To find the coordinates after reflecting over the line $x = 1$, follow these steps:

1. Identify the original coordinates of the point, let’s call it $(x, y)$.

2. Calculate the distance of $x$-coordinate from the line $x = 1$. This distance is $|x - 1|$.

3. Determine the reflected $x$-coordinate by moving the same distance to the opposite side of the line $x = 1$.

   • If $x > 1$, the new $x$-coordinate will be $x - 2 \times |x - 1|$, which simplifies to $2 - x$.
   • If $x < 1$, the new $x$-coordinate will be $x + 2 \times |x - 1|$, which also simplifies to $2 - x$.

Thus, for any point $(x, y)$, the coordinates after reflecting over the line $x = 1$ are:

$(2 - x, y)$

Example

If the original point is $(3, 4)$:

• The distance from $x = 1$ is $3 - 1 = 2$.
• Moving the same distance to the other side, the new $x$-coordinate is $1 - 2 = -1$.

So, the reflected coordinates of $(3, 4)$ over $x = 1$ are $(-1, 4)$.

Would you like further details or have any questions about this reflection process?

Here are some related questions for practice:

1. What are the coordinates of $(5, -3)$ after reflecting over $x = 1$?
2. How would the coordinates of $(2, 7)$ change if reflected over $x = -1$?
3. Can you generalize the formula for reflections over any vertical line $x = k$?
4. How would the transformation differ for a reflection over the y-axis?
5. What are the properties of reflections over non-horizontal or non-vertical lines, such as $y = x$?

Tip: To confirm your reflection results, plot the points on a coordinate plane to visualize the distance symmetry relative to the line.