Increasing And Decreasing - Exponentials
Objective: The learner will state the meaning of an increasing function and a decreasing function.
Objective: The learner will determine the intervals over which an exponential function is increasing and decreasing.
Definition: A function is said to be increasing on the interval (a, b) if whenever x_{1} and x_{2} are two numbers in the interval such that x_{1} < x_{2} then f(x_{1}) < f(x_{2}). Another way to look at this is: As you trace the graph from a to b (that is from left to right) the graph should go up.
Definition: A function is said to be decreasing on the interval (a, b) if whenever x_{1} and x_{2} are two numbers in the interval such that x_{1} < x_{2} then f(x_{1}) > f(x_{2}). Another way to look at this is: As you trace the graph from a to b (that is from left to right) the graph should go down.
Of course, as the functions become more complicated so do determining the intervals over which the function is increasing and decreasing. In general, to determine the intervals of increasing and decreasing:
Draw a picture of the graph using some type of graphing device.
Make sure that the picture you are viewing is complete. That is, all the important data about the graph can be seen visually. This would mean all zeros, maximums and minimums, asymptotes, and end behavior.
Determine the location of all maximums, minimums, and asymptotes.
From the information gathered in the previous step, and by visually inspecting the graph, write out the intervals of increasing and decreasing.
Note: If the endpoint of an increasing or decreasing interval is in the domain of the function, then that endpoint is included in both the increasing and decreasing solution. A minimum point is the last point where the function is decreasing, and the first point where the function begins to increase. Similarly, a maximum point is the last point where the function is increasing and the first point where the function begins to decrease.
Practice
As mentioned earlier, determining the intervals of increasing and decreasing of a function often require the use of a graphing utility. For this reason it is suggested that you use a graphing calculator or some other graphing utility, such as the one provided below. There are ten problems in the Practice Problems set that you can test your skills with. The answers are also provided.
Instructions Using Program (Word Document)