Name: how to identify a decreasing function
Uploaded: 2026-01-22T05:52:42.000Z
Duration: 1 min 27 s
Channel: JoVE

A function is decreasing when its output decreases as the input increases, meaning that as one moves from left to right along the x-axis, the f(x)-values on the graph get smaller.This behavior is identified by observing whether the graph slopes downward from left to right.Consider a man running on a track. The time taken and the distance covered for each lap are recorded to determine changes in speed over different intervals.The average speed&#8212;or rate of change&#8212;between intervals is determined by calculating the change in distance and dividing it by the change in time between two recorded points.Next, to identify whether the speed is increasing or decreasing, each lap&#8217;s speed is calculated by dividing the distance covered by the time taken for that lap. This helps analyze how the runner&#8217;s pace changes from one lap to the next.When plotted as a speed-versus-time graph, the data shows a consistent decline in speed. This represents a decreasing function, confirming that the runner slows down with each successive lap.The concept of decreasing functions models various situations where outputs decrease with increasing input, such as battery life or cooling temperature.

decreasing function

A decreasing function describes a relationship where the output consistently declines as the input increases. This means that for any two input values, if one is greater than the other, the corresponding output is smaller. Mathematically, a function f is decreasing on an interval I if for&#160; every x1 &lt; x2&#8203; in I, f (x1) &gt; f (x2). This type of behavior is visually identified on a graph that slopes downward from left to right.The nature of a function can be analyzed by calculating its rate of change. For a function defined at discrete points, the average rate of change over an interval is the ratio of the change in the output to the change in the input:&#160;&#160;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:53.09%;'><img style='aspect-ratio:1445/184;' src='https://cloudfront.jove.com/files/ftp_upload/40049/40049_Equation_1_editor_40049.svg' alt='40049_Equation_1.svg' /></figure>If this value is negative across intervals, the function is decreasing. In continuous functions, the derivative f&#8242;(x) serves as an indicator&#8212;if f&#8242;(x) &lt; 0 for all x in an interval, the function is decreasing on that interval.Decreasing functions appear in many natural and technological contexts. Examples include the temperature of a cooling object, the voltage of a discharging battery, and the height of a falling object after its peak. These scenarios involve quantities that reduce as time or another input progresses, making decreasing functions essential for modeling and analyzing such phenomena.

Decreasing Function

A decreasing function describes a relationship where the output consistently declines as the input increases. This means that for any two input values, if ...

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JoVE Core

Math Fundamentals

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Functions and Their Graphs

KEY TERMS AND CONCEPTS

how to identify a decreasing function

A decreasing function shows a drop in output as the input increases. If one input value is larger than another, the function gives a smaller output. For an interval I, a function f is decreasing when x1 f(x2) for every pair of values in that interval.

On a graph, a decreasing function slopes downward from left to right. The same idea can be checked with rate of change. For discrete points, the average rate of change is the change in output divided by the change in input. When that value is negative over intervals, the function is decreasing.

For continuous functions, the derivative helps show the same pattern. If f'(x) &lt; 0 for all x in an interval, then the function is decreasing on that interval. This gives a clear test for how a function changes across a range of values.

Decreasing functions also describe many real situations. A cooling object loses temperature over time. A discharging battery loses voltage. A falling object also shows a decrease in height after its peak. These examples make decreasing functions useful for modeling change in science and technology.

How to Identify a Decreasing Function

introduction to functions

Functions in Tables, Graphs, and Formulas

Functions describe input-output relationships that follow a clear rule. In a function, each input matches one corresponding output. This makes functions useful for showing how quantities change and for making predictions in math and science. A parking garage fee system is one common example. The input is the number of non-negative hours a car stays inside. Negative time does not make sense in this setting. The total cost starts with a flat entry charge and then rises by a steady hourly rate,...

types of functions i

Polynomial, Rational, and Radical Functions

Functions describe how one quantity changes in relation to another. They are key math tools for showing relationships between variables. Because functions can model real-world patterns, they are used across many types of problems. Algebraic functions are built from basic operations such as addition, subtraction, multiplication, division, and root extraction. The main algebraic types are polynomial, rational, and radical functions. Each one has its own form and use. Polynomial functions use...

