Plotting and Naming Points in All Four Quadrants

Hello! This course will teach you how shapes can slide, flip, and turn without changing their size or shape. Before we move any shapes, we need to know how to give every corner of a shape an exact “address” on a grid.

Today you will learn to read and plot ordered pairs such as $(4,5)$ in all four quadrants of a coordinate plane. These point-addresses will be the language we use for every transformation later.

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The coordinate plane: a map made from two number lines

A coordinate plane is a flat grid made by crossing two number lines.

• The x-axis goes left and right. It is the horizontal line.
• The y-axis goes up and down. It is the vertical line.
• The place where they cross is the origin, written $(0,0)$.

The origin is your home base. Every point begins with directions from there.

A helpful memory clue:

• x is for moving across
• y is for moving up or down

Positive numbers tell you to move right or up:

• Positive $x$: move right.
• Negative $x$: move left.
• Positive $y$: move up.
• Negative $y$: move down.

The negative sign does not mean “bad.” It just means move in the opposite direction.

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Ordered pairs: two directions in a special order

A point’s address is called an ordered pair. It looks like this:

$$(x,y)$$

The order is always important:

1. The first number is $x$: move left or right.
2. The second number is $y$: move up or down.

For example, $(4,5)$ means:

1. Start at $(0,0)$.
2. Move $4$ spaces right.
3. Move $5$ spaces up.
4. Put a dot there.

So the point is at $(4,5)$.

But $(5,4)$ is a different point:

1. Move $5$ spaces right.
2. Move $4$ spaces up.

Same two numbers, different order, different location. The parentheses and comma are part of the address: $(4,5)$ is one ordered pair, not two separate answers.

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Finding points in every direction

Here is the full movement rule:

Coordinate part	Positive value	Negative value
First number, $x$	right	left
Second number, $y$	up	down

Let’s use the rule with four points.

• $(4,5)$: right $4$, up $5$
• $(-3,1)$: left $3$, up $1$
• $(-5,-5)$: left $5$, down $5$
• $(6,-2)$: right $6$, down $2$

Notice that you always do the sideways trip first, even when the first number is negative.

The picture also shows an important way to read a point already on a grid. Suppose a dot is at $(-3,1)$.

1. Look straight down or up from the dot to find its horizontal position: $x=-3$.
2. Look straight across from the dot to find its vertical position: $y=1$.
3. Write the point in the correct order:

$$(-3,1)$$

When a point has a letter name, you can write both its name and address. For example:

$$A=(-3,1)$$

Later, the corners of polygons will be called vertices, often with capital letters such as $A$, $B$, and $C$. Knowing each vertex’s ordered pair lets us track it when the whole shape moves.

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The four quadrants

The two axes split the plane into four regions called quadrants. They are numbered in this pattern:

• Quadrant I: top right
• Quadrant II: top left
• Quadrant III: bottom left
• Quadrant IV: bottom right

The numbering begins in the top-right corner and goes around to the left.

Quadrant	Location	Signs of its points
I	top right	$(+,+)$
II	top left	$(-,+)$
III	bottom left	$(-,-)$
IV	bottom right	$(+,-)$

You do not need to memorize the table as a mysterious code. You can rebuild it from directions:

• Top means $y$ is positive.
• Bottom means $y$ is negative.
• Right means $x$ is positive.
• Left means $x$ is negative.

So, for example, $(-5,3)$ is left and up. Left means negative $x$; up means positive $y$. Therefore it is in Quadrant II.

Use the four labeled points in the image:

• $(4,5)$ is in Quadrant I.
• $(-3,1)$ is in Quadrant II.
• $(-5,-5)$ is in Quadrant III.
• $(6,-2)$ is in Quadrant IV.

A fast quadrant check is to look only at the signs. You do not need to count squares once you know the coordinate pair.

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Points on the axes are special

Not every point belongs in a quadrant.

If one coordinate is $0$, the point sits directly on an axis:

• $(0,4)$ is on the y-axis. There is no right-left movement.
• $(-2,0)$ is on the x-axis. There is no up-down movement.
• $(0,0)$ is the origin, where both axes meet.

The axes are borders between quadrants, not parts of any quadrant. So it is not correct to say that $(0,4)$ is in Quadrant I or Quadrant II. It is on the y-axis between them.

A reliable plotting routine is:

1. Find the origin.
2. Read the first number, $x$, and move left or right.
3. Read the second number, $y$, and move up or down.
4. Place the dot and label it carefully.

A reliable reading routine reverses that idea:

1. Find how far left or right the point is: that is $x$.
2. Find how far up or down it is: that is $y$.
3. Write $(x,y)$, with $x$ first every time.

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Key takeaways

A coordinate plane gives each point a precise address.

• The horizontal axis is the x-axis; the vertical axis is the y-axis.
• The origin is $(0,0)$.
• An ordered pair is written $(x,y)$: sideways first, then vertical.
• Positive $x$ means right and negative $x$ means left.
• Positive $y$ means up and negative $y$ means down.
• Quadrants I, II, III, and IV are, respectively: top right, top left, bottom left, and bottom right.
• Points with a zero coordinate lie on an axis, not inside a quadrant.

Next, you will use this grid language to tell whether a move is a rigid transformation—a slide, flip, or turn that keeps a shape exactly the same—or a change that stretches, shrinks, or distorts it.