transformations: unit 1 quiz 1 review
1. list characteristics of the following: then make a sketch (example) of each.
translation

rotation

reflection

apply the given transformations, label each image appropriately.
2. translate the polygon left 3 and up 2 units
3. rotate the polygon 90° about the origin

4. translate polygon right 4 and down 5 units
5. reflect △xyz across y = -x

Question

transformations: unit 1 quiz 1 review
1. list characteristics of the following: then make a sketch (example) of each.
translation

rotation

reflection

apply the given transformations, label each image appropriately.
2. translate the polygon left 3 and up 2 units
3. rotate the polygon 90° about the origin

4. translate polygon right 4 and down 5 units
5. reflect △xyz across y = -x

Accepted Answer

### Question 2: Translate the polygon left 3 and up 2 units
# Explanation:
## Step 1: Identify coordinates of point L
First, we need to find the coordinates of point $ L $ from the grid. Let's assume from the grid (looking at the position of $ L $): if we consider the origin $(0,0)$ at the intersection of the axes, and each grid square is 1 unit. Let's say $ L $ has coordinates $(x, y)$. From the diagram, let's assume $ L $ is at $(2, 1)$ (we need to check the grid: moving from the origin, right 2, up 1? Wait, maybe better to look at the position. Wait, the polygon is on the right side of the y-axis. Let's re-examine: Let's say the original coordinates of $ L $ are $(2, 1)$? Wait, no, maybe the grid: let's suppose the x-axis (horizontal) and y-axis (vertical). Let's say $ L $ is at $(2, 1)$? Wait, maybe I made a mistake. Wait, the first polygon: let's look at the points. Let's assume $ L $ is at $(2, 1)$ (but maybe it's $(2, 1)$ or another position. Wait, maybe the correct coordinates: let's suppose $ L $ is at $(2, 1)$. Wait, no, let's think again. Let's say the original coordinates of $ L $ are $(2, 1)$. Then, translating left 3 units means subtracting 3 from the x-coordinate, and up 2 units means adding 2 to the y-coordinate.

## Step 2: Apply translation rule
The translation rule for left 3 (subtract 3 from x) and up 2 (add 2 to y) is $(x, y) \to (x - 3, y + 2)$.

So if $ L $ is at $(2, 1)$, then $ L' $ would be $(2 - 3, 1 + 2) = (-1, 3)$. Wait, but maybe the original coordinates of $ L $ are different. Wait, maybe the correct original coordinates of $ L $ are $(2, 1)$? Wait, perhaps I need to check the grid again. Alternatively, maybe the original coordinates of $ L $ are $(2, 1)$, so:

Original $ L $: Let's say $ L = (2, 1) $ (we need to confirm from the grid, but since it's a diagram, let's proceed with the rule).

## Step 3: Write the rule and new coordinates
The translation rule is $(x, y) \to (x - 3, y + 2)$ (left 3: $ x $ decreases by 3, up 2: $ y $ increases by 2).

So if $ L $ is at $(2, 1)$, then $ L' = (2 - 3, 1 + 2) = (-1, 3) $.

# Answer:
- $ L(2, 1) $ (assuming original coordinates)
- $ L'(-1, 3) $
- Rule: $ (x, y) \to (x - 3, y + 2) $

### Question 3: Rotate the polygon 90° about the origin
# Explanation:
## Step 1: Recall 90° rotation rule
The rule for rotating a point $(x, y)$ 90° counterclockwise about the origin is $(x, y) \to (-y, x)$. If it's clockwise, the rule is $(x, y) \to (y, -x)$, but typically, in math, 90° rotation about the origin is counterclockwise unless specified otherwise. Let's assume counterclockwise.

## Step 2: Identify coordinates of point T
Let's assume the coordinates of $ T $ from the grid. Let's say $ T $ is at $(4, 2)$ (looking at the diagram: the polygon has points $ M, T, H, $ etc. Let's say $ T $ is at $(4, 2)$.

## Step 3: Apply rotation rule
Using the 90° counterclockwise rule $(x, y) \to (-y, x)$, for $ T(4, 2) $, the new coordinates $ T' $ would be $(-2, 4)$.

## Step 4: Write the rule
The rule for 90° counterclockwise rotation about the origin is $(x, y) \to (-y, x)$.

# Answer:
- $ T(4, 2) $ (assuming original coordinates)
- $ T'(-2, 4) $
- Rule: $ (x, y) \to (-y, x) $ (90° counterclockwise about origin)

### Question 4: Translate polygon right 4 and down 5 units
# Explanation:
## Step 1: Identify coordinates of point B
Let's find the coordinates of $ B $ from the grid. Looking at the triangle, $ B $ is at $(-1, 4)$ (assuming: from the origin, left 1, up 4).

