Question

for 9 and 10, reflect each figure over the x-axis and then over the y-axis.
9.
10.

1. reflect the figure over the y-axis and then over the line y = -x.
2. reflect the figure over the x-axis and then over the line y = x.
3. draw a glide reflection using the translation <x+0, y-4> and the line of reflection x = 1.
4. draw a glide reflection using the translation <x-3, y=0> and the line of reflection y = -1.

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p.9

Explanation:

Step1: Analyze the problem (Question 9)

We have a line segment \( \overline{LM} \) on a coordinate grid. First, we need to reflect it over the \( x \)-axis. The rule for reflecting a point \( (x,y) \) over the \( x \)-axis is \( (x,y) \to (x, -y) \). Let's find the coordinates of \( L \) and \( M \). From the grid, assume \( L = (-4, 2) \) and \( M = (-1, 3) \).

Step2: Reflect over the \( x \)-axis

Applying the reflection rule to \( L \): \( (-4, 2) \to (-4, -2) \).
Applying the reflection rule to \( M \): \( (-1, 3) \to (-1, -3) \). Now we have the segment after \( x \)-axis reflection.

Step3: Reflect the new segment over the \( y \)-axis

The rule for reflecting a point \( (x,y) \) over the \( y \)-axis is \( (x,y) \to (-x, y) \).
Reflecting \( (-4, -2) \) over \( y \)-axis: \( (4, -2) \).
Reflecting \( (-1, -3) \) over \( y \)-axis: \( (1, -3) \).

Step4: Draw the final figure

Plot the points \( (4, -2) \) and \( (1, -3) \) and draw the segment connecting them. This is the figure after reflecting over \( x \)-axis then \( y \)-axis.

(Note: For other questions, similar steps apply. For example, in Question 10, we would identify the vertices of the two figures, apply the reflection rules (over \( x \)-axis then \( y \)-axis) step by step. For reflection over lines like \( y = -x \) (Question 11, 12), the rule is \( (x,y) \to (-y, -x) \) after the initial reflection. For glide reflections (Questions 13,14), we first perform the translation then the reflection or vice versa depending on the definition, but typically glide reflection is translation followed by reflection. For Question 13, translation \( \langle x + 0, y - 4
angle \) means we move each point down 4 units (since \( x \) doesn't change, \( y \) decreases by 4), then reflect over \( x = 1 \). The rule for reflecting over \( x = a \) is \( (x,y) \to (2a - x, y) \). So for a point \( (x,y) \), after translation \( (x, y - 4) \), then reflect over \( x = 1 \): \( (2(1)-x, y - 4)=(2 - x, y - 4) \). We would apply this to each vertex of \( \triangle ABC \) (Question 13) and draw the final figure. Similarly for Question 14, translation \( \langle x - 3, y + 0
angle \) (move left 3 units) then reflect over \( y = -1 \). The rule for reflecting over \( y = b \) is \( (x,y) \to (x, 2b - y) \). So after translation \( (x - 3, y) \), reflect over \( y = -1 \): \( (x - 3, 2(-1)-y)=(x - 3, -2 - y) \). Apply this to each vertex of the given quadrilateral and draw the final figure.)

Answer:

(For each question, the final figure is drawn as per the steps above. Since the problem is to draw the figures, the answer is the visual representation obtained by following the reflection/translation rules for each question. For example, for Question 9, the reflected segment has endpoints \( (4, -2) \) and \( (1, -3) \); for Question 10, the two figures (the upper curve and the lower triangle) are reflected over \( x \)-axis then \( y \)-axis following similar coordinate transformation rules, etc.)