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

How do you find if a function is increasing or decreasing?

Read the graph from left to right. Where the graph rises, the function is increasing; where it falls, it is decreasing; where it is flat, it is constant. In symbols, f is increasing on an interval if f(x₂) > f(x₁) whenever x₂ > x₁ there. For example, f(x) = 3x − 2 is increasing everywhere because its slope is positive.

How do you find increasing and decreasing intervals?

Locate the turning points, then check the behavior between them. For y = (x − 2)², the vertex is at x = 2, so the function decreases on (−∞, 2) and increases on (2, ∞). Write intervals using x-values, not y-values, and use parentheses, because a function is neither increasing nor decreasing at the single turning point.

How do you find increasing intervals on a graph?

Trace the graph from left to right and mark every x-value where it changes direction. The stretches where the curve climbs are the increasing intervals. If a graph rises from x = −3 to x = 1, falls until x = 4, and then rises again, the increasing intervals are (−3, 1) and (4, ∞), assuming the right end keeps going up.

How do you determine if a function is increasing or decreasing using derivatives?

Find f′(x) and look at its sign: f′(x) > 0 means increasing and f′(x) < 0 means decreasing. For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 3(x − 1)(x + 1). The derivative is positive when x < −1 or x > 1, and negative between −1 and 1, so f decreases only on (−1, 1).

How do you identify an increasing or decreasing function from its equation?

For many common functions you can tell at a glance. A linear function y = mx + b increases when m > 0 and decreases when m < 0. An exponential function y = a·bˣ with a > 0 increases when b > 1 and decreases when 0 < b < 1. A parabola always does both, switching direction at its vertex.

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