Question

c(-3, -1) b(-4, -3) a(0, -5) c(2, 1) a translation 5 units left a reflection across x = 0, then a translation 5 units left a reflection across y = 0, then a translation 5 units left a rotation 90° about the origin

Explanation:

1. Analyze the coordinates of point \( C(2,1) \) and its image \( C'(-3,-1) \), \( B \) (not labeled but \( B'(-4,-3) \)) and \( A \) (not labeled but \( A'(0,-5) \)).
2. First, check reflection across \( y = 0 \) (x - axis): Reflecting \( C(2,1) \) over \( y = 0 \) gives \( (2,-1) \). Then translate 5 units left: \( 2 - 5=-3 \), \( -1\) stays, so \( (-3,-1) \), which matches \( C' \). Check other points (e.g., if original \( B \) had coordinates, reflecting over \( y = 0 \) and translating left 5 would match \( B' \), same for \( A \)).
3. Option 1: Translation 5 units left on \( C(2,1) \) would be \( 2 - 5=-3 \), but \( y \) - coordinate would stay 1, not - 1. So wrong.
4. Option 2: Reflection across \( x = 0 \) (y - axis) on \( C(2,1) \) gives \( (-2,1) \), translating 5 left gives \( -2-5 = -7

eq - 3 \). Wrong.

1. Option 4: Rotation \( 90^{\circ} \) about origin: For \( (x,y)\), rotation \( 90^{\circ} \) counter - clockwise is \( (-y,x) \), clockwise is \( (y,-x) \). For \( C(2,1) \), counter - clockwise rotation gives \( (-1,2) \), clockwise gives \( (1,-2) \), neither is \( (-3,-1) \). Wrong.

Answer:

C. a reflection across \( y = 0 \), then a translation 5 units left