Translation

A graph can be translated in two ways, horizontally and vertically.


Horizontal Translation: When a graph is translated horizontally, each point on the graph moves left or right along the x-axis. The equation of the graph remains the same, but a constant value is added or subtracted from the x-coordinate of each point. For example, if we translate the graph of y = f(x) by c units to the right, the new equation becomes y = f(x − c).

Vertical Translation: When a graph is translated vertically, each point on the graph moves up or down along the y-axis. The equation of the graph remains the same, but a constant value is added or subtracted from the y-coordinate of each point. For example, if we translate the graph of y = f(x) by d units upward, the new equation becomes y = f(x) + d.

Graph Transformations

Graph transformations involve changing the appearance or position of graphs by shifting them horizontally or vertically, stretching or compressing them, reflecting them across axes, or rotating them around a fixed point. These modifications help visualize how functions change under different conditions or transformations.

In this article, we will learn the meaning of graph transformations, the types of graph transformations, and properties of graph translations.

Table of Content

  • What is Transformations of Graph?
  • Types of Graph Transformations
  • Translation
  • Reflection
  • Scaling (Dilation)
  • Rotation
  • Shearing
  • Properties of Graph Transformations

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What is Transformations of Graph?

The transformations of a graph are changes made to its appearance or position. These changes can include shifting the graph up, down, left, or right, stretching or compressing it, and flipping it. These transformations help us understand how the graph changes based on different factors, such as adding or subtracting numbers to the function, multiplying or dividing it, or reflecting it across an axis....

Types of Graph Transformations

Translation (Shift): Moving the entire graph horizontally, vertically, or both without changing its shape or orientation.Vertical Shift (Vertical Translation): Moving the entire graph up or down by a certain amount.Horizontal Shift (Horizontal Translation): Moving the entire graph left or right by a certain amount.Vertical Stretch or Compression: Scaling the graph vertically by a factor, making it taller (stretch) or shorter (compression).Horizontal Stretch or Compression: Scaling the graph vertically by a factor, making it taller (stretch) or shorter (compression).Reflection over the x-axis: Scaling the graph vertically by a factor, making it taller (stretch) or shorter (compression).Reflection over the y-axis: Flipping the graph left to right across the y-axis, changing the sign of the x-coordinates.Rotation: Rotating the graph around a point, usually the origin, by a certain angle.Horizontal Shearing: Slanting the graph horizontally by a certain factor.Vertical Shearing: Slanting the graph vertically by a certain factor....

Translation

A graph can be translated in two ways, horizontally and vertically....

Reflection

Horizontal Reflection: A horizontal reflection, also known as a reflection across the y-axis, flips the graph of a function over the y-axis. This is achieved by replacing x with −x in the equation of the function. For example, if we reflect the graph of y=f(x) horizontally, the new equation becomes y=f(−x)....

Scaling (Dilation)

Scaling, also known as dilation, is a geometric transformation that involves stretching or compressing an object in one or more directions. In the context of graphs, scaling refers to the process of enlarging or shrinking a graph along the x-axis, y-axis, or both. There are two types of scaling:...

Rotation

Clockwise Rotation: Clockwise rotation turns the graph of a function clockwise around a fixed point (usually the origin) by a certain angle. This is achieved by applying a rotation matrix to the coordinates of each point on the graph....

Shearing

Shearing is a geometric transformation that involves shifting one part of an object or figure in a fixed direction while keeping the rest of the object stationary. This results in a deformation of the object, where the shape is changed but the size and orientation remain the same....

Properties of Graph Transformations

Graph transformations exhibit several properties that are important to understand when manipulating graphs. Here are some key properties:...

Solved Examples on Graph Transformation

Example 1: Horizontal Translation...

Conclusion

In conclusion, graph transformations play a crucial role in altering the appearance and position of graphs while preserving certain properties. Understanding the various types of transformations, such as translation, rotation, reflection, scaling, and shearing, enables us to manipulate graphs to visualize and analyze mathematical concepts effectively. By applying transformation rules, we can see how graphs change in response to different factors and scenarios, enhancing our understanding of mathematical relationships and functions....

FAQs about Graph Transformations

What are the 4 types of transformation?...