Question 147790
Here's a graph of your equation.
 {{{ graph( 300, 300, -5, 5, -10, 10, 2.718^x+3) }}}
Let me see if I can decipher what you need. 
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Horizontal asymptote. 
From the equation, 
{{{g(x)=e^x+3}}}
as x becomes large (negative), the exponent term quickly goes to 0,
{{{g(x)=0+3}}}
{{{g(x)=3}}} as x approaches {{{-infinity}}}
As x goes to negative infinity, the horizontal asymptote is y=3.
{{{ drawing(300,300,-5,5,-10,10,blue(line(-10,3,10,3)),graph( 300, 300, -5, 5, -10, 10, 2.718^x+3)) }}}
It's shown here as the blue line. 

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The y intercept in (x,y) form is the point that the graph crosses the y axis.
In that case 
{{{x=0}}} and 
{{{g(x)=e^0+3}}}
{{{g(x)=1+3}}}
{{{g(x)=4}}}
In (x,y) format that would be (0,4). 
{{{ drawing(300,300,-5,5,-10,10,blue(line(-10,3,10,3)),green(circle(0,4,.1)),green(circle(0,4,.15)),green(circle(0,4,.2)),graph( 300, 300, -5, 5, -10, 10, 2.718^x+3)) }}}

The y-intercept is shown as the green circle.