD><TD>146.413</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing( 900, 900, -10, 10, -10, 10,
  grid( 1 ),circle(-5,-1.993,0.05),
circle(-5,-1.993,0.08),
circle(-4,-1.982,0.05),
circle(-4,-1.982,0.08),
circle(-3,-1.95,0.05),
circle(-3,-1.95,0.08),
circle(-2,-1.865,0.05),
circle(-2,-1.865,0.08),
circle(-1,-1.632,0.05),
circle(-1,-1.632,0.08),
circle(0,-1,0.05),
circle(0,-1,0.08),
circle(1,0.718,0.05),
circle(1,0.718,0.08),
circle(2,5.389,0.05),
circle(2,5.389,0.08),
circle(3,18.086,0.05),
circle(3,18.086,0.08),
circle(4,52.598,0.05),
circle(4,52.598,0.08),
circle(5,146.413,0.05),
circle(5,146.413,0.08))}}}



Connect the points to form the graph {{{y=e^x-2}}}

{{{drawing( 900, 900, -10, 10, -10, 10,
grid( 1 ),
graph( 900, 900, -10, 10, -10, 10, 1*exp(x)+-2),
circle(-5,-1.993,0.05),
circle(-5,-1.993,0.08),
circle(-4,-1.982,0.05),
circle(-4,-1.982,0.08),
circle(-3,-1.95,0.05),
circle(-3,-1.95,0.08),
circle(-2,-1.865,0.05),
circle(-2,-1.865,0.08),
circle(-1,-1.632,0.05),
circle(-1,-1.632,0.08),
circle(0,-1,0.05),
circle(0,-1,0.08),
circle(1,0.718,0.05),
circle(1,0.718,0.08),
circle(2,5.389,0.05),
circle(2,5.389,0.08),
circle(3,18.086,0.05),
circle(3,18.086,0.08),
circle(4,52.598,0.05),
circle(4,52.598,0.08),
circle(5,146.413,0.05),
circle(5,146.413,0.08))}}} Graph of {{{y=e^x-2}}} through the points solved for earlier


If this seems like a lot of work, you can reduce the number of points to work with