Translations: find the coordinates

Learning Objectives Define a translation and identify its horizontal and vertical components. Apply a translation rule in the form (x, y) -> (x + a, y + b) to find the coordinates of an image point. Determine the coordinates of the vertices of a translated polygon on the coordinate plane. Work backwards to find the coordinates of a pre-image given the image and the translation rule. Derive the translation rule given the coordinates of a pre-image and its corresponding image. Interpret and use vector notation <a, b> to describe a translation. Ever played a video game where your character slides across the screen to a new position? 🎮 That's a translation! Let's learn how the game calculates that new spot. This tutorial focuses on translations, which are a...

TermDefinitionExample TranslationA transformation that moves every point of a figure or space by the same distance in the same direction. It is often called a 'slide'.If point P(2, 1) is translated 3 units right and 4 units up, its new position is P'(5, 5). Pre-imageThe original figure before a transformation is applied.In the transformation of triangle ABC to triangle A'B'C', triangle ABC is the pre-image. ImageThe resulting figure after a transformation has been applied to the pre-image.If point P(3, 4) is translated to P'(5, 6), then P' is the image of P. Coordinate RuleAn algebraic rule that describes the transformation of a point's coordinates (x, y) to its image's coordinates (x', y').The rule T(x, y) = (x - 2, y + 5) descr...

Coordinate Rule for Translation P(x, y) \rightarrow P'(x+a, y+b) To find the image coordinates (x', y'), add the horizontal shift 'a' to the original x-coordinate and the vertical shift 'b' to the original y-coordinate. A positive 'a' means a shift right, negative 'a' means left. A positive 'b' means a shift up, negative 'b' means down. Finding the Pre-image (Working Backwards) P(x, y) = P'(x' - a, y' - b) To find the original pre-image coordinates (x, y), subtract the horizontal shift 'a' from the image's x'-coordinate and the vertical shift 'b' from the image's y'-coordinate. Deriving the Translation Rule a = x' - x \quad \text{and} \...