Question

1. a circle is graphed on a coordinate grid with its center at (-4, 7). the circle will be translated p units to the right and v units down. which rule describes the center of the new circle after this translation?

options:

• $(x, y) \to (-4 + p, 7 + v)$
• $(x, y) \to (-4 + p, 7 - v)$
• $(x, y) \to (-4 - p, 7 - v)$
• $(x, y) \to (-4 - p, 7 + v)$

Explanation:

Step1: Recall translation rules

In coordinate geometry, translating a point \((x,y)\) \(h\) units right (or left) and \(k\) units up (or down) follows the rule: \((x,y)\to(x + h,y + k)\) (right/up) or \((x,y)\to(x + h,y - k)\) (right/down), \((x,y)\to(x - h,y + k)\) (left/up), \((x,y)\to(x - h,y - k)\) (left/down). Here, right translation affects the \(x\)-coordinate (add \(p\)) and down translation affects the \(y\)-coordinate (subtract \(v\)).

Step2: Apply to the center \((-4,7)\)

The original center is \((-4,7)\). Translating \(p\) units right: new \(x\)-coordinate is \(-4 + p\). Translating \(v\) units down: new \(y\)-coordinate is \(7 - v\). So the translation rule for the center (and any point on the circle) is \((x,y)\to(-4 + p,7 - v)\).

Answer:

\(\boldsymbol{(x, y) \to (-4 + p, 7 - v)}\) (the option with this rule)