Question

which of the following represents a translation? show your work here (x,y)→(x + 3,y + 4) (x,y)→(\frac{1}{2}x,\frac{1}{2}y) (x,y)→(y,-x) (x,y)→(-x,y)

Explanation:

Step1: Recall translation definition

In a translation in the coordinate - plane, every point of a figure is moved the same distance in the same direction. A translation of a point $(x,y)$ is of the form $(x + a,y + b)$ where $a$ and $b$ are real numbers.

Step2: Analyze each option

• For the transformation $(x,y)\to(x + 3,y + 4)$, it moves the point $x$ - coordinate 3 units to the right (if $a=3>0$) and $y$ - coordinate 4 units up (if $b = 4>0$). This is a translation.
• For the transformation $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$, it is a dilation (scaling) since the coordinates are multiplied by a non - one factor $\frac{1}{2}$.
• For the transformation $(x,y)\to(y,-x)$, it is a rotation (a 90 - degree counter - clockwise rotation about the origin).
• For the transformation $(x,y)\to(-x,y)$, it is a reflection across the $y$ - axis.

Answer:

$(x,y)\to(x + 3,y + 4)$