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

Eine Transformation ändert die Gleichung einer Funktion, um die Position des Diagramms zu verschieben, ohne seine Form zu ändern.Eine gängige Transformation ist eine horizontale Verschiebung, bei der das Diagramm basierend auf Eingabeänderungen nach links oder rechts verschoben wird.Wenn x durch x plus eine Zahl ersetzt wird, verschiebt sich das Diagramm nach links. Das fühlt sich kontraintuitiv an, da ein Pluszeichen auf richtig hindeutet. In f von x plus 5 tritt die Ausgabe, die ursprünglich bei x gleich 0 erschien, jetzt auf, wenn x gleich -5 ist.Visuell bewegt sich das Diagramm um 5 Einheiten nach links.Wenn x durch x minus einer Zahl ersetzt wird, verschiebt sich der Graph nach rechts. Hier erfolgt nun die Ausgabe bei x gleich 0, wenn x gleich 5 ist.Visuell bewegt sich das Diagramm um 5 Einheiten nach rechts.Bei periodischen Funktionen wird eine horizontale Verschiebung der Wellenform von einem bestimmten Referenzpunkt als Phasenverschiebung bezeichnet.Phasenverschiebungen treten in Wechselstromkreisen auf, in denen Strom- und Spannungswellenformen nicht synchron sind. In einer Induktivität hinkt der Strom der Spannung hinterher, während in einem Kondensator der Strom die Spannung leitet.Diese Phasenunterschiede erscheinen als Wellenformverschiebungen, die zeigen, wie verschiedene Schaltungskomponenten zu unterschiedlichen Zeiten reagieren.

transformationen von funktionen ii

Mathematische Transformationen ver&#228;ndern die Lage oder Orientierung des Graphen einer Funktion, ohne ihre grundlegende Form zu ver&#228;ndern. Eine wichtige Transformationsart ist die horizontale Verschiebung, bei der die Eingangsvariable innerhalb der Funktionsgleichung modifiziert wird. Diese Operation beeinflusst die Stellen entlang der horizontalen Achse, an denen Funktionswerte auftreten, ver&#228;ndert jedoch nicht die Gesamtstruktur der Funktion.
Eine horizontale Verschiebung wird erreicht, indem die Variable x durch x + c oder x &#8722; c ersetzt wird, wobei c eine Konstante ist. Bei der Ersetzung von x durch x + c verschiebt sich der Graph um c Einheiten nach links, da jeder Funktionswert bereits bei einem kleineren Eingabewert erreicht wird. Wird x durch x &#8722; c ersetzt, verschiebt sich der Graph um c Einheiten nach rechts, wodurch das Auftreten der Funktionswerte verz&#246;gert wird. Diese Transformation verlagert das Verhalten der Funktion entlang der x-Achse, ohne ihre Form, Skalierung oder Orientierung zu &#228;ndern.
Horizontale Verschiebungen spielen in verschiedenen wissenschaftlichen und technischen Anwendungen eine wesentliche Rolle, insbesondere in zeitabh&#228;ngigen Systemen. Bei solchen Ph&#228;nomenen k&#246;nnen sie zeitliche Verz&#246;gerungen oder Vorl&#228;ufe von Systemreaktionen darstellen. In der Signalverarbeitung oder in oszillatorischen Systemen spiegeln solche Verschiebungen Phasen- bzw. Zeit&#228;nderungen zwischen verwandten Signalen wider; h&#228;ufig werden sie mit Sinusfunktionen modelliert, die einen Phasenterm enthalten, um diese Verschiebungen zu ber&#252;cksichtigen. Das Erkennen und Interpretieren horizontaler Verschiebungen ist daher entscheidend f&#252;r die Analyse von Systemen, in denen Zeitabstimmung und Synchronisation eine zentrale Rolle spielen.

Transformationen von Funktionen II

Mathematische Transformationen ver&#228;ndern die Lage oder Orientierung des Graphen einer Funktion, ohne ihre grundlegende Form zu ver&#228;ndern. Eine ...

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Math Fundamentals

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Functions and Their Graphs

Funktionen und ihre Graphen

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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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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...

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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...

piecewise-deﬁned functions

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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

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decreasing function

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...

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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