Question

1. which rule best represents the transformation applied to △abc to create △abc? a) (x, y)→(−x, y) b) (x, y)→(x - 9, y - 7) c) (x, y)→(x, −y) d) (x, y)→(x + 9, y - 7) 2) triangle lmn is plotted on a coordinate plane. it is then dilated by a factor of 3 using the origin as the center of dilation, and translated 5 units right and 3 units up. how is this sequence of transformations written in coordinate notation to correctly map triangle lmn to its image, triangle lmn? a) (x, y)→(3x + 5, 3y - 5) b) (x, y)→(3x - 3, 3y + 3) c) (x, y)→(3x + 5, 3y + 3) d) (x, y)→(3x + 3, 3y + 5)

Explanation:

Step1: Analyze transformation in question 1

Look at the change in coordinates of $\triangle ABC$ to $\triangle A'B'C'$. Point $A$ moves from left - hand side of y - axis to right - hand side and moves down, point $B$ also moves right and down, point $C$ moves right and down. The x - coordinate increases and y - coordinate decreases.

Step2: Check the rules

For a point $(x,y)$:

• Option a) $(x,y)\to(-x,y)$ is a reflection over the y - axis.
• Option b) $(x,y)\to(x - 9,y - 7)$ moves the point left 9 units and down 7 units.
• Option c) $(x,y)\to(x,-y)$ is a reflection over the x - axis.
• Option d) $(x,y)\to(x + 9,y - 7)$ moves the point right 9 units and down 7 units which is correct for the transformation of $\triangle ABC$ to $\triangle A'B'C'$.

Step3: Analyze transformation in question 2

Dilation by a factor of 3 about the origin multiplies the x and y coordinates by 3, so $(x,y)\to(3x,3y)$. Then translation 5 units right and 3 units up adds 5 to the x - coordinate and 3 to the y - coordinate. So the combined transformation is $(x,y)\to(3x+5,3y + 3)$.

Answer:

1. d) $(x,y)\to(x + 9,y - 7)$
2. c) $(x,y)\to(3x + 5,3y + 3)$