# Question Video: Finding the Image of a Shape After a Given Transformation Mathematics

Which of the following represents the image of △𝐴𝐵𝐶, where 𝐴(1, 3), 𝐵(3, 3), and 𝐶(3, 7), after a transformation (𝑥, 𝑦) ⟶ (𝑥, −𝑦)? [A] 𝐴′(−1, 3), 𝐵′(−3, 3), and 𝐶′(−3, 7) [B] 𝐴′(−1, −3), 𝐵′(−3, −3), and 𝐶′(−3, −7) [C] 𝐴′(1, −3), 𝐵′(3, −3), and 𝐶′(3, −7) [D] 𝐴′(3, 1), 𝐵′(3, 3), and 𝐶′(7, 3)

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### Video Transcript

Which of the following represents the image of triangle 𝐴𝐵𝐶, where 𝐴 has coordinates one, three; 𝐵 has coordinates three, three; and 𝐶 has coordinates three, seven, after a transformation 𝑥, 𝑦 is mapped to 𝑥, negative 𝑦? (a) 𝐴 prime negative one, three; 𝐵 prime negative three, three; and 𝐶 prime negative three, seven. (b) 𝐴 prime negative one, negative three; 𝐵 prime negative three, negative three; and 𝐶 prime negative three, negative seven. (c) 𝐴 prime one, negative three; 𝐵 prime three, negative three; and 𝐶 prime three, negative seven. Or (d) 𝐴 prime three, one; 𝐵 prime three, three; and 𝐶 prime seven, three.

We’re given the rule that describes this transformation. Every point 𝑥, 𝑦 is mapped to the point 𝑥, negative 𝑦. In other words, the 𝑥-coordinate stays the same, and the 𝑦-coordinate changes sign or is multiplied by negative one. We can apply this mapping to each vertex of triangle 𝐴𝐵𝐶.

The point 𝐴 with coordinates one, three is mapped to the point 𝐴 prime with coordinates one, negative three. The point 𝐵 with coordinates three, three is mapped to three, negative three. And the point 𝐶 with coordinates three, seven is mapped to the point three, negative seven. Looking carefully at the four options given, we can see that this set of coordinates is option (c).

We can also visualize the effect of this transformation graphically. Here, we have plotted triangle 𝐴𝐵𝐶 on a coordinate grid. If we also plot triangle 𝐴 prime 𝐵 prime 𝐶 prime, we can see that the two triangles appear to be reflections of one another. The mirror line is the 𝑥-axis. So, this tells us that we can represent the transformation of reflection in the 𝑥-axis as the mapping 𝑥, 𝑦 is mapped to 𝑥, negative 𝑦, although it’s beyond the current scope to recall this.