Graphing and sketching exponential functions: step by step tutorial. The properties such as domain, range, horizontal asymptotes and intercepts of the graphs of these functions are also examined in details. Free graph paper is available.
We first start with the properties of the graph of the basic
of base a,
f (x) = ax , 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
2x > 0
Multiply both sides by 2-2
which is positive.
2x2-2 > 0
Use exponential properties
2(x- 2) > 0
This last statement suggests that f(x) > 0. The range of f is
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
f(x) = 0
2(x - 2) = 0
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.