Question

a figure is translated using the mapping (x, y) → (x + a, y + b). if the value of a is negative and the value of b is negative, which best describes the translation? the figure moves left and up. the figure moves left and down. the figure moves right and up. the figure moves right and down.

Explanation:

Step1: Analyze x - direction translation

In the mapping \((x,y)\to(x + a,y + b)\), for the x - coordinate: when \(a>0\), the figure moves right (since we add a positive number to \(x\), which increases the x - value, moving the point to the right on the coordinate plane); when \(a < 0\), \(x+a=x-|a|\), so the x - coordinate decreases, which means the figure moves left.

Step2: Analyze y - direction translation

For the y - coordinate: when \(b>0\), the figure moves up (since we add a positive number to \(y\), increasing the y - value, moving the point up); when \(b < 0\), \(y + b=y-|b|\), so the y - coordinate decreases, which means the figure moves down.

Step3: Combine the directions

Given that \(a\) is negative and \(b\) is negative. From Step 1, when \(a\) is negative, the figure moves left in the x - direction. From Step 2, when \(b\) is negative, the figure moves down in the y - direction. So the figure moves left and down.

Answer:

The figure moves left and down. (Corresponding to the option: The figure moves left and down.)