Name: horizontal shifts in function graphs
Uploaded: 2025-11-26T05:44:02.000Z
Duration: 1 min 29 s
Channel: JoVE

Une transformation modifie l’équation d’une fonction pour décaler la position du graphique, sans changer sa forme.Une transformation courante est un décalage horizontal, qui consiste à déplacer le graphique vers la gauche ou la droite en fonction des modifications d’entrée.Si x est remplacé par x plus un nombre, le graphique se décale vers la gauche. Cela semble contre-intuitif, car un signe plus suggère le bien. Dans f de x plus 5, le résultat qui apparaissait à l’origine en x est égal à 0 se produit maintenant lorsque x est égal à moins 5.Visuellement, le graphique se déplace vers la gauche de 5 unités.Si x est remplacé par x moins un nombre, le graphique se décale vers la droite. Ici, la sortie en x est égale à 0 se produit maintenant lorsque x est égal à 5.Visuellement, le graphique se déplace vers la droite de 5 unités.Pour les fonctions périodiques, un décalage horizontal de la forme d’onde à partir d’un point de référence spécifié est appelé déphasage.Les déphasages se produisent dans les circuits CA, où les formes d’onde de courant et de tension ne sont pas synchronisées. Dans une inductance, le courant est en retard sur la tension, tandis que dans un condensateur, le courant est en retard sur la tension.Ces différences de phase apparaissent sous forme de décalages de forme d’onde, montrant comment les différents composants du circuit réagissent à différents moments.

transformations de fonctions ii

En math&#233;matiques, les transformations modifient la position ou l&#8217;orientation du graphe d&#8217;une fonction tout en pr&#233;servant sa forme fondamentale. Un type important de transformation est le d&#233;calage horizontal, qui consiste &#224; modifier la variable ind&#233;pendante dans l&#8217;expression d&#8217;une fonction. Cette op&#233;ration affecte l&#8217;emplacement des images de la fonction le long de l&#8217;axe horizontal, sans toutefois alt&#233;rer la structure globale de la fonction.
Un d&#233;calage horizontal s&#8217;obtient en rempla&#231;ant la variable ind&#233;pendante x par x + c ou x &#8722; c, o&#249; c est une constante. Si l&#8217;on remplace par x + c, le graphe se d&#233;place vers la gauche de c unit&#233;s, puisque chaque image est atteinte avec un argument plus petit. Si l&#8217;on remplace par x &#8722; c, le graphe se d&#233;cale vers la droite de c unit&#233;s, ce qui retarde l&#8217;obtention des images. Cette transformation repositionne la fonction le long de l&#8217;axe des abscisses sans modifier sa forme, son &#233;chelle ni son orientation.
Les d&#233;calages horizontaux jouent un r&#244;le d&#233;terminant dans diverses applications scientifiques et techniques, en particulier dans les syst&#232;mes qui &#233;voluent au cours du temps. Dans les ph&#233;nom&#232;nes d&#233;pendant du temps, ces d&#233;calages peuvent repr&#233;senter des retards ou des avances dans les r&#233;ponses du syst&#232;me. Par exemple, dans le traitement du signal ou les syst&#232;mes oscillatoires, ils refl&#232;tent des variations de phase ou de synchronisation entre des signaux li&#233;s. Ils sont souvent mod&#233;lis&#233;s &#224; l&#8217;aide de fonctions sinuso&#239;dales qui int&#232;grent un terme de phase pour rendre compte de ces d&#233;calages. Reconna&#238;tre et interpr&#233;ter les d&#233;calages horizontaux est essentiel pour analyser des syst&#232;mes o&#249; le calage temporel et la synchronisation constituent des facteurs cl&#233;s.

Transformations de fonctions II

En math&#233;matiques, les transformations modifient la position ou l&#8217;orientation du graphe d&#8217;une fonction tout en pr&#233;servant sa forme ...

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Math Fundamentals

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Functions and Their Graphs

Fonctions et représentations graphiques

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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 graph moves right by c units because the output is delayed until a larger input value.

Horizontal shifts change where a function happens, but not its scale or orientation. They are important in systems that change over time. In signal processing and oscillatory systems, these shifts can show a delay or an advance in timing. Sinusoidal functions often use a phase term to model these offsets. Understanding horizontal shifts helps with timing and synchronization in math, science, and engineering.

Horizontal Shifts in Function Graphs

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,...

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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...

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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.

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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...

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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

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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

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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 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

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key terms and concepts

Learn how horizontal shifts move function graphs left or right by changing the input variable, and see how they model timing delays in real systems.

Horizontal Shifts in Function Graphs | 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

Horizontal Shifts in Function Graphs

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