Mathematics
Grade 6
15 min
Translations: graph the image
Translations: graph the image
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define translation as a geometric transformation.
Identify the pre-image and image of a figure after a translation.
Describe a translation using directional words and the number of units.
Apply translation rules to find the new coordinates of a point or figure.
Graph the image of a 2D figure after a given translation on a coordinate plane.
Distinguish between horizontal and vertical components of a translation.
Have you ever slid a book across a table without turning or flipping it? 📚 That's exactly what a mathematical translation is!
In this lesson, you'll discover how shapes move on a grid called a coordinate plane. We'll learn to slide shapes (translate them) to new positions and draw their 'after' picture, which is called th...
2
Key Concepts & Vocabulary
TermDefinitionExample
TransformationA change in the position, size, or orientation of a geometric figure.Sliding a square, flipping a triangle, or making a circle bigger are all transformations.
TranslationA type of transformation that slides a figure from one position to another without turning, flipping, or changing its size. Every point of the figure moves the same distance in the same direction.Moving a triangle 3 units to the right and 2 units up.
Pre-imageThe original figure before any transformation is applied.If you start with a square, that square is the pre-image.
ImageThe new figure after a transformation has been applied. It is often denoted with a prime symbol (e.g., A' is the image of point A).After sliding the square, its new position is the image (e.g., Square A'...
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Core Formulas
Horizontal Translation Rule
For a point $(x, y)$:
- To translate right by $h$ units: $(x, y) \rightarrow (x+h, y)$
- To translate left by $h$ units: $(x, y) \rightarrow (x-h, y)$
When a figure slides horizontally (left or right), only the x-coordinate changes. Add to x for a rightward shift, subtract from x for a leftward shift. The y-coordinate remains the same.
Vertical Translation Rule
For a point $(x, y)$:
- To translate up by $k$ units: $(x, y) \rightarrow (x, y+k)$
- To translate down by $k$ units: $(x, y) \rightarrow (x, y-k)$
When a figure slides vertically (up or down), only the y-coordinate changes. Add to y for an upward shift, subtract from y for a downward shift. The x-coordinate remains the same.
Combined Translation Rule
For a point $(x, y)$ translated $...
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Challenging
Point F is at (3, 5). It is translated 2 units left and 3 units down to get F'. Then, F' is translated 4 units right and 1 unit up to get F''. What are the final coordinates of F''?
A.(1, 2)
B.(5, 3)
C.(9, 9)
D.(-3, 1)
Challenging
A quadrilateral has vertices L(1,1), M(3,1), N(3,3), and P(?,?). After a translation of 4 units right and 2 units down, the image has vertices L'(5,-1), M'(7,-1), N'(7,1), and P'(3,1). What were the original coordinates of vertex P?
A.(-1, 3)
B.(7, -1)
C.(1, 5)
D.(-1, -1)
Challenging
A student translates point G(2, -1) using the rule '3 units right, 4 units up'. They graph the image G' at (6, 2). Which statement best explains their error?
A.They confused the x and y shifts; they moved 4 units right and 3 units up.
B.They moved left instead of right.
C.They moved down instead of up.
D.They miscounted the units on the graph for both directions.
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