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Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Understanding Signs of Numbers in Ordered Pairs and Reflections on the Coordinate Plane 🧮

Imagine you are playing an open-world video game where your character moves around a big map. The map has a center, and you can move left, right, up, and down. The game needs a way to know exactly where your character is standing at any moment. That is what a coordinate plane does in math: it gives every point an address using two numbers. 🎯

In this lesson, you explore how the signs (positive and negative) of numbers in an ordered pair tell you which part of the coordinate plane a point is in, and how changing those signs creates reflections across the axes.

The Coordinate Plane and Its Axes

The coordinate plane is made of two number lines that cross at zero:

• The x-axis is the horizontal number line (left and right).
• The y-axis is the vertical number line (up and down).

The place where they meet is called the origin, at the point \( (0, 0) \). As shown in [Figure 1], the axes divide the plane into four regions called quadrants.

Ordered Pairs: The Address of a Point

Each point in the coordinate plane has an address called an ordered pair. It looks like this: \( (x, y) \).

• \( x \) is the x-coordinate: how far you move left or right.
• \( y \) is the y-coordinate: how far you move up or down.
1. Start at the origin \( (0, 0) \).
2. Move along the x-axis: right if \( x \) is positive, left if \( x \) is negative.
3. From there, move along the vertical direction: up if \( y \) is positive, down if \( y \) is negative.
4. Mark the point.
• Move 3 units to the right (because 3 is positive).
• Then 2 units down (because -2 is negative).

You land at \( (3, -2) \).

Quadrants and Signs of Coordinates

The coordinate plane is split into four quadrants, numbered counterclockwise, as you see in [Figure 1]. Each quadrant has a different combination of signs for \( x \) and \( y \):

• Quadrant I: \( x > 0, y > 0 \) (both positive)
• Quadrant II: \( x < 0, y > 0 \) (x negative, y positive)
• Quadrant III: \( x < 0, y < 0 \) (both negative)
• Quadrant IV: \( x > 0, y < 0 \) (x positive, y negative)

On the axes themselves:

• Points on the x-axis have \( y = 0 \).
• Points on the y-axis have \( x = 0 \).

So just by looking at the signs of \( x \) and \( y \), you can tell which quadrant or axis a point is on—before even drawing it. 💡

Classifying Points by Quadrant: Examples

Decide which quadrant each point is in, just by its signs:

• \( (4, 5) \): both positive → Quadrant I.
• \( (-3, 6) \): x negative, y positive → Quadrant II.
• \( (-2, -7) \): both negative → Quadrant III.
• \( (5, -4) \): x positive, y negative → Quadrant IV.
• \( (0, -3) \): x is 0, y negative → on the negative part of the y-axis.
• \( (-6, 0) \): y is 0, x negative → on the negative part of the x-axis.

Rational Numbers as Points on Axes

Both coordinates can be rational numbers, not just whole numbers. Rational numbers are numbers that can be written as a fraction of two integers, like \( \frac{1}{2}, -\frac{3}{4}, 2, -5, 0 \).

For example:

• \( \left( \frac{1}{2}, -2 \right) \): x is a positive fraction, y is a negative integer → Quadrant IV.
• \( \left( -1.5, \frac{3}{4} \right) \): x negative decimal, y positive fraction → Quadrant II.

As shown in [Figure 2], the grid can include fractional tick marks to show these rational coordinates.



Reflections Across the Axes

Now the amazing part: when ordered pairs look the same except for signs, the points are connected by reflections across the x-axis, y-axis, or both. Think of a reflection like looking in a mirror. The shape is the same, but flipped.

There are three main types of reflections you use here:

1. Reflection across the x-axis
2. Reflection across the y-axis
3. Reflection across both axes

1. Reflection Across the x-axis

When you reflect a point across the x-axis, it flips from above to below the axis, or from below to above, but stays the same distance from it.

If a point is \( (x, y) \), its reflection across the x-axis is \( (x, -y) \).

