Finding Coordinates And Image Points Under Translation In Math

Hey guys! Let's dive into a super interesting topic in coordinate geometry: translations. If you're scratching your head about how points move around on a graph when we shift them, you're in the right place. We're going to break down a problem step-by-step so you can see exactly how it works. Imagine you're moving a piece on a chessboard – that’s kind of what a translation is in math, but on a coordinate plane. Let's get started and make this crystal clear!

Problem Overview: Unveiling Point A's Journey

We’ve got a point, A, sitting somewhere on our coordinate plane with coordinates (x, y). Now, this point undergoes a translation, which we'll call T. Think of T as a set of instructions telling A exactly how to move. In this case, T is represented by a column matrix: T = egin{pmatrix} -5 \ 4 end{pmatrix}. This means we're shifting point A 5 units to the left (because of the -5) and 4 units upwards (because of the 4). After this move, A ends up at a new spot, A', with coordinates (-2, 5). Our mission? First, we need to figure out the original coordinates of point A. Then, we're going to take point A and apply a different translation, T = egin{pmatrix} 2 \ 7

end{pmatrix}, to see where it lands this time. Sounds like a plan? Let’s jump into it!

Decoding the Translation: Finding the Original Point A

Okay, so the first part of our quest is to find the original coordinates of point A. We know that a translation shifts a point without changing its orientation. It’s like sliding a shape across a table – it’s still the same shape, just in a new location. Mathematically, a translation works by adding the translation vector to the original point's coordinates. So, if A(x, y) is translated by T = egin{pmatrix} -5 \ 4

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