12. The rule \( T_{\langle 4,-1\rangle} \) is used for point \( (2,-7) \). Where is the translated point in the coordinate system? \( \begin{array}{ll}\text { (A) Quadrant I } & \text { (C) Quadrant III } \\ \text { (B) Quadrant II } & \text { (D) Quadrant IV }\end{array} \)

To find the translated point using the rule \( T_{\langle 4,-1\rangle} \), you need to add the translation vector to the original point \( (2,-7) \). This means you will sum the x-coordinates and y-coordinates separately: \[ \text{New } x = 2 + 4 = 6 \] \[ \text{New } y = -7 + (-1) = -8 \] Thus, the translated point is \( (6,-8) \). Since the x-coordinate is positive and the y-coordinate is negative, this point lies in Quadrant IV. The translated point \( (6, -8) \) is located in Quadrant IV, where x values are positive and y values are negative, making it a lower right corner of the coordinate system. This quadrant is known for hosting points where the joys of mathematics meet the challenges of real-world applications like navigation or optimizing business analytics, always keeping a twinkle of adventure in its corner!