• The x-coordinate stays the same.
• The sign of the y-coordinate changes (positive becomes negative, negative becomes positive).

Example ideas:

• \( (3, 4) \) → reflect across x-axis → \( (3, -4) \).
• \( (-2, -5) \) → reflect across x-axis → \( (-2, 5) \).

Notice the pattern: only the sign of y changes.

2. Reflection Across the y-axis

When you reflect across the y-axis, you flip left to right or right to left, but stay the same distance from the y-axis.

If a point is \( (x, y) \), its reflection across the y-axis is \( (-x, y) \).

• The y-coordinate stays the same.
• The sign of the x-coordinate changes.

Examples:

• \( (4, -1) \) → reflect across y-axis → \( (-4, -1) \).
• \( (-3, 2) \) → reflect across y-axis → \( (3, 2) \).

Again, only the sign of x changes.

3. Reflection Across Both Axes (Through the Origin)

If you reflect a point across both axes (or rotate it 180° around the origin), both coordinates change sign.

If a point is \( (x, y) \), its reflection across both axes is \( (-x, -y) \).

Examples:

• \( (2, 5) \) → reflect across both axes → \( (-2, -5) \).
• \( (-4, -3) \) → reflect across both axes → \( (4, 3) \).

Here, both x and y change signs, and the point ends up in the opposite quadrant.

How Reflections Relate Quadrants

Reflections move points between quadrants in predictable ways, as you see in [Figure 3]:

• Quadrant I ↔ Quadrant IV by reflecting across the x-axis (change y sign).
• Quadrant I ↔ Quadrant II by reflecting across the y-axis (change x sign).
• Quadrant I ↔ Quadrant III by reflecting across both axes (change both signs).
• And similarly for other quadrants.



Recognizing Related Points by Their Signs

Look at these sets of ordered pairs and see how they are related:

• \( (4, 3), (4, -3) \)
• \( (4, 3), (-4, 3) \)
• \( (4, 3), (-4, -3) \)

Comparing \( (4, 3) \) and \( (4, -3) \): only the y changes sign, so they are reflections across the x-axis.

Comparing \( (4, 3) \) and \( (-4, 3) \): only the x changes sign, so they are reflections across the y-axis.

Comparing \( (4, 3) \) and \( (-4, -3) \): both x and y change sign, so they are reflections through the origin (across both axes).

This is the key idea: When two ordered pairs differ only by signs, their points are related by reflections across one or both axes. 🤔

Solved Example 1: Classifying and Reflecting a Point

Problem: A point has coordinates \( (-5, 2) \).

1. Which quadrant is this point in?
2. What is its reflection across the x-axis?
3. What is its reflection across the y-axis?
4. What is its reflection across both axes?

Step-by-step solution:

Step 1: Decide the quadrant.

• x is -5 (negative).
• y is 2 (positive).

Negative x and positive y means Quadrant II.

Step 2: Reflection across the x-axis.

Reflection across the x-axis: \( (x, y) → (x, -y) \).

Start with \( (-5, 2) \):

• x stays -5.
• y becomes -2.

So the reflection is \( (-5, -2) \), which is in Quadrant III (both negative).

Step 3: Reflection across the y-axis.

Reflection across the y-axis: \( (x, y) → (-x, y) \).

• x becomes 5.
• y stays 2.

So the reflection is \( (5, 2) \), which is in Quadrant I (both positive).

Step 4: Reflection across both axes.

Reflection across both axes: \( (x, y) → (-x, -y) \).

• x becomes 5.
• y becomes -2.

So the reflection is \( (5, -2) \), which is in Quadrant IV (x positive, y negative).

Solved Example 2: Working With Rational Numbers

Problem: Consider the point \( \left( -1.5, -\frac{3}{2} \right) \).

1. Which quadrant is this point in?
2. Find its reflection across the x-axis.
3. Find its reflection across the y-axis.
4. Find its reflection across both axes.

Step-by-step solution:

Step 1: Decide the quadrant.

