SOLUTION: Reflections of shapes - Studypool - Free Printable

It looks like you've uploaded a worksheet about reflections in the coordinate plane. The task involves:

1. Finding the coordinates of the vertices of a figure after a given reflection.
2. Describing each transformation using a rule.

Let's go through each part carefully and solve it step by step.

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🔷 Part 1: Find the coordinates of the vertices after the given transformations

#### 1. Reflection across the x-axis
Given: $ P(-3, 4), Q(0, 2), R(3, 4) $

> Rule for reflection over the x-axis:
> $(x, y) \rightarrow (x, -y)$

Apply this:
- $P(-3, 4) \rightarrow P'(-3, -4)$
- $Q(0, 2) \rightarrow Q'(0, -2)$
- $R(3, 4) \rightarrow R'(3, -4)$

✔ Answer: $P'(-3, -4), Q'(0, -2), R'(3, -4)$

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#### 2. Reflection across $x = 2$
Given: $P(-1, 3), Q(1, 5), R(4, 3)$

> Rule for reflection over vertical line $x = a$:
> Distance from point to line is preserved on the opposite side.

For $x = 2$, the formula is:
If a point has $x$-coordinate $x$, then its image has $x'$ such that:
$$
x' = 2a - x = 2(2) - x = 4 - x
$$
So: $(x, y) \rightarrow (4 - x, y)$

Now apply:
- $P(-1, 3) \rightarrow (4 - (-1), 3) = (5, 3)$
- $Q(1, 5) \rightarrow (4 - 1, 5) = (3, 5)$
- $R(4, 3) \rightarrow (4 - 4, 3) = (0, 3)$

✔ Answer: $P'(5, 3), Q'(3, 5), R'(0, 3)$

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#### 3. Reflection across $x = -1$
Given: $P(-3, 1), Q(-1, 4), R(-2, 6)$

> Use rule: $(x, y) \rightarrow (2(-1) - x, y) = (-2 - x, y)$

Apply:
- $P(-3, 1) \rightarrow (-2 - (-3), 1) = (1, 1)$
- $Q(-1, 4) \rightarrow (-2 - (-1), 4) = (-1, 4)$ → stays same (on line!)
- $R(-2, 6) \rightarrow (-2 - (-2), 6) = (0, 6)$

✔ Answer: $P'(1, 1), Q'(-1, 4), R'(0, 6)$

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#### 4. Reflection across $y = 2$
Given: $P(1, -1), Q(3, 2), R(5, 1)$

> Rule for reflection over horizontal line $y = b$:
> $(x, y) \rightarrow (x, 2b - y)$

Here, $b = 2$, so: $(x, y) \rightarrow (x, 4 - y)$

Apply:
- $P(1, -1) \rightarrow (1, 4 - (-1)) = (1, 5)$
- $Q(3, 2) \rightarrow (3, 4 - 2) = (3, 2)$ → stays same (on line)
- $R(5, 1) \rightarrow (5, 4 - 1) = (5, 3)$

✔ Answer: $P'(1, 5), Q'(3, 2), R'(5, 3)$

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🔷 Part 2: Write a rule to describe each transformation

We are given four graphs with original figures (in black) and reflected images (in blue). Let’s analyze each one.

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#### (a) Reflection across $x = 2$

From graph: Original triangle is on left side of $x=2$, image is symmetric on the right.

We already used this rule earlier:
✔ Rule: $(x, y) \rightarrow (4 - x, y)$

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#### (b) Reflection across the x-axis

Original triangle is above x-axis, image is below — flipped vertically.

✔ Rule: $(x, y) \rightarrow (x, -y)$

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#### (c) Reflection across $x = -2$

The line $x = -2$ is vertical. From graph, the figure is reflected over this line.

Use formula:
$(x, y) \rightarrow (2(-2) - x, y) = (-4 - x, y)$

✔ Rule: $(x, y) \rightarrow (-4 - x, y)$

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#### (d) Reflection across $y = 2$

Image is flipped over horizontal line $y = 2$.
As before: $(x, y) \rightarrow (x, 4 - y)$

✔ Rule: $(x, y) \rightarrow (x, 4 - y)$

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✔ Final Answers Summary:

#### Part 1: Coordinates After Reflection

1. Reflect over x-axis:
$P'(-3, -4), Q'(0, -2), R'(3, -4)$

2. Reflect over $x = 2$:
$P'(5, 3), Q'(3, 5), R'(0, 3)$

3. Reflect over $x = -1$:
$P'(1, 1), Q'(-1, 4), R'(0, 6)$

4. Reflect over $y = 2$:
$P'(1, 5), Q'(3, 2), R'(5, 3)$

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#### Part 2: Rules for Transformations

(a) Reflection across $x = 2$:
→ $(x, y) \rightarrow (4 - x, y)$

(b) Reflection across x-axis:
→ $(x, y) \rightarrow (x, -y)$

(c) Reflection across $x = -2$:
→ $(x, y) \rightarrow (-4 - x, y)$

(d) Reflection across $y = 2$:
→ $(x, y) \rightarrow (x, 4 - y)$

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