types of functions ii

Sine, Cosine, and Exponential Models

Trigonometric and exponential functions help model two major patterns in science and math: repeating cycles and fast growth or decay. These functions extend basic algebra by describing behavior that changes in ways a straight line cannot. They are useful in many real-world settings, especially in science and engineering. Trigonometric functions such as sine and cosine are best for periodic phenomena, which means patterns that repeat over and over. Their smooth curves continue for all real...

types of functions iii

Logarithmic and Piecewise Functions

Logarithmic and piecewise functions help model real-world situations in different ways. A logarithmic function is the inverse of an exponential function. Its graph rises quickly for small x values and then flattens as x gets larger. This makes logarithms useful when data spans several orders of magnitude. Logarithmic scales compress large ranges into compact numbers. One common example is the pH scale in chemistry. It is written as pH = −log10[H+], where [H+] is the hydrogen ion concentration.

rules for defining functions

How to Tell If a Relation Is a Function

A relation is a function when each input x has exactly one output y. This rule helps students decide whether an equation, graph, or table represents a function. The equation y = 2x + 5 is a function because every x-value gives one unique y-value. But x = y^2 + 1 is not a function of x. When x = 2, for example, the equation gives two possible y-values: y = 1 and y = -1. Graphs can be checked with the vertical line test. If a vertical line crosses a curve more than once, the graph fails the...

piecewise-deﬁned functions

Graphing Piecewise Function Intervals

Piecewise defined functions use different rules over different parts of the domain. Each rule applies to a specific interval, so one function can change its behavior when the input changes. These functions are useful for modeling systems that do not stay the same across all input values. A piecewise function can combine more than one expression and still count as a single function. For example, one rule may be linear for inputs less than or equal to -1. Another rule may be quadratic for values...

graphs of functions

Function Graphs and Their Shapes

Function graphs show how output values change when input values change. Each point on the graph is an ordered pair. The x-coordinate is the independent variable, and it sets the horizontal position. The y-coordinate is the dependent variable, and it sets the vertical position. Linear functions such as y = x make a straight line. This shape shows a constant rate of change. Nonlinear functions behave differently and can curve in more complex ways. Even power functions form U-shaped curves.

A decreasing function shows a drop in output as the input increases. If one input value is larger than another, the function gives a smaller output. For an interval I, a function f is decreasing when x1 f(x2) for every pair of values in that interval. On a graph, a decreasing function slopes downward from left to right. The same idea can be checked with rate of change. For discrete points, the average rate of change is the change in output divided by the change in input. When that value is...

increasing function

Intervals Where a Function Rises

A function can rise as its input values increase. This pattern is called an increasing function. On a graph, it appears as a line or curve that slopes upward from left to right. An increasing function follows a clear rule. If x1 is less than x2, then f(x1) is less than f(x2). In other words, larger inputs give larger output values. This idea helps describe growth in situations such as population dynamics, financial investments, and resource consumption. The average rate of change shows how...

transformations of functions i

Shifting Function Graphs on the Plane

Shifting function graphs changes their position on the coordinate plane without changing their overall shape. These transformations help move a graph while keeping its pattern and structure the same. Shifting is one of the most common graph transformations. When a constant is added to or subtracted from the output of a function, the graph shifts vertically. A positive change moves the graph upward. A negative change moves it downward. This is like raising or lowering the height of a telescopic...

transformations of functions ii

Horizontal Shifts in Function Graphs

Horizontal shifts move a function’s graph left or right without changing its shape. In this type of transformation, the input variable x is replaced inside the equation. The graph keeps the same overall structure, but its position on the x-axis changes. A shift to the left happens when x is replaced by x + c, where c is a constant. The outputs then appear after a smaller input value, so the graph moves left by c units. A shift to the right happens when x is replaced by x - c. In that case, the...

transformations of functions iii

Graph Reflections and Stretching

Function graphs can be changed without altering their basic shape. These transformations include reflections, vertical stretching, and horizontal compression. They help show how a function changes on a graph while keeping its fundamental form. A reflection flips a graph across an axis. When every output value is multiplied by negative one, the graph mirrors over the horizontal axis. This reverses peaks and troughs, much like signal inversion in electrical systems, where a waveform is flipped...

combining functions

Function Operations and Domain Rules

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introduction to one-to-one functions

Graphing One-to-One Functions and Inverses

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key terms and concepts

Learn how to identify a decreasing function by its graph, negative rate of change, and derivative. See real examples like cooling, batteries, and falling objects.

How to Identify a Decreasing Function | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

How to Identify a Decreasing Function

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