This is the standard
exponential, except that the "+
4" pushes
the graph up
so it is four units higher than usual.

First I compute
some points:

Then I plot those
points:

Student very
often only compute y-values
for x-values
that are close to zero. Then either they have no idea where
the graph goes on the left-hand side, and leave it hanging there:

incorrect
graph

...or else they
take the graph down to the x-axis,
as is usual for the standard exponential graph:

incorrect
graph

But this isn't
the standard exponential graph; it is the standard exponential
graph raised
by four units.
When x
is negative, y
= 2^{x}
+ 4 won't
be very close to zero; instead, it will be very close to 4,
because the values will be "a teensy-tiny little number,
plus four", which works out to be a teensy-tiny bit more
than four.

To help me with
my graph, and to indicate that I know that y
= 2^{x}
+ 4 never
goes below (or even touches, for that matter) the line y
= 4, I will
drawn a dashed line at y
= 4:

drawing
an asymptote can be helpful

This dashed-in
line, indicating where the graph goes as x
heads off to the side, is called a "horizontal asymptote",
or just an "asymptote".
It is not required that you draw it in, but it can be helpful,
and can point out to your teacher on the test that you do know
what you're doing.

Then I'll draw
the exponential:

graph
of y
= 2^{x}
+ 4

Graph y
= 5^{–x}

I need to remember that
the "negative" exponent reverses the location (along the x-axis)
in which the power on 5
is negative. When the x-values
are negative (that is, when I'm on the left-hand side of the graph),
the value of –x
will be positive, so the graph will grow quickly on the left-hand side.
On the other hand, when the x-values
are positive (that is, on the right-hand side of the graph), the value
of –x
will be negative, so the graph will stay very close to the x-axis.

In other words,
the standard values are reversed:

Then y
= 5^{–x}
graphs as:

Any graph that looks like
the above (big on the left and crawling along the x-axis
on the right) displays exponential decay, rather than exponential
growth. For a graph to display exponential decay, either the exponent
is "negative" or else the base is between 0
and 1.
You should expect to need to be able to identify the type of exponential
equation from the graph. The first two worked examples displayed exponential
growth; the last example above displays exponential decay; and the following
displays exponential growth again.

Graph
y = 2^{(x
+ 3)}

This is not the same
as "2^{x}
+ 3". In "2^{x}
+ 3", the standard
exponential is shifted up three units. In this case, the shift in "inside"
the exponential. Instead of the "+
3" shifting
the "2^{x}"
up by three, the "+
3" shifts the
"2^{x}"
over sideways by three. The only question is: shifts sideways which
way, left or right? The way I keep it straight is to consider one of
the basic points on any exponential. When the power is zero, the exponential
is 1.
For "2^{(x
+ 3)}",
when is the power zero? When x
+ 3 = 0, so x
= –3. That is, the
basic plot point (0,
1) has been shifted
to the point (–3,
1), so the graph
has been shifted three points to the left: