Contents

# Slide**|Definition & Meaning**

**Definition**

In math and geometry, **sliding** is when we **move** a **set** **of** **points** **defining** some **shape** by an equal amount **in** **any** **direction**. Therefore, the **shape** **remains** **exactly** the **same** in every way; it **just** **moves** to a **different** **place**. In other words, to slide a shape means to **move** it **without** **rescaling**, **turning**, or **flipping** it in any way. **Formally**, sliding is **known** as **translation**.

**Demonstration of the Concept of Slide**

In the study of mathematics, the **real** **world** is **represented** by **symbols**, **numbers**, and **equations**. It is a universal language that facilitates understanding complicated ideas, problem-solving, and prediction.Â

In mathematics, the term “**slide**,” also known as “**translation**,” is used to **describe** the **movement** of a **figure** **over** a **specific** **distance** and in a **specific** **direction**. In this article, we’ll talk about the mathematical idea of slides and how it’s used in subjects like geometry, algebra, and trigonometry.

Figure 1 – Concept of Slide in Geometry

**Geometric Slide**

The study of the **characteristics** and **connections** between **points**, **lines**, **angles**, and **shapes** in space is known as **Geometry**. A slide is a **type** of **transformation** used in geometry that involves **moving** a **figure** a **specific** **amount** in one direction. To accomplish this, **move** **each** **point** of the **figure** in the **same** **direction** and over the **same** **distance**.

As a result, **only** the **figure’s** **position** is **altered**; its **size** and **shape** are **unaltered**.

When **comparing** or **analyzing** **figures** that are similar but not congruent, **slides** are **helpful** in geometry. For instance, we can **use** a **slide** **transformation** to **move** a **square** to a **different** **location** if we have one and want to. The figure is moved without altering its size, orientation, or shape. This **makes** it **simple** for us to **evaluate** **how** the **figures** are **situated** and **interconnected**.

**Algebraic Slide**

Figure 2 – Slide in Algebraic Domain

In the field of mathematics known as algebra, variables, equations, and functions are all addressed. In **algebra**, **slides** can be used to **translate** a **function’s** **graph** in a **specific** **direction** and over a **specific** **distance**.

This is accomplished by **increasing** the **independent** **variable** of the **function** **by** a **constant**. As an illustration, to translate the **function** **y = x** **two** **units** to the **right**, we would add 2 to x, creating the **new** **function** **y = x + 2.**

The new function’s **graph** will **resemble** that of the **original** **function**, with the **exception** that **it** **will** be **two** **units** to the **right**. The behavior of the function can then be compared and analyzed as it is translated in various directions and distances.

**Trigonometric Slide**

Figure 3 – Slide in the context of Trigonometry

The relationships between the sides and angles of triangles are the subject of the mathematical discipline of trigonometry. **Slides** can be used in trigonometry to **translate** the **graph** of a **trigonometric** **function** in a **specific** **direction** and over a **specific** **distance**. This is accomplished by increasing the independent variable of the function by a constant.

As an illustration, if we want to **translate** the **sine** function **y = sin(x) pi/2** units to the right, we would add pi/2 to x, creating the new function **y = sin(x + pi/2).**

The **graph** of the **new** **function** will merely be **pi/2 units** to the **right** of the **graph** of the **original** **function**. The behavior of the function can then be compared and analyzed as it is translated in various directions and distances.

## Procedure To Find Slide

Finding a slide of a function, also referred to as a translation, is done as follows:

- The
**original****function’s**equation, f, should be**written down**Â**(x)**. **Decide**on the**translation’s****magnitude**and direction. A**slide**to the**right**is**represented**by the**equation’s x-term**having a**positive****constant****added**to it. A**slide**to the**left**is represented by the**equation’s x-term**having a**negative****constant****added**to it. The**constant’s**value**determines****how****large**a**translation**will be.**Add**the**constant**to the**x-term**in the**original****function’s**equation. The**equation**of the**translated****function**, g, is what is produced as a result (x).**To****see**the**impact**of the translation,**plot****both**the**original**function, f(x), and the**translated****function**, g(x), on the same coordinate plane.

The equation of the translated function would be g(x) = 2(x-2), for instance, if f(x) = 2x and we wanted to translate the function 2 units to the right.

**Properties of Slide**

The following are the characteristics of a slide (or translation) of a function:

**The Shape is Preserved:**A slide does not alter the design of a function’s graph. The graph is only moved horizontally or vertically.**The Direction of the Slide:**direction determines whether a slide is moving up, down, to the right, to the left, or in any other direction.**The Magnitude of the Slide:**The value of the constant added to the x- or y-term of the original equation determines the magnitude of a slide.**Commutative Property:**The result is unaffected by the order in which the slides are executed. In other words, if f(x) is shifted by a constant c to get g(x), and g(x) is shifted by a constant d to get h(x), then h(x) = g(x + d) = f(x + c + d).**Additivity Property:**The combined effect of two slides is equal to the combined effect of one slide. In other words, if f(x) is shifted by a constant c to get g(x) and f(x) is shifted by a constant d to get h(x), then h(x) = g(x + d) = f(x + c + d).

These characteristics can be helpful in resolving issues in mathematics and other related fields, as well as in understanding the impact of a slide on the graph of a function.

**Solved Example of an Equation With a Slide Offset**

**Example 1**

**Find** the **equation** of the **translated** **function** that is moved **2 units** to the **right** and** 1 unit up**, given the **function** **f(x) = x ^{2}**.

**Solution**

Figure 4 – Example of Slide

The **direction** and **magnitude** of the **translation** must be **specified** to determine the equation of the translated function. We want to translate the function, **in** **this** **case**, **2 units** to the **right** and **1 unit up**.

The original equation’s x-term is subtracted by 2, and the y-term is increased by 1 to produce the equation for the translated function:

**g(x) = (x – 2) ^{2} + 1**

The graph of **g(x)** is **identical** to the graph of **f(x)**, with a **2 rightward** and **1 upward shift**.

*All mathematical drawings and images were created with GeoGebra.*