Review
We first start with the properties of the graph of the basic
exponential function
of base a,
f (x) = a^{x} , a > 0 and a not
equal to 1.
The domain of function f is the set of all real numbers. The range of f is the interval (0 , +infinity).
The graph of f has a horizontal asymptote given by y = 0. Function f has a y intercept at (0 , 1). f is an increasing function if a is greater than 1 and a decreasing function if
a is smaller than 1 .
You may want to review all the above properties of the exponential function interactively .
Example 1: f is a function given by
f (x) = 2^{(x  2)}
 Find the domain and range of f.
 Find the horizontal asymptote of the graph.
 Find the x and y intercepts of the graph.
of f if there are any.
 Sketch the graph of f.
Answer to Example 1

The domain of f is
the set of all real numbers. To find the range of f,we start with
Multiply both sides by 2^{2}
which is positive.
Use exponential properties
This last statement suggests that f(x) > 0. The range of f is
(0, +inf).

As x decreases without bound,
f(x) = 2^{(x} ^{ 2)} approaches 0. The graph of f has a horizontal asymptote at y = 0.

To find the x intercept we need to solve the equation
This equation does not have a solution, see range above, f(x) > 0. The graph of f does not have an x intercept. The y intercept is given by
(0 , f(0)) = (0,2^{(0  2)}) = (0 , 1/4).
So far we have the domain, range, y intercept and the horizontal asymptote. We need extra points.
(4 , f(4)) = (4, 2^{(4  2)}) = (4 , 2^{2}) = (4 , 4)
(1 , f(2)) = (1, 2^{(1  2)}) = (1 , 2^{3}) = (1 , 1/8)

Let us now use all the above information
to graph f.
Matched Problem to Example1: f is a function given by
f (x) = 2^{(x + 2)}
 Find the domain and range of f.
 Find the horizontal asymptote of the graph of f.
 Find the x and y intercepts of the graph of f if there are any.
 Sketch the graph of f.
Example 2: f is a function given by
f (x) = 3^{(x + 1)}  2
 Find the domain and range of f.
 Find the horizontal asymptote of the graph of f.
 Find the x and y intercepts of the graph of f if there are any.
 Sketch the graph of f.
Answer to Example 2

The domain of f is the set of all real numbers. To find the range of f,
we start with
Multiply both sides by 3 which is positive.
Use exponential properties
Subtract 2 to both sides
This last statement suggests that f(x) > 2. The range of f is
(2, +inf).

As x decreases without bound, f(x) = 3^{(x} ^{+ 1)} 2 approaches 2. The graph of f has a horizontal asymptote y = 2.

To find the x intercept we need to solve the equation f(x) = 0
Add 2 to both sides of the equation
Rewrite the above equation in Logarithmic form
Solve for x
The y intercept is given by
(0 , f(0)) = (0,3^{(0 + 1)}  2) = (0 , 1).

So far we have the domain, range, x and y intercepts and the horizontal asymptote. We need extra points.
(2 , f(2)) = (2, 3^{(2 + 1)}  2) = (4 , 1/32) = (4 , 1.67)
(4 , f(4)) = (4, 3^{(4 + 1)}  2) = (4 , 2^{3}) = (4 , 1.99)
Let us now use all the above information to graph f.
Matched Problem to Example2: f is a function given by
f (x) = 2^{(x  2) }+ 1

Find the domain and range of f.

Find the horizontal asymptote of the graph of f.

Find the x and y intercepts of the graph of f if there are any.

Sketch the graph of f.
More references and links exponential functions and graphing.