• x is \( -1.5 \), which is negative.
• y is \( -\frac{3}{2} \), which is also negative.

Both negative → Quadrant III.

Step 2: Reflection across the x-axis.

Reflection: \( (x, y) → (x, -y) \).

So:

• x stays \( -1.5 \).
• y becomes \( \frac{3}{2} \).

The reflection is \( \left( -1.5, \frac{3}{2} \right) \), in Quadrant II (x negative, y positive).

Step 3: Reflection across the y-axis.

Reflection: \( (x, y) → (-x, y) \).

• x becomes \( 1.5 \).
• y stays \( -\frac{3}{2} \).

The reflection is \( \left( 1.5, -\frac{3}{2} \right) \), in Quadrant IV.

Step 4: Reflection across both axes.

Reflection: \( (x, y) → (-x, -y) \).

• x becomes \( 1.5 \).
• y becomes \( \frac{3}{2} \).

The reflection is \( \left( 1.5, \frac{3}{2} \right) \), in Quadrant I.

Solved Example 3: Finding a Missing Coordinate

Problem: A point \( B \) is the reflection of point \( A(7, -4) \) across the y-axis.

1. What are the coordinates of \( B \)?
2. In which quadrants are \( A \) and \( B \)?

Step-by-step solution:

Step 1: Use the reflection rule.

Across the y-axis: \( (x, y) → (-x, y) \).

For \( A(7, -4) \):

• x is 7, so the new x is -7.
• y is -4, and it stays -4.

So \( B = (-7, -4) \).

Step 2: Decide the quadrants.

• Point \( A(7, -4) \): x positive, y negative → Quadrant IV.
• Point \( B(-7, -4) \): x negative, y negative → Quadrant III.

So reflecting across the y-axis moved the point from Quadrant IV to Quadrant III.

Real-World Applications 🌍

1. Video Games and Animation

In many games, your character’s position is tracked using coordinates. If a game wants to mirror a map or flip a character’s movement from right to left, it can use reflections similar to changing \( (x, y) \) to \( (-x, y) \) or \( (x, -y) \). Knowing how signs affect position helps programmers control movement and symmetry.

2. Graphics and Design

When a designer creates a logo and wants a perfectly symmetrical reflection on the other side of a line, they use the same idea: reflect points across an axis. Changing signs of coordinates creates that mirror image exactly.

3. Maps and Navigation

On some maps or grid systems, north-south and east-west directions can be treated like positive and negative axes. For example, east might be positive x, west negative x, north positive y, and south negative y. Moving to the “mirror” location on the map is like reflecting across an axis.

4. Physics and Engineering

In physics, when objects move with direction (like left/right or up/down), those directions are often modeled with positive and negative numbers. Reflecting paths or forces across an axis in a diagram can show what happens if something bounces off a wall or flips direction.

Key Ideas to Remember ⭐

• The coordinate plane has two axes: horizontal x-axis and vertical y-axis, crossing at the origin \( (0, 0) \).
• An ordered pair \( (x, y) \) tells you how far to move along x (left/right) and then along y (up/down).
• The signs of x and y decide the quadrant:
  • Quadrant I: \( (+, +) \)
  • Quadrant II: \( (-, +) \)
  • Quadrant III: \( (-, -) \)
  • Quadrant IV: \( (+, -) \)
• Points can also lie on the axes when one coordinate is 0.
• Reflection across the x-axis: \( (x, y) → (x, -y) \) (change y sign).
• Reflection across the y-axis: \( (x, y) → (-x, y) \) (change x sign).
• Reflection across both axes (through the origin): \( (x, y) → (-x, -y) \) (change both signs).
• When two ordered pairs differ only by signs, their points are related by reflections across one or both axes.
• Rational numbers (fractions and decimals) can be used as coordinates just like whole numbers, and their signs work the same way.

Understanding how signs in ordered pairs connect to quadrants and reflections gives you a powerful way to visualize and predict where points are, how they move, and how they relate to each other on the coordinate plane. 🎉