Question

reflect the image over the y - axis and then translate it 4 to the right and down 3. which quadrants will the image be located in? quadrant iii and iv, quadrant ii and iii, quadrant i and iv, quadrant i and ii

Explanation:

Step1: Analyze y - axis reflection rule

When reflecting a point $(x,y)$ over the $y$-axis, the transformation is $(-x,y)$. If the original image is in some quadrants, after reflection over the $y$-axis, the quadrants change. For example, points in Quadrant I ($x>0,y > 0$) go to Quadrant II ($x<0,y>0$) and vice - versa, and points in Quadrant IV ($x > 0,y<0$) go to Quadrant III ($x<0,y < 0$) and vice - versa.

Step2: Analyze right - translation rule

Translating a point $(x,y)$ 4 units to the right changes the $x$ - coordinate: $(x,y)\to(x + 4,y)$. Translating 3 units down changes the $y$ - coordinate: $(x,y)\to(x,y-3)$. After reflection over the $y$ - axis, assume a point $(-x,y)$ from the reflection, then after translation 4 units to the right and 3 units down, it becomes $(-x + 4,y-3)$.
Let's assume the original image is in Quadrant I. After reflection over the $y$ - axis, it goes to Quadrant II. Then, after translating 4 units to the right (increasing the $x$ - value) and 3 units down (decreasing the $y$ - value), the new image will be in Quadrant I and IV.

Answer:

Quadrant I and IV