Transformations Lesson in Math: Definition, Types, Rules, and Example

Lesson Overview

• What are Transformations In Math?
• Types of Transformations
• Rules of Transformations
• Transformations Formula
• Transformations Examples

Transformation in math refers to the process of manipulating a geometric figure's location, size, or orientation without fundamentally altering its shape. 

These operations, performed on a coordinate plane, are crucial for understanding geometric principles and have applications in fields like computer graphics and animation.

What are Transformations In Math?

In mathematics, transformations are ways to change the position, size, or orientation of a shape. These changes are applied to geometric figures on a coordinate plane. 

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general term of Transformations Quiz

Types of Transformations

Geometric transformations are categorized into four main types: translation, reflection, rotation, and dilation. Each type alters the position, size, or orientation of a shape in a specific way. 

Rules of Transformations

The rules of transformations describe how a function or graph can be shifted, flipped, or stretched/compressed. These transformations help understand how changes to the function's equation affect its visual representation.

1. Vertical Transformation

A vertical transformation shifts the entire graph of a function up or down along the y-axis.

• Shifting Up: Adding a constant to the function shifts it upwards. For example, if we have the function f(x) and add 'a' to it, the new function becomes f(x) + a. This means every point (x, y) on the original graph moves to (x, y + a).
• Shifting Down: Subtracting a constant from the function shifts it downwards. So, f(x) - a shifts the graph down by 'a' units, and each point (x, y) becomes (x, y - a).

• Example: Consider the function f(x) = x³ + 2x². If we vertically transform it by 4 units upwards, the new function is f(x) + 4 = x³ + 2x² + 4. Each point on the original graph is shifted 4 units up.

2. Horizontal Transformation

A horizontal transformation shifts the graph left or right along the x-axis.

• Shifting Right: Replacing 'x' with 'x - a' in the function shifts the graph to the right by 'a' units. This means a point (x, y) on the original graph becomes (x + a, y).
• Shifting Left: Replacing 'x' with 'x + a' shifts the graph to the left by 'a' units. A point (x, y) becomes (x - a, y).

• Example: Let's take the function f(x) = 2x + 3. If we shift it 2 units to the left, the new function is f(x + 2) = 2(x + 2) + 3. Each point on the original graph is shifted 2 units to the left.

3. Flipped Transformation About the X-axis

This transformation flips the graph over the x-axis, creating a mirror image.

• Flipping: To flip a function f(x) about the x-axis, we change it to -f(x). This negates the y-values of all the points. So, a point (x, y) becomes (x, -y).

• Example: If we flip the function f(x) = 3x + 2 across the x-axis, it becomes -f(x) = -(3x + 2). Each point on the graph is reflected across the x-axis.

4. Mirror Transformation About the Y-axis

This transformation reflects the graph over the y-axis.

• Mirroring: To mirror a function f(x) about the y-axis, we replace 'x' with '-x' in the function, resulting in f(-x). This negates the x-values of all the points. A point (x, y) becomes (-x, y).

• Example: Consider the function f(x) = 5x + 1. Mirroring it about the y-axis gives us f(-x) = 5(-x) + 1 = -5x + 1. Each point on the graph is reflected across the y-axis.

5. Stretched/Compressed Vertical Transformation

This transformation stretches or compresses the graph vertically.

• Stretching (c > 1): Multiplying the function by a constant 'c' where c > 1 stretches the graph vertically. The point (x, y) becomes (x, cy).
• Compressing (0 < c < 1): Multiplying the function by a constant 'c' where 0 < c < 1 compresses the graph vertically. The point (x, y) becomes (x, cy).

• Example: Let's take the function f(x) = x². If we stretch it vertically by a factor of 3, the new function is 3f(x) = 3x². A point (1, 1) on the original graph would become (1, 3) on the stretched graph.
• 6. Stretched/Compressed Horizontal Transformation

This transformation stretches or compresses the graph horizontally.

• Stretching (0 < c < 1): Replacing 'x' with 'cx' in the function, where 'c' is a constant between 0 and 1, stretches the graph horizontally. The point (x, y) becomes (x/c, y).
• Compressing (c > 1): Replacing 'x' with 'cx' in the function, where 'c' is a constant greater than 1, compresses the graph horizontally. The point (x, y) becomes (x/c, y).

Example: The graph shows both the stretched function, and the original function. so now a point (1, 1) on the original curve has now become a point (3, 1) on the stretched curve.

Transformations Formula

The transformation of any function or graph follows a general formula:

y = −a * f(−(b * (x + h))) + k

• −a: Reflects the graph across the x-axis.
• −b: Reflects the graph across the y-axis.
• +h: Shifts the graph horizontally to the left.
• −h: Shifts the graph horizontally to the right.
• +k: Moves the graph vertically upward.
• −k: Moves the graph vertically downward.

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Math12 1 Transformations 1

Transformations Examples

1. Translate the point A(2, 3) 4 units to the right and 1 unit down.

• Solution: To translate right, add to the x-coordinate. To translate down, subtract from the y-coordinate.
  • New x-coordinate: 2 + 4 = 6
  • New y-coordinate: 3 - 1 = 2
  • The translated point is A'(6, 2).

2. Reflect the point B(-1, 5) across the x-axis.

• Solution: To reflect across the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate.
  • The reflected point is B'(-1, -5).

3. Rotate the point C(4, -2) 90 degrees clockwise about the origin.

• Solution: To rotate 90 degrees clockwise, swap the x and y coordinates and change the sign of the new y-coordinate.
  • The rotated point is C'(-2, -4).

4. Dilate the point D(3, 1) by a scale factor of 2 with the center of dilation at the origin.

• Solution: To dilate, multiply both the x and y coordinates by the scale factor.
  • New x-coordinate: 3 × 2 = 6
  • New y-coordinate: 1 × 2 = 2
  • The dilated point is D'(6, 2).

5. Translate the point E(-3, 2) 2 units to the left, then reflect it across the y-axis.

• Solution:
  • Subtract 2 from the x-coordinate: -3 - 2 = -5. The translated point is (-5, 2).
  • To reflect across the y-axis, change the sign of the x-coordinate: (-5, 2) becomes (5, 2).
  • The final point after both transformations is E'(5, 2).