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

A transformation modifies a function's equation to shift the graph's position, without changing its shape.A common transformation is a horizontal shift, moving the graph left or right based on input changes.If x is replaced with x plus a number, the graph shifts left. This feels counterintuitive, since a plus sign suggests right. In f of x plus 5, the output that originally appeared at x equals 0 now occurs when x equals negative 5.Visually, the graph moves left by 5 units.If x is replaced with x minus a number, the graph shifts right. Here, the output at x equals 0 now occurs when x equals 5.Visually, the graph moves right by 5 units.For periodic functions, a horizontal shift of the waveform from a specified reference point is called a phase shift.Phase shifts occur in AC circuits, where current and voltage waveforms are out of sync. In an inductor, current lags voltage, while in a capacitor, current leads voltage.These phase differences appear as waveform shifts, showing how different circuit components respond at different times.

transformaties van functies ii

Transformaties in de wiskunde veranderen de positie of ori&#235;ntatie van de grafiek van een functie, terwijl de fundamentele vorm behouden blijft. Een belangrijk type transformatie is de horizontale verschuiving, waarbij de invoervariabele binnen het functievoorschrift wordt aangepast. Deze bewerking be&#239;nvloedt waar de functiewaarden langs de horizontale as optreden, maar verandert de algehele structuur van de functie niet.
Een horizontale verschuiving wordt gerealiseerd door de invoervariabele x te vervangen door x + c of x &#8722; c, waarbij c een constante is. Als de vervanging x + c is, verschuift de grafiek c eenheden naar links, omdat elke functiewaarde met een kleinere invoer wordt bereikt. Bij de invoer x &#8722; c verschuift de grafiek c eenheden naar rechts, waardoor het optreden van functiewaarden wordt vertraagd. Deze transformatie verplaatst het gedrag van de functie langs de x-as zonder de vorm, schaal of ori&#235;ntatie te wijzigen.
Horizontale verschuivingen spelen een cruciale rol in diverse wetenschappelijke en technische toepassingen, met name in systemen die in de tijd evolueren. Bij tijdsafhankelijke verschijnselen kunnen dergelijke verschuivingen temporele vertragingen of vervroegingen in systeemresponsen representeren. In de signaalverwerking en in oscillerende systemen weerspiegelen zij bijvoorbeeld veranderingen in fase of timing tussen gerelateerde signalen. Dit wordt vaak gemodelleerd met sinuso&#239;dale functies die een faseverschuivingsterm (fasehoek) bevatten om deze verschuivingen te modelleren. Het herkennen en interpreteren van horizontale verschuivingen is essentieel voor de analyse van systemen waarbij timing en synchronisatie een sleutelrol spelen.

Transformaties van functies II

Transformaties in de wiskunde veranderen de positie of ori&#235;ntatie van de grafiek van een functie, terwijl de fundamentele vorm behouden blijft. Een ...

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JoVE Core

Math Fundamentals

Wiskundige basisbeginselen

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Eenheid

Functions and Their Graphs

Functies en hun grafieken

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN BEGRIPPEN

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,...

types of functions i

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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...

piecewise-deﬁned functions

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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.

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

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...

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introduction to one-to-one functions

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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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