How to Find the Increasing or Decreasing Functions?

Knowing whether a function is increasing or decreasing tells you how the output behaves as the input grows. A function that goes up as you move left to right on the graph is increasing; one that goes down is decreasing. Identifying these intervals from a graph or equation is an essential Algebra 1 skill and a stepping stone to understanding maximum and minimum values.

What Are Increasing and Decreasing Functions?

• A function f is increasing on an interval if, for any two values a < b in that interval, f(a) < f(b). The graph rises from left to right.
• A function f is decreasing on an interval if, for any two values a < b in that interval, f(a) > f(b). The graph falls from left to right.
• A function is constant on an interval if \(\color{blue}{f(a) = f(b)}\) for all a and b in that interval — the graph is flat.

How to Identify Increasing and Decreasing Intervals

From a graph

Read the graph from left to right. Where the graph goes up, the function is increasing; where it goes down, it is decreasing; where it is flat, it is constant. Describe each interval using open intervals (parentheses, not brackets) in x-values only.

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Example: f(x) = −\(\color{blue}{x^{2} + 4}\).
\(\color{blue}{f(-1) = 3}\), \(\color{blue}{f(0) = 4}\), \(\color{blue}{f(1) = 3}\), \(\color{blue}{f(2) = 0}\). The graph rises to \(\color{blue}{x = 0}\), then falls.
Increasing on (−∞, 0), decreasing on (0, ∞).

From a formula (using the concept of slope)

For linear functions, the slope tells you everything: positive slope → increasing, negative slope → decreasing, zero slope → constant. For non-linear functions, find where the function values rise or fall by analyzing key points or the vertex.

Example: \(\color{blue}{f(x) = x}\)\(\color{blue}{^{3} – 3x}\).
Values: f(−2) = −2, \(\color{blue}{f(-1) = 2}\), \(\color{blue}{f(0) = 0}\), f(1) = −2, \(\color{blue}{f(2) = 2}\).
The function rises on (−∞, −1), falls on (−1, 1), then rises on (1, ∞).
Increasing on (−∞, −1) ∪ (1, ∞), decreasing on (−1, 1).

Step-by-Step Summary

1. Sketch or examine the graph of the function from left to right.
2. Locate the turning points (where the graph switches from rising to falling or vice versa).
3. Write the intervals using the x-coordinates of the turning points.
4. Use open parentheses for the endpoints of each interval.
5. Label each interval as increasing, decreasing, or constant.

Watch: Increasing, Decreasing, and Constant Intervals (Video Lesson)

Khan Academy explains how to read increasing and decreasing intervals from a graph with visual examples:

Increasing and Decreasing Functions – Worked Examples

Example 1: \(\color{blue}{f(x) = 2x + 3}\). Determine whether f is increasing or decreasing.

This is a linear function with slope 2 > 0. The function is increasing on (−∞, ∞).

Example 2: f(x) = −\(\color{blue}{x^{2} + 4}\). Find the increasing and decreasing intervals.

The parabola opens downward with vertex at \(\color{blue}{x = 0}\). Left of \(\color{blue}{x = 0}\) the graph rises; right of \(\color{blue}{x = 0}\) it falls.
Increasing on (−∞, 0); decreasing on (0, ∞).

Example 3: \(\color{blue}{f(x) = x}\)\(\color{blue}{^{2} – 4}\). Find the increasing and decreasing intervals.

The parabola opens upward with vertex at \(\color{blue}{x = 0}\). Left of \(\color{blue}{x = 0}\) the graph falls; right it rises.
Decreasing on (−∞, 0); increasing on (0, ∞).

Example 4: \(\color{blue}{f(x) = x}\)\(\color{blue}{^{3} – 3x}\). Find the increasing and decreasing intervals.

Compute key values: f(−2) = −2, \(\color{blue}{f(-1) = 2}\), \(\color{blue}{f(0) = 0}\), f(1) = −2, \(\color{blue}{f(2) = 2}\).
The function has local max at x = −1 and local min at \(\color{blue}{x = 1}\).
Increasing on (−∞, −1) ∪ (1, ∞); decreasing on (−1, 1).

More Practice: Increasing vs Decreasing Functions Video

Professor Leonard provides a thorough walkthrough of increasing, decreasing, and constant intervals using multiple function types:

Exercises for Increasing and Decreasing Functions

Identify the increasing and decreasing intervals for each function.

1. f(x) = −\(\color{blue}{3x + 1}\)
2. \(\color{blue}{f(x) = x}\)² + \(\color{blue}{2x – 3}\)
3. f(x) = −\(\color{blue}{x^{2} + 6x}\)
4. \(\color{blue}{f(x) = x}\)³ (use key values at x = −2, −1, 0, 1, 2)
5. f(x) = |x| (absolute value)

Answers

1. Slope = −3 < 0: decreasing on (−∞, ∞)
2. Vertex at x = −\(\color{blue}{\frac{b}{(2a)}}\) = −\(\color{blue}{\frac{2}{2}}\) = −1: decreasing on (−∞, −1), increasing on (−1, ∞)
3. Vertex at \(\color{blue}{x = \frac{6}{2} = 3}\): increasing on (−∞, 3), decreasing on (3, ∞)
4. f is always increasing: increasing on (−∞, ∞)
5. Decreasing on (−∞, 0), increasing on (0, ∞)

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Want More Practice?

We haven’t published a worksheet built specifically for Increasing and Decreasing Functions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Characteristics of Quadratic Functions Worksheet
• Download Interpreting Functions and Parameters Worksheet
• Download Graphing Functions and Transformations Worksheet

Frequently Asked Questions

Should I use open or closed brackets for increasing/decreasing intervals?

Mathematicians typically use open parentheses for the endpoints of increasing/decreasing intervals, because at a single turning-point the function is technically neither increasing nor decreasing. However, some textbooks use closed brackets; follow your instructor’s convention.

Can a function be increasing and decreasing at the same point?

No. A function can only increase or decrease (or be constant) at a given point — it cannot do both simultaneously. A turning point separates an increasing interval from a decreasing one.

How do increasing/decreasing intervals relate to maxima and minima?

A local maximum occurs where the function changes from increasing to decreasing; a local minimum occurs where it changes from decreasing to increasing. That connection is covered in the Maxima and Minima topic.

Related Topics

• Function Notation
• Maxima and Minima of a Function
• Finding Domain and Range of a Function
• Average Rate of Change of a Function
• Graphing Rational Functions