Mathematics
Grade 10
15 min
Translations: write the rule
Translations: write the rule
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Introduction & Learning Objectives
Learning Objectives
Define a translation and identify its key properties as a rigid transformation.
Distinguish between a pre-image and its corresponding image.
Determine the horizontal and vertical components of a translation from a graph or from coordinates.
Write the algebraic rule for a translation in coordinate notation, (x, y) -> (x + a, y + b).
Express a translation using a component vector, <a, b>.
Derive the rule for a translation given the coordinates of a single pre-image point and its image.
Ever used a chess app where the pieces slide smoothly across the board? ♟️ That's a translation, and we can describe that exact move with a simple mathematical rule!
In this tutorial, you will learn how to describe a translation, which is a 'slide' in...
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Key Concepts & Vocabulary
TermDefinitionExample
TranslationA transformation that slides every point of a figure or graph the same distance in the same direction. It is a type of rigid transformation (or isometry), meaning it preserves size, shape, and orientation.If triangle ABC is translated 3 units right and 4 units down, every single point on the triangle moves exactly 3 units right and 4 units down.
Pre-imageThe original figure in a transformation.If point P(2, 5) is translated, P is the pre-image.
ImageThe figure that results from a transformation. The image of a point is often denoted with a prime symbol (').If point P(2, 5) is translated to P'(6, 3), then P' is the image of P.
Coordinate NotationA notation that uses an arrow to show how the coordinates of a point (x, y) change after a transfo...
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Core Formulas
Translation Rule in Coordinate Notation
(x, y) \rightarrow (x + a, y + b)
This is the primary algebraic rule for a translation. 'a' represents the horizontal shift (positive for right, negative for left) and 'b' represents the vertical shift (positive for up, negative for down).
Formula to Find the Translation Components
a = x' - x \quad \text{and} \quad b = y' - y
To find the values of 'a' and 'b' for the rule, subtract the coordinates of the pre-image point (x, y) from the coordinates of its corresponding image point (x', y').
Translation using a Vector
T_{<a, b>}(x, y) = (x + a, y + b)
This notation shows that a translation T with vector <a, b> is applied to a point (x, y), resulting in the im...
4 more steps in this tutorial
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Challenging
A translation is defined by the vector <3k, k - 2>. This translation maps point P(1, 4) to P'(13, 6). What is the rule for this translation in coordinate notation?
A.(x, y) -> (x + 4, y + 2)
B.(x, y) -> (x + 12, y + 2)
C.(x, y) -> (x + 9, y + 1)
D.(x, y) -> (x + 13, y + 6)
Challenging
A translation maps the midpoint of the line segment with endpoints A(2, 9) and B(6, 1) to the point M'(7, 8). What is the component vector for this translation?
A.<3, 3>
B.<1, 7>
C.<4, 5>
D.<3, 4>
Challenging
A translation T maps the circle with equation (x - 4)² + (y + 1)² = 25 to the circle (x + 3)² + (y + 6)² = 25. What is the rule for translation T?
A.(x, y) -> (x + 7, y + 5)
B.(x, y) -> (x - 1, y - 7)
C.(x, y) -> (x - 7, y - 5)
D.(x, y) -> (x + 1, y + 7)
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