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.
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.
By applying these transformations, we can visualize how the original graph is modified to represent different situations or scenarios.
- 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.
Let's discuss these in detail as follows:
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.
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).
Vertical Reflection: A vertical reflection, also known as a reflection across the x-axis, flips the graph of a function over the x-axis. This is achieved by replacing y with −y in the equation of the function. For example, if we reflect the graph of y=f(x) vertically, 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:
Horizontal Scaling: Horizontal scaling stretches or compresses the graph of a function horizontally. If the scaling factor is greater than 1, the graph is stretched horizontally, and if the scaling factor is between 0 and 1, the graph is compressed horizontally. This is achieved by multiplying the x-coordinates of each point on the graph by the scaling factor. For example, if we scale the graph of y = f(x) horizontally by a factor of a, the new equation becomes y = f(ax).
Vertical Scaling: Vertical scaling stretches or compresses the graph of a function vertically. This is achieved by multiplying the y-coordinates of each point on the graph by the scaling factor. For example, if we scale the graph of y = f(x) vertically by a factor of b, the new equation becomes y = bf(x).
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.
Example: If we rotate the graph of y = f(x) clockwise by an angle θ, the new coordinates (x', y') are obtained using the transformation:
(x', y') = \begin{cases}
x' = x \cos(\theta) + y \sin(\theta) \\
y' = -x \sin(\theta) + y \cos(\theta)
\end{cases}
Therefore, the new equation after rotating y = f(x) clockwise by θ is:
y = f \left( \frac{x \cos(\theta) + y \sin(\theta)}{\cos(\theta) - \sin(\theta)} \right)
Counterclockwise Rotation: Counterclockwise rotation turns the graph of a function counterclockwise around a fixed point (usually the origin) by a certain angle. This is achieved by applying the inverse rotation matrix to the coordinates of each point on the graph.
Example: If we rotate the graph of y = f(x) anti-clockwise by an angle θ, the new coordinates (x', y') are obtained using the transformation:
(x', y') = \begin{cases}
x' = x \cos(\theta) - y \sin(\theta) \\
y' = -x \sin(\theta) + y \cos(\theta)
\end{cases}
Therefore, the new equation after rotating y = f(x) anti-clockwise by θ is:
y = f \left( \frac{x \cos(\theta) - y \sin(\theta)}{\cos(\theta) + \sin(\theta)} \right)
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.
In the context of graphs, shearing involves stretching or compressing a graph in one direction while leaving the other direction unchanged. There are two types of shearing:
Horizontal Shearing
Horizontal shearing stretches or compresses the graph of a function horizontally in one direction while keeping the other direction unchanged. This is achieved by adding a multiple of the y-coordinate to the x-coordinate of each point on the graph.
Example: If we horizontally shear the graph of y = f(x) by a factor of k , the new coordinates (x', y') are obtained using the transformation:
(x', y') = \begin{cases}
x' = x + ky \\
y' = y
\end{cases}
Therefore, the new equation after horizontally shearing y = f(x) by a factor of k is:
y = f(x - ky)
Vertical Shearing
Vertical shearing stretches or compresses the graph of a function vertically in one direction while keeping the other direction unchanged. This is achieved by adding a multiple of the x-coordinate to the y-coordinate of each point on the graph.
Example: If we vertically shear the graph of y = f(x) by a factor of k , the new coordinates (x', y') are obtained using the transformation:
(x', y') = \begin{cases}
x' = x \\
y' = y + kx
\end{cases}
Therefore, the new equation after vertical shearing y = f(x) by a factor of k is:
y = f(x) - kx
Graph transformations exhibit several properties that are important to understand when manipulating graphs. Here are some key properties:
- Preservation of Shape: Most graph transformations preserve the basic shape of the original graph. For instance, a translation or rotation does not change the overall shape of the graph, but simply moves or rotates it.
- Preservation of Slope: Some transformations, such as translations and scalings, preserve the slope of the original graph. For example, if a function has a constant slope, it will maintain that slope after a translation or scaling.
- Preservation of Length: Certain transformations, like translations and rotations, preserve the lengths of line segments and curves on the graph. This means that distances between points on the original graph remain unchanged.
- Effect on Intercepts: Transformations can affect the intercepts of a graph. For example, a vertical translation will change the y-intercept of a function, while a horizontal translation will change the x-intercept.
- Symmetry: Some transformations introduce symmetry into the graph. For example, a reflection across the x-axis or y-axis creates symmetry with respect to that axis.
- Orientation: Transformations like rotations change the orientation of the graph. For example, a clockwise rotation will change the orientation of the graph compared to a counterclockwise rotation.
- Effect on Domain and Range: Certain transformations may affect the domain and range of the function represented by the graph. For example, a vertical scaling may stretch or compress the range of the function.
Solved Examples on Graph Transformation
Example 1: Horizontal Translation
Solution:
Original Function: f(x) = x2
Transformed Function: f(x - 3) = (x - 3)2
Result: The graph of f(x) = x2 is shifted 3 units to the right.
Example 2: Vertical Translation
Solution:
Original Function: f(x) = sin(x)
Transformed Function: f(x) + 2 = sin(x) + 2
Result: The graph of f(x) = sin(x) is shifted 2 units up.
Example 3: Reflection over the x-axis
Solution::
Original Function: f(x) = √(x)
Transformed Function: -f(x) = -√(x)
Result: The graph of f(x) = √(x) is reflected over the x-axis.
Example 4: Vertical Stretch
Solution:
Original Function: f(x) = cos(x)
Transformed Function: 2f(x) = 2cos(x)
Result: The graph of f(x) = cos(x) is stretched vertically by a factor of 2.
Example 5: Combination of Transformations
Solution:
Original Function: f(x) = x3
Transformed Function: 2f(3x - 1) + 4 = 2(3x - 1)3 + 4
Result: The graph undergoes a horizontal compression by a factor of 1/3, a horizontal translation by 1/3 units to the right, a vertical stretch by a factor of 2, and a vertical translation by 4 units up.
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.
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