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) → (-4 + p, 7 + v)
• (x, y) → (-4 + p, 7 - v)
• (x, y) → (-4 - p, 7 - v)
• (x, y) → (-4 - p, 7 + v)

Explanation:

Step1: Analyze horizontal translation

Moving \( p \) units right means adding \( p \) to the \( x \)-coordinate. The original \( x \)-coordinate is \( -4 \), so new \( x \)-coordinate is \( -4 + p \).

Step2: Analyze vertical translation

Moving \( v \) units down means subtracting \( v \) from the \( y \)-coordinate. The original \( y \)-coordinate is \( 7 \), so new \( y \)-coordinate is \( 7 - v \).

Step3: Determine the translation rule

Combining the horizontal and vertical changes, the translation rule for the center \((x,y)\) (here the center is \((-4,7)\), but the rule is general for any point \((x,y)\) being translated) is \((x, y) \to (-4 + p, 7 - v)\) when applying the translation to the center \((-4,7)\) (treating the center as the point \((x,y)\) for the translation rule). Wait, actually, the center is a point \((h,k)=(-4,7)\). When translating a point \((h,k)\) \( p \) units right and \( v \) units down, the new point is \((h + p, k - v)\). So substituting \( h=-4 \), \( k = 7 \), we get \((-4 + p, 7 - v)\). So the correct rule is \((x, y) \to (-4 + p, 7 - v)\).

Answer:

The correct option is the second one: \(\boldsymbol{(x, y) \to (-4 + p, 7 - v)}\) (the option with the circle selected next to \((x, y) \to (-4 + p, 7 - v)\))