Question 181968
To graph ANY function, simply follow this basic routine:


1) Plug in any x value to find it's corresponding function value (or y value). This gives you an ordered pair (x,y)


2) Plot the points that you calculated from step 1


3) Draw a smooth connected through ALL of the points that you plotted in step 2




I'll do the first two to get you started. The other two follow the same basic outline.


# 1




In order to graph {{{y=5^x}}}, we need to plot a few points.



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



{{{f(-2)=5^(-2)}}} Plug in {{{x=-2}}} (note: you can start at any x-value).



{{{f(-2)=0.04}}} Raise 5 to the -2nd power to get {{{0.04}}}.



So when {{{x=-2}}}, {{{y=0.04}}}. So we have the point (-2,0.04).



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



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



{{{f(-1)=5^(-1)}}} Plug in {{{x=-1}}}.



{{{f(-1)=0.2}}} Raise 5 to the -1st power to get {{{0.2}}}.



So when {{{x=-1}}}, {{{y=0.2}}}. So we have the point (-1,0.2).



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



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



{{{f(0)=5^(0)}}} Plug in {{{x=0}}}.



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



So when {{{x=0}}}, {{{y=1}}}. So we have the point (0,1).



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



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



{{{f(1)=5^(1)}}} Plug in {{{x=1}}}.



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



So when {{{x=1}}}, {{{y=5}}}. So we have the point (1,5).



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



Now let's make a table of the values we just found.



<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<TR><TD>x</TD><TD>y</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>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(900, 900, -7, 7, -7, 7,
grid(1),
graph(900, 900, -7, 7, -7, 7, 0),
circle(-2,0.04,0.08),circle(-2,0.04,0.10),circle(-2,0.04,0.05),
circle(-1,0.2,0.08),circle(-1,0.2,0.10),circle(-1,0.2,0.05),
circle(0,1,0.08),circle(0,1,0.10),circle(0,1,0.05),
circle(1,5,0.08),circle(1,5,0.10),circle(1,5,0.05)

)}}}


<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=5^x}}}:



{{{ drawing(900, 900, -7, 7, -7, 7,
grid(1),
graph(900, 900, -7, 7, -7, 7, 5^x),
circle(-2,0.04,0.08),circle(-2,0.04,0.10),circle(-2,0.04,0.05),
circle(-1,0.2,0.08),circle(-1,0.2,0.10),circle(-1,0.2,0.05),
circle(0,1,0.08),circle(0,1,0.10),circle(0,1,0.05),
circle(1,5,0.08),circle(1,5,0.10),circle(1,5,0.05)

)}}} Graph of {{{y=5^x}}}




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


# 2




In order to graph {{{y=4^(x+2)}}}, we need to plot a few points.



{{{f(x)=4^(x+2)}}} Start with the given equation.



{{{f(-4)=4^(-4+2)}}} Plug in {{{x=-4}}} (note: you can start at any x-value).



{{{f(-4)=4^(-2)}}} Add.



{{{f(-4)=0.063}}} Raise 4 to the -2nd power to get {{{0.063}}}.



So when {{{x=-4}}}, {{{y=0.063}}}. So we have the point (-4,0.063).



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



{{{f(x)=4^(x+2)}}} Start with the given equation.



{{{f(-3)=4^(-3+2)}}} Plug in {{{x=-3}}}.



{{{f(-3)=4^(-1)}}} Add.



{{{f(-3)=0.25}}} Raise 4 to the -1st power to get {{{0.25}}}.



So when {{{x=-3}}}, {{{y=0.25}}}. So we have the point (-3,0.25).



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



{{{f(x)=4^(x+2)}}} Start with the given equation.



{{{f(-2)=4^(-2+2)}}} Plug in {{{x=-2}}}.



{{{f(-2)=4^(0)}}} Add.



{{{f(-2)=1}}} Raise 4 to the 0th power to get {{{1}}}.



So when {{{x=-2}}}, {{{y=1}}}. So we have the point (-2,1).



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



{{{f(x)=4^(x+2)}}} Start with the given equation.



{{{f(-1)=4^(-1+2)}}} Plug in {{{x=-1}}}.



{{{f(-1)=4^(1)}}} Add.



{{{f(-1)=4}}} Raise 4 to the 1st power to get {{{4}}}.



So when {{{x=-1}}}, {{{y=4}}}. So we have the point (-1,4).



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



Now let's make a table of the values we just found.



<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<TR><TD>x</TD><TD>y</TD></TR><tr><td>-4</td><td>0.063</td></tr>
<tr><td>-3</td><td>0.25</td></tr>
<tr><td>-2</td><td>1</td></tr>
<tr><td>-1</td><td>4</td></tr>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(900, 900, -7, 7, -7, 7,
grid(1),
graph(900, 900, -7, 7, -7, 7, 0),
circle(-4,0.063,0.08),circle(-4,0.063,0.10),circle(-4,0.063,0.05),
circle(-3,0.25,0.08),circle(-3,0.25,0.10),circle(-3,0.25,0.05),
circle(-2,1,0.08),circle(-2,1,0.10),circle(-2,1,0.05),
circle(-1,4,0.08),circle(-1,4,0.10),circle(-1,4,0.05)

)}}}


<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=4^(x+2)}}}:



{{{ drawing(900, 900, -7, 7, -7, 7,
grid(1),
graph(900, 900, -7, 7, -7, 7, 4^(x+2)),
circle(-4,0.063,0.08),circle(-4,0.063,0.10),circle(-4,0.063,0.05),
circle(-3,0.25,0.08),circle(-3,0.25,0.10),circle(-3,0.25,0.05),
circle(-2,1,0.08),circle(-2,1,0.10),circle(-2,1,0.05),
circle(-1,4,0.08),circle(-1,4,0.10),circle(-1,4,0.05)

)}}} Graph of {{{y=4^(x+2)}}}