reflect the figure over the line y = 3. plot all of the points of the reflected figure. you may click a…

Explanation:

Step1: Recall Reflection Over Horizontal Line

To reflect a point ((x, y)) over the horizontal line (y = k), the formula for the reflected point ((x, y')) is (y' = 2k - y). Here, (k = 3), so the formula becomes (y' = 2(3)-y=6 - y).

Step2: Identify Original Points

First, we need to find the coordinates of the vertices of the original triangle. Let's assume the original points (from the graph, estimating coordinates):

• Let's say the first point is (A(1, 6)) (since it's on (x = 1), (y = 6)), the second point (B(3, 6)) ( (x = 3), (y = 6) ), and the third point (C(6, 4)) ( (x = 6), (y = 4) ). Wait, maybe better to check the original graph: looking at the pink triangle, let's re - identify: Wait, the original triangle has vertices: Let's see the grid. Let's take the left - most point: (x = 1), (y = 6) (so ((1,6))), then a point at (x = 3), (y = 6) ( ((3,6)) ), and the third point at (x = 6), (y = 4) ( ((6,4)) ), and another? Wait, maybe it's a triangle with three vertices: Let's correct, maybe the three vertices are ((1,6)), ((3,6)), and ((6,4)) (or maybe ((6,8)) and ((6,4))? Wait, the top vertex seems to be at (x = 7), (y = 9)? Wait, maybe I mis - estimated. Let's do it properly. Let's assume the original vertices are: Let’s denote the original points as (P(x_1,y_1)), (Q(x_2,y_2)), (R(x_3,y_3)). Suppose from the graph:
• Point 1: ((1,6)) ( (x = 1), (y = 6) )
• Point 2: ((3,6)) ( (x = 3), (y = 6) )
• Point 3: ((6,4)) ( (x = 6), (y = 4) )
• Wait, maybe a fourth point? No, it's a triangle. Wait, maybe the three vertices are ((1,6)), ((3,6)), and ((6,8)) (top) and ((6,4))? No, the pink triangle: Let's look at the y - axis. The line (y = 3) is horizontal. Let's take each vertex:

1. For a point ((x,y)), apply (y'=6 - y):

• Let's take the first vertex: Let's say the left vertex is ((1,6)). Then reflected (y'=6 - 6 = 0), so the reflected point is ((1,0)).
• The middle vertex (same y as the first) ((3,6)): reflected (y'=6 - 6 = 0), so ((3,0)).
• The third vertex: Let's say the bottom - right vertex is ((6,4)): reflected (y'=6 - 4 = 2), so ((6,2)). Wait, no, maybe the top - right vertex is ((6,8)): then (y'=6 - 8=-2), so ((6, - 2)), and the bottom - right is ((6,4)): (y'=6 - 4 = 2). Wait, maybe the original triangle has vertices ((1,6)), ((3,6)), ((6,8)), and ((6,4))? No, it's a triangle, so three vertices. Let's re - examine the graph. The pink triangle: one vertex at ((1,6)), one at ((3,6)), and one at ((6,8)) (top) and ((6,4)) (bottom)? No, it's a triangle, so three vertices: ((1,6)), ((3,6)), ((6,8)) (top) and ((6,4)) is a mistake. Wait, the vertical side: from ((6,4)) to ((6,8)), and the horizontal side from ((1,6)) to ((3,6)), and the slant side from ((3,6)) to ((6,8)) and from ((1,6)) to ((6,8))? No, maybe it's a triangle with vertices ((1,6)), ((3,6)), and ((6,8)).

Now, apply the reflection formula (y' = 6 - y) to each vertex:

• For ((1,6)): (y'=6 - 6 = 0), so reflected point is ((1,0)).
• For ((3,6)): (y'=6 - 6 = 0), so reflected point is ((3,0)).
• For ((6,8)): (y'=6 - 8=-2), so reflected point is ((6, - 2)).
• Wait, but there is also a point at ((6,4)) (the bottom of the vertical side). For ((6,4)): (y'=6 - 4 = 2), so reflected point is ((6,2)). Wait, maybe the original figure is a quadrilateral? But the problem says "the figure", maybe a triangle or a quadrilateral. Anyway, the key is to apply (y'=6 - y) to each vertex.

Step3: Plot Reflected Points

Once we have the reflected coordinates for each vertex using (y' = 6 - y), we plot these points ((x,6 - y)) on the coordinate plane. For example, if a vertex is ((x,y)), the reflected vertex is ((x,6 - y)). So we calculate the new y - coordinate for each x - coordinate of the original vertex and then plot the points.

Answer:

To reflect the figure over (y = 3), for each vertex ((x,y)) of the original figure, the reflected vertex is ((x,6 - y)). Calculate the new (y) - coordinates using (y'=6 - y) and plot the points ((x,6 - y)) on the coordinate plane. (The actual plotting involves marking the points with the new coordinates, e.g., if original points are ((1,6)), ((3,6)), ((6,8)), ((6,4)), the reflected points are ((1,0)), ((3,0)), ((6, - 2)), ((6,2)) and then connecting them to form the reflected figure.)