Lesson Objectives
- Learn how to determine the intervals where a function is increasing
- Learn how to determine the intervals where a function is decreasing
- Learn how to determine the intervals where a function is constant
How to Determine the Intervals Where a Function is Increasing, Decreasing, or Constant
In this lesson, we want to learn how to determine where a function is increasing, decreasing, or constant from its graph. Let's begin with something simple, the linear function. We know when we look at the graph of a line with a positive slope, the graph rises as we move from left to right. In other words, the y-values are always increasing as the x-values are increasing. $$y=2x - 1$$ $$m=2$$ 
Additionally, if the line has a negative slope, the graph falls as we move from left to right. In this case, the y-values are always decreasing as the x-values are increasing. $$y=-5x + 2$$ $$m=-5$$ 
When working with lines, the process is very easy, for a positively sloped line, we will increase across the entire domain. On the other hand, for a negatively sloped line, we will decrease across the entire domain. In most cases, we will deal with a graph that is much more complex than a simple line. To deal with harder situations, let's think about a simple rule:
A function f is increasing on any interval if, for any (x1) and (x2), we have: $$x_1 < x_2$$ $$f(x_1) < f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are getting larger.
A function f is decreasing on any interval if, for any (x1) and (x2), we have: $$x_1 < x_2$$ $$f(x_1) > f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are getting smaller. 
A function f is constant on any interval if, for every (x1) and (x2), we have: $$f(x_1)=f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are constant. 
When we get the question of whether a function is increasing, decreasing, or constant on an interval, think about what happens to the y-values as the x-values go from left to right. Is the graph climbing (increasing), falling (decreasing), or flat (constant).
Example #1: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^3 - 3x^2$$
From the graph we see that f is increasing on the intervals (-∞, 0) and (2, ∞) and is decreasing on (0, 2). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help. 
Example #2: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^4 - 2x^2$$ 
From the graph we see that f is increasing on the intervals (-1, 0) and (1, ∞) and is decreasing on (-∞, -1) and (0, 1). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help. 
Additionally, if the line has a negative slope, the graph falls as we move from left to right. In this case, the y-values are always decreasing as the x-values are increasing. $$y=-5x + 2$$ $$m=-5$$ When working with lines, the process is very easy, for a positively sloped line, we will increase across the entire domain. On the other hand, for a negatively sloped line, we will decrease across the entire domain. In most cases, we will deal with a graph that is much more complex than a simple line. To deal with harder situations, let's think about a simple rule: A function f is increasing on any interval if, for any (x1) and (x2), we have: $$x_1 < x_2$$ $$f(x_1) < f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are getting larger.
A function f is decreasing on any interval if, for any (x1) and (x2), we have: $$x_1 < x_2$$ $$f(x_1) > f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are getting smaller. A function f is constant on any interval if, for every (x1) and (x2), we have: $$f(x_1)=f(x_2)$$ This tells us as we move right from x1 to x2, the y-values are constant. When we get the question of whether a function is increasing, decreasing, or constant on an interval, think about what happens to the y-values as the x-values go from left to right. Is the graph climbing (increasing), falling (decreasing), or flat (constant). Increasing, Decreasing, and Constant Functions
- A function is increasing on an open interval, if f(x1) < f(x2) whenever x1 < x2 for any x1 and x2 in the interval
- A function is decreasing on an open interval, if f(x1) > f(x2) whenever x1 < x2 for any x1 and x2 in the interval
- A function is constant on an open interval, if f(x1) = f(x2) for any for any x1 and x2 in the interval
- Some textbooks use closed intervals when discussing this topic.
- A good article for reference: Math Doctors Article
- Additionally, some resources may use the terms strictly increasing and strictly decreasing.
- A good article for reference: Math is Fun Article
Example #1: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^3 - 3x^2$$
From the graph we see that f is increasing on the intervals (-∞, 0) and (2, ∞) and is decreasing on (0, 2). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help. Example #2: Find the intervals on which f is increasing and on which f is decreasing. $$f(x)=x^4 - 2x^2$$ From the graph we see that f is increasing on the intervals (-1, 0) and (1, ∞) and is decreasing on (-∞, -1) and (0, 1). These intervals are always given in terms of the x-values. A common mistake is to try to give the intervals in terms of the y-values. If you are having trouble with this concept, drawing in a person walking may help. Skills Check:
Example #1
Determine if the following is true or false. $$f(x)=x^2 - 3x$$ 
Decreasing on: $$\left(-\infty, \frac{3}{2}\right)$$ Increasing on: $$\left(\frac{3}{2}, \infty\right)$$
Please choose the best answer.
A
True
B
False
Example #2
Determine if the following is true or false. $$f(x)=\large{x}^{\frac{2}{3}}- 1$$ 
Decreasing on: $$\left(-\infty, -1\right)$$ Increasing on: $$\left(-1, \infty\right)$$
Please choose the best answer.
A
True
B
False
Example #3
Sketch the graph and find the interval on which the function is increasing. $$\ f(x)=\begin{cases}-2x + 3 & \text{if}\hspace{.2em}x ≤ 5 \\ 3x + 1 & \text{if}\hspace{.2em}x > 5 \end{cases}$$
Please choose the best answer.
A
$$(-\infty, 5)$$
B
$$(0, 5)$$
C
$$(-\infty, 0)$$
D
$$(5, \infty)$$
E
$$(-\infty, \infty)$$
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