Question

graph each figure and its image under the given reflection.

1. rectangle abcd with vertices a(-5, 2), b(1, 2), c(1, -1), and d(-5, -1) in the line y = -2
2. △fgh with vertices f(-3, 2), g(-4, -1), and h(-6, -1) in the line y = x
3. △stu with vertices s(-3, -2), t(-2, 3), and u(2, 2) in the line y = x
4. △cde with vertices c(-3, 6), d(-1, 1), and e(3, 5) in the line y = x

Explanation:

Step1: Reflect over \( y = -2 \) (Problem 10)

For a point \((x, y)\) reflected over \( y = k \), the new \( y \)-coordinate is \( 2k - y \). Here, \( k = -2 \), so \( y' = 2(-2) - y = -4 - y \).

• \( A(-5, 2) \): \( y' = -4 - 2 = -6 \) → \( A'(-5, -6) \)
• \( B(1, 2) \): \( y' = -4 - 2 = -6 \) → \( B'(1, -6) \)
• \( C(1, -1) \): \( y' = -4 - (-1) = -3 \) → \( C'(1, -3) \)
• \( D(-5, -1) \): \( y' = -4 - (-1) = -3 \) → \( D'(-5, -3) \)

Step2: Reflect over \( y = x \) (Problem 11)

For a point \((x, y)\) reflected over \( y = x \), swap \( x \) and \( y \): \((y, x)\).

• \( F(-3, 2) \) → \( F'(2, -3) \)
• \( G(-4, -1) \) → \( G'(-1, -4) \)
• \( H(-6, -1) \) → \( H'(-1, -6) \)

Step3: Reflect over \( y = x \) (Problem 12)

Swap \( x \) and \( y \) for each vertex:

• \( S(-3, -2) \) → \( S'(-2, -3) \)
• \( T(-2, 3) \) → \( T'(3, -2) \)
• \( U(2, 2) \) → \( U'(2, 2) \) (since \( x = y \))

Step4: Reflect over \( y = x \) (Problem 13)

Swap \( x \) and \( y \):

• \( C(-3, 6) \) → \( C'(6, -3) \)
• \( D(-1, 1) \) → \( D'(1, -1) \)
• \( E(3, 5) \) → \( E'(5, 3) \)

Step5: Graph the Figures

Plot the original vertices and their reflected images on the respective coordinate planes. Connect the vertices to form the original figure and its image.

Answer:

For each problem, the reflected vertices are:

1. \( A'(-5, -6) \), \( B'(1, -6) \), \( C'(1, -3) \), \( D'(-5, -3) \)
2. \( F'(2, -3) \), \( G'(-1, -4) \), \( H'(-1, -6) \)
3. \( S'(-2, -3) \), \( T'(3, -2) \), \( U'(2, 2) \)
4. \( C'(6, -3) \), \( D'(1, -1) \), \( E'(5, 3) \)

(Graphing involves plotting these points and connecting them to show the original and reflected figures.)