Question

multiple-choice question what notation do you use to show the new translated points? prime notation algebraic notation same letters as pre-image rewatch
multiple-choice question when reflecting from y=x, we count vertically first horizontally first rewatch

Explanation:

First Question (Notation for Translated Points)

In geometry (a subfield of Mathematics), when we have a pre - image (original) point and its image after a transformation like translation, we use prime notation. For example, if we have a point \( E \), after translation, its image is \( E' \). The other options: algebraic notation is more about variables and equations, and "same letters as pre - image" doesn't show the new (transformed) point clearly. So prime notation is used for the new translated points.

When reflecting a point \((x,y)\) over the line \( y=x \), the rule is that the coordinates of the image are \((y,x)\). To find the reflection, we can think of counting the distance from the point to the line \( y = x \) both vertically and horizontally, but the order doesn't matter in terms of "first" because the reflection over \( y=x \) swaps the \( x \) and \( y \) coordinates. However, if we consider the process of moving from the original point to the line and then to the image, we can count either vertically or horizontally first, but in the context of reflection over \( y = x \), the key is that the reflection swaps \( x \) and \( y \). But if we have to choose between vertically first or horizontally first, when reflecting over \( y=x \), we can count vertically first (the distance in the \( y \) - direction to the line and then move the same distance in the \( x \) - direction, or vice - versa). But more accurately, for a point \((a,b)\), the reflection over \( y = x \) is \((b,a)\). If we are counting the steps, we can move vertically (change in \( y \)) and then horizontally (change in \( x \)) or the other way, but the standard way is that the reflection over \( y=x \) is a swap. However, if the options are vertically first or horizontally first, and we consider the distance from the point to the line \( y = x \), we can count vertically first (the vertical distance from the point to the line \( y=x \) and then move the same horizontal distance, or vice - versa). But in most cases, when reflecting over \( y=x \), the order of vertical or horizontal counting is not a strict "first" in a way that one is more correct, but if we have to choose from the given options (vertically first or horizontally first), and considering the reflection rule, we can say that we can count vertically first (for example, for a point above the line \( y=x \), we move down (vertically) to the line and then right (horizontally) to the image, or similar for other positions).

Answer:

prime notation

Second Question (Reflection over \( y = x \))