Question

reflections of shapes
graph the image of the figures using the transformation given.

1. reflection across the x - axis
2. reflection across y = 5
3. reflection across y = 1
4. reflection across the x - axis
5. reflection across the x - axis

t(2, 2), c(2, 5), z(5, 4), v(5, 0)

1. reflection across y = -2

h(-1, -5), m(-1, -4), r(1, -2), c(3, -3)

Explanation:

Step1: Recall Reflection Rules

For reflection across the \( x \)-axis: The rule is \( (x, y) \to (x, -y) \). For reflection across a horizontal line \( y = k \): The rule is \( (x, y) \to (x, 2k - y) \).

Step2: Solve Problem 5 (Reflection across \( x \)-axis)

Given points: \( T(2, 2) \), \( C(2, 5) \), \( Z(5, 4) \), \( V(5, 0) \).

• For \( T(2, 2) \): Apply \( (x, y) \to (x, -y) \), so \( T'(2, -2) \).
• For \( C(2, 5) \): Apply \( (x, y) \to (x, -y) \), so \( C'(2, -5) \).
• For \( Z(5, 4) \): Apply \( (x, y) \to (x, -y) \), so \( Z'(5, -4) \).
• For \( V(5, 0) \): Apply \( (x, y) \to (x, -y) \), so \( V'(5, 0) \) (since \( -0 = 0 \)).

Plot these new points \( T'(2, -2) \), \( C'(2, -5) \), \( Z'(5, -4) \), \( V'(5, 0) \) and connect them to get the reflected figure.

Step3: Solve Problem 6 (Reflection across \( y = -2 \))

Given points: \( H(-1, -5) \), \( M(-1, -4) \), \( R(1, -2) \), \( C(3, -3) \).
Use the rule \( (x, y) \to (x, 2(-2) - y) = (x, -4 - y) \).

• For \( H(-1, -5) \): \( -4 - (-5) = -4 + 5 = 1 \), so \( H'(-1, 1) \).
• For \( M(-1, -4) \): \( -4 - (-4) = -4 + 4 = 0 \), so \( M'(-1, 0) \).
• For \( R(1, -2) \): \( -4 - (-2) = -4 + 2 = -2 \), so \( R'(1, -2) \) (since \( -4 - (-2) = -2 \), and the point is on the line \( y = -2 \), so it reflects to itself).
• For \( C(3, -3) \): \( -4 - (-3) = -4 + 3 = -1 \), so \( C'(3, -1) \).

Plot these new points \( H'(-1, 1) \), \( M'(-1, 0) \), \( R'(1, -2) \), \( C'(3, -1) \) and connect them to get the reflected figure.

(Note: For the graph - based problems 1 - 4, identify the coordinates of each vertex of the figure, apply the reflection rule (across \( x \)-axis: \( (x,y)\to(x, - y) \); across \( y = k \): \( (x,y)\to(x, 2k - y) \) for \( y = 1 \) or \( y = 5 \)), find the reflected vertices, and plot them to draw the reflected image.)

Answer:

For problem 5, the reflected points are \( T'(2, -2) \), \( C'(2, -5) \), \( Z'(5, -4) \), \( V'(5, 0) \). For problem 6, the reflected points are \( H'(-1, 1) \), \( M'(-1, 0) \), \( R'(1, -2) \), \( C'(3, -1) \). (Graphs are drawn by plotting these points and connecting them as per the original figure's shape.)