Question 130693
In order to graph {{{f(x)=2^x-4}}}, we need to plot some points. To do that, we need to plug in some x values to get some y values



So let's find the first point:




{{{f(x)=2^x-4}}} Start with the given function



{{{f(0)=2^(0)-4}}} Plug in {{{x=0}}}



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



{{{f(0)=-3}}} Subtract 4 from 1 to get -3



So when {{{x=0}}}, we have {{{y=-3}}}



So our 1st point is (0,-3)



--------------  Let's find another point  --------------



{{{f(x)=2^x-4}}} Start with the given function



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



{{{f(1)=2-4}}} Raise 2 to the 1st power to get 2



{{{f(1)=-2}}} Subtract 4 from 2 to get -2



So when {{{x=1}}}, we have {{{y=-2}}}



So our 2nd point is (1,-2)



--------------  Let's find another point  --------------



{{{f(x)=2^x-4}}} Start with the given function



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



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



{{{f(2)=0}}} Subtract 4 from 4 to get 0



So when {{{x=2}}}, we have {{{y=0}}}



So our 3rd point is (2,0)



--------------  Let's find another point  --------------



{{{f(x)=2^x-4}}} Start with the given function



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



{{{f(3)=8-4}}} Raise 2 to the 3rd power to get 8



{{{f(3)=4}}} Subtract 4 from 8 to get 4



So when {{{x=3}}}, we have {{{y=4}}}



So our 4th point is (3,4)



--------------  Let's find another point  --------------



{{{f(x)=2^x-4}}} Start with the given function



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



{{{f(4)=16-4}}} Raise 2 to the 4th power to get 16



{{{f(4)=12}}} Subtract 4 from 16 to get 12



So when {{{x=4}}}, we have {{{y=12}}}



So our 5th point is (4,12)



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> 0</TD><TD>-3</TD></TR> 
<TR><TD> 1</TD><TD>-2</TD></TR> 
<TR><TD> 2</TD><TD>0</TD></TR> 
<TR><TD> 3</TD><TD>4</TD></TR> 
<TR><TD> 4</TD><TD>12</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-15,15,-15,15,
  grid( 1 ),
circle(0,-3,0.05),
circle(0,-3,0.08),
circle(0,-3,0.05),
circle(0,-3,0.1),
circle(0,-3,0.05),
circle(0,-3,0.12),
circle(1,-2,0.05),
circle(1,-2,0.08),
circle(1,-2,0.05),
circle(1,-2,0.1),
circle(1,-2,0.05),
circle(1,-2,0.12),
circle(2,0,0.05),
circle(2,0,0.08),
circle(2,0,0.05),
circle(2,0,0.1),
circle(2,0,0.05),
circle(2,0,0.12),
circle(3,4,0.05),
circle(3,4,0.08),
circle(3,4,0.05),
circle(3,4,0.1),
circle(3,4,0.05),
circle(3,4,0.12),
circle(4,12,0.05),
circle(4,12,0.08),
circle(4,12,0.05),
circle(4,12,0.1),
circle(4,12,0.05),
circle(4,12,0.12)
)}}}



Now connect the points to graph {{{y=2^x-4}}}  (note: the more points you plot, the easier it is to draw the graph)

{{{drawing(900,900,-15,15,-15,15,
grid( 1 ),
graph(900,900,-15,15,-15,15, 2^x-4),
circle(0,-3,0.05),
circle(0,-3,0.08),
circle(0,-3,0.05),
circle(0,-3,0.1),
circle(0,-3,0.05),
circle(0,-3,0.12),
circle(1,-2,0.05),
circle(1,-2,0.08),
circle(1,-2,0.05),
circle(1,-2,0.1),
circle(1,-2,0.05),
circle(1,-2,0.12),
circle(2,0,0.05),
circle(2,0,0.08),
circle(2,0,0.05),
circle(2,0,0.1),
circle(2,0,0.05),
circle(2,0,0.12),
circle(3,4,0.05),
circle(3,4,0.08),
circle(3,4,0.05),
circle(3,4,0.1),
circle(3,4,0.05),
circle(3,4,0.12),
circle(4,12,0.05),
circle(4,12,0.08),
circle(4,12,0.05),
circle(4,12,0.1),
circle(4,12,0.05),
circle(4,12,0.12)
)}}}



By looking at the graph, we can note the following



x-intercept: (2,0)


y-intercept: (0,-3)


Horizontal Asymptote(s): {{{y=-4}}} (ie a horizontal line at the y value of -4)


Vertical Asymptote(s): none (ie the graph continues on forever in both directions along the x-axis)