Question

find the coordinates of the image.
$r_{x=2}(u)=\square$ (type an ordered pair.)

Explanation:

Step1: Identify coordinates of U

From the graph, the coordinates of point \( U \) are \( (4, -5) \) (assuming the grid and position, x - coordinate is 4, y - coordinate is - 5).

Step2: Reflect over \( x = 2 \)

The formula for reflecting a point \( (x,y) \) over the vertical line \( x = a \) is \( (2a - x,y) \). Here, \( a = 2 \), \( x = 4 \), \( y=-5 \).
Substitute into the formula: \( 2\times2 - 4=-0 \)? Wait, no, \( 2\times2 - 4 = 0 \)? Wait, no, \( 2a - x=2\times2 - 4 = 0 \)? Wait, no, let's recalculate. If the original x - coordinate is \( x = 4 \), and the line of reflection is \( x = 2 \). The distance between \( x = 4 \) and \( x = 2 \) is \( 4 - 2=2 \). So the reflected x - coordinate is \( 2 - 2 = 0 \)? Wait, no, the distance from the point to the line is \( |x - a|=|4 - 2| = 2 \). So the reflected point is \( a - (x - a)=2a - x \). So \( 2\times2-4 = 0 \). The y - coordinate remains the same. So the reflected point \( R_{x = 2}(U) \) has coordinates \( (0,-5) \)? Wait, no, maybe I misread the coordinates of U. Let's re - examine the graph. Looking at the grid, if the x - axis and y - axis are as shown, let's find the correct coordinates of U. Let's assume each grid square is 1 unit. From the origin (0,0), moving right 4 units (x = 4) and down 5 units (y=-5), so U is (4, - 5). The line of reflection is \( x = 2 \). The formula for reflection over \( x=a \) is \( (2a - x,y) \). So \( a = 2 \), \( x = 4 \), so \( 2\times2-4=0 \), \( y=-5 \). So the image is \( (0,-5) \)? Wait, no, maybe I made a mistake in the coordinates of U. Wait, maybe the y - coordinate is positive? Wait, the graph has y - axis with positive upwards? Wait, the original graph: the y - axis has 10 at the top and - 10 at the bottom. Wait, maybe the coordinates of U are (4,5)? Wait, the dot is below the x - axis? Wait, no, the y - axis: if the arrow is upwards, then the lower part is negative. Wait, maybe I messed up the y - coordinate. Let's re - check. If the point U is above or below? Wait, the graph shows the y - axis with 10 at the top (positive) and - 10 at the bottom (negative). The point U is below the x - axis, so y is negative. Let's re - calculate the reflection. The distance between \( x = 4 \) and \( x = 2 \) is 2 units to the right of the line \( x = 2 \). So the reflected point should be 2 units to the left of \( x = 2 \), so \( 2-2 = 0 \). So the x - coordinate is 0, y - coordinate is - 5. So \( R_{x = 2}(U)=(0,-5) \)? Wait, no, maybe the coordinates of U are (4,5). Let's assume that the y - coordinate is positive. If U is (4,5), then reflecting over \( x = 2 \): \( 2\times2 - 4=0 \), so the image is (0,5). But the original problem's box is for an ordered pair. Wait, maybe I misread the coordinates. Let's look again. The graph: the x - axis (horizontal) and y - axis (vertical). Let's count the grid. From the x = 2 line, the point U is 2 units to the right (x = 4) and let's say y = 5 (above the x - axis). Then reflecting over x = 2: the distance from x = 4 to x = 2 is 2, so the reflected x is 2 - 2 = 0, y remains 5. So \( R_{x = 2}(U)=(0,5) \)? Wait, the user's graph: the point U is marked, let's assume the correct coordinates of U are (4, - 5) or (4,5). Wait, maybe the initial coordinates of U are (4,5). Let's re - do the reflection. If U is (4,5), then reflection over \( x = 2 \): \( x'=2\times2 - 4 = 0 \), \( y' = 5 \). So the image is (0,5). But maybe I made a mistake in the sign of y. Let's check the graph again. The y - axis: the top is 10 (positive), bottom is - 10 (negative). The point U is below the x - axis? No, looking at the dot, it's above? Wait, the dot is in the fourth quadrant? No, x positive, y negative? Or x positive, y positive? Wait, the grid: the x - axis (horizontal) with positive to the right, y - axis (vertical) with positive upwards. The point U is at (4, - 5) (x = 4, y=-5). Then reflection over x = 2: x'=2*2 - 4 = 0, y'=-5. So the image is (0, - 5). But maybe the correct coordinates are (4,5). Let's assume that the y - coordinate is positive. Then x'=0, y' = 5. But the problem says "Type an ordered pair". Let's re - confirm the formula. The reflection of a point (x,y) over x = a is (2a - x,y). So if a = 2, x = 4, then 2*2 - 4 = 0, y remains the same. So regardless of y (as long as we have the correct x), the x - coordinate of the image is 0. So if U is (4, - 5), the image is (0, - 5). If U is (4,5), the image is (0,5). Let's check the graph again. The point U: looking at the grid, if we count the x - axis: from x = 2, moving right 2 units (x = 4), and y - axis: moving down 5 units (y=-5). So U is (4, - 5). Then reflection over x = 2: (0, - 5).

Answer:

\((0, - 5)\)