## Step 2: Apply translation rule
The translation rule for right 4 (add 4 to x) and down 5 (subtract 5 from y) is $(x, y) \to (x + 4, y - 5)$.

## Step 3: Calculate new coordinates of B'
For $ B(-1, 4) $, applying the rule: $ x' = -1 + 4 = 3 $, $ y' = 4 - 5 = -1 $. So $ B' $ is $(3, -1)$.

## Step 4: Write the rule
The rule is $(x, y) \to (x + 4, y - 5)$.

# Answer:
- $ B(-1, 4) $ (assuming original coordinates)
- $ B'(3, -1) $
- Rule: $ (x, y) \to (x + 4, y - 5) $

### Question 5: Reflect $ \triangle XYZ $ across $ y = -x $
# Explanation:
## Step 1: Recall reflection rule over $ y = -x $
The rule for reflecting a point $(x, y)$ across the line $ y = -x $ is $(x, y) \to (-y, -x)$.

## Step 2: Identify coordinates of point Z
Let's find the coordinates of $ Z $ from the grid. Looking at the triangle, $ Z $ is at $(2, -2)$ (assuming: from the origin, right 2, down 2).

## Step 3: Apply reflection rule
Using the rule $(x, y) \to (-y, -x)$ for $ Z(2, -2) $:

$ x' = -(-2) = 2 $? Wait, no: wait, the rule is $(x, y) \to (-y, -x)$. So for $ Z(2, -2) $:

$ x' = -(-2) = 2 $? Wait, no: $ y = -2 $, so $ -y = 2 $; $ x = 2 $, so $ -x = -2 $. So $ Z' $ is $(2, -2)$? Wait, that can't be. Wait, maybe I made a mistake. Wait, let's recheck the rule: reflection over $ y = -x $ is $(x, y) \to (-y, -x)$. So if $ Z $ is at $(2, -2)$, then:

$ x' = -y = -(-2) = 2 $

$ y' = -x = -2 $

So $ Z'(2, -2) $? Wait, that's the same as $ Z $. That would mean $ Z $ is on the line $ y = -x $, because if $ y = -x $, then $ -2 = -2 $ (since $ x = 2 $, $ -x = -2 $, so $ y = -2 = -x $). So $ Z $ is on the line of reflection, so its image is itself.

## Step 4: Write the rule
The rule for reflection across $ y = -x $ is $(x, y) \to (-y, -x)$.

# Answer:
- $ Z(2, -2) $ (assuming original coordinates)
- $ Z'(2, -2) $ (since $ Z $ is on $ y = -x $)
- Rule: $ (x, y) \to (-y, -x) $ (reflection over $ y = -x $)

### Question 1: List characteristics and sketch (brief)
#### Translation:
- **Characteristics**: A translation (slide) moves every point of a figure or space by the same distance in a given direction. The shape, size, and orientation of the figure remain unchanged. It is a rigid transformation (isometry).
- **Sketch Example**: Draw a triangle, then draw another triangle parallel to it, shifted left/right/up/down by a few units, with the same shape and size.

#### Rotation:
- **Characteristics**: A rotation turns a figure around a fixed point (center of rotation) by a given angle. The shape and size remain the same, but the orientation changes. It is a rigid transformation.
- **Sketch Example**: Draw a square, then draw another square rotated 90° around a point (like the center of the square), showing the angle of rotation.

#### Reflection:
- **Characteristics**: A reflection flips a figure over a line (line of reflection). The shape and size remain the same, and the figure is a mirror image across the line. It is a rigid transformation.
- **Sketch Example**: Draw a triangle, then draw its mirror image across a vertical or horizontal line, showing the line of reflection.

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QUESTION IMAGE

Question

Explanation:

Question 2: Translate the polygon left 3 and up 2 units

Step 1: Identify coordinates of point L

Step 2: Apply translation rule

Step 3: Write the rule and new coordinates

Step 1: Recall 90° rotation rule

Step 2: Identify coordinates of point T

Step 3: Apply rotation rule

Step 4: Write the rule

Step 1: Identify coordinates of point B

Step 2: Apply translation rule

Step 3: Calculate new coordinates of B'

Step 4: Write the rule

Answer:

Question 3: Rotate the polygon 90° about the origin