Question

transformations test (topic 4)
the point (1,8) was translated in the coordinate plane and then reflected across the x-axis. its final image had the coordinates (4,-3). which of the following best describes the translation that occurred?
a shift of 3 units to the right and 11 units down
a shift of 3 units to the right and 5 units down
a shift of 8 units left and 2 units down
a shift of 2 units left and 4 units up

Explanation:

Step1: Reverse the reflection

To reverse the reflection across the x - axis, we use the rule that reflecting a point \((x,y)\) across the x - axis gives \((x, -y)\). So, if the final point after reflection is \((4,-3)\), the point before reflection (after translation) should be \((4,3)\) (since to reverse the reflection, we change the sign of the y - coordinate: if \(y'=-y\), then \(y=-y'\), so for \(y' = - 3\), \(y = 3\)).

Step2: Find the translation vector

Let the original point be \((x_1,y_1)=(1,8)\) and the point after translation (before reflection) be \((x_2,y_2)=(4,3)\). The horizontal translation \(a=x_2 - x_1\) and the vertical translation \(b=y_2 - y_1\).

For the x - coordinate: \(a = 4-1=3\) (a positive value means a shift to the right).

For the y - coordinate: \(b=3 - 8=-5\) (a negative value means a shift down, and the magnitude is 5 units).

So the translation is a shift of 3 units to the right and 5 units down.

Answer:

a shift of 3 units to the right and 5 units down