Increasing or Decreasing Functions over an Interval
On this page you will learn to use algebra to determine if a function is increasing or decreasing over a given interval. The idea itself is pretty simple when you remember that increasing means that as x values increase so do the output values (y). Decreasing means that as the inputs (x values) increase, the outputs decrease.
Read through the notes below, watch the video, then try the practice problems. This is a key concept that will play major roles later on in more advanced mathematics. Now is a great time to learn it as it will also help you better understand polynomial functions and the algebra involved.
The intervals where a graph is increasing or decreasing are pretty easy to find given a graph. Increasing basically means the graph is going up and decreasing means the graph is going down, when read from left to right.
But what does going up, or down, really mean? In this section we will break that down and help you understand how to determine if a function is increasing or decreasing in a given function, algebraically. Let’s start with a graph. The graph below is the function f(x) = –x4 + x3 + 2x2.
We can see that this graph is increasing from negative infinity until -0.693. Keep in mind that when discussing intervals of increasing or decreasing behavior, we use the x – coordinates, the inputs, not the outputs.
There is a local maximum at (-0.693, -.397), after which the graph is decreasing. The graph them decreases from -0.693 until 0. At (0, 0) we have another local extrema, a local minimum. After that point, the graph increases until x = 1.443, which is another local maximum, and actually is the global maximum, as the graph decreases after that point. The degree is 4 and we have three turning points, which means the graph will not turn back up again. The global maximum is (1.443, 2.833).
Determining Increasing Algebraically
Instructions: Show that the function f(x) = –x4 + x3 + 2x2 is increasing between -1 and -0.9.
Idea: If as consecutive x – inputs increase their outputs (y – coordinates) also increase, the function is increasing. Another way to think of it is that if the x – coordinates decrease, so will the y – coordinates.
How To: Plug in the two values of x into the function. Then, compare their outputs.
Let’s see an example. Let’s use x = -1 and x = – 0.9. Notice that from those two inputs, the x values are increasing.
Plug those values into the function and compare the outputs. If the outputs are also increasing, the function is increasing between those two points.
Algebraically we can see that as x increased, so did y. Or, if you chose -0.9 to be 
then as x values decreased, so did the y values.
Let’s see another example, a little more concisely. Let’s show that the function f(x) = 4x3 – 6x2 + 3 is decreasing between 0 and 1. To be consistent, let’s assign as the smaller value, the one furthest left on the graph. So, 
So we can see that the x values are increasing. If their corresponding outputs decrease, then the function is decreasing.
So 
is decreasing, while 
is increasing. Therefore, the function is decreasing over that interval.
To grade your work, click the link here. It will download a PDF file.
