🎯⭐ INTERACTIVE LESSON

The Coordinate Plane and Graphing

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The Coordinate Plane and Graphing - Complete Interactive Lesson

Part 1: Building the Plane: Axes, Origin & Ordered Pairs

📍 The Coordinate Plane and Graphing

Part 1 of 5 — Building the Plane: Axes, Origin & Ordered Pairs

Topics in This Part

Section
The Two Number Lines
The Origin and the Axes
Ordered Pairs $(x, y)$
Reading the Address of a Point

🔑 Key Concept: The coordinate plane is just two number lines crossing at right angles. Every point on it has a unique "address" written as an ordered pair $(x, y)$ — and the order matters.

The Two Number Lines

A single number line lets you locate one number. To locate a position in 2-D space, we need two number lines:

The $x$ -axis runs horizontally (left–right). Right is positive, left is negative.
The $y$ -axis runs vertically (up–down). Up is positive, down is negative.

They cross at a single point called the origin, written $(0, 0)$ .

Axis	Direction	Positive way	Negative way

Concept Check 🎯

Ordered Pairs: $(x, y)$

A point's address is an ordered pair. The word ordered is the whole point — you always list the numbers in the same order:

$(\,\underbrace{x}_{\text{across first}},\ \underbrace{y}_{\text{up second}}\,)$

Read the Movement 🔽

For the point $(-4, 6)$ , describe how to travel from the origin.

Name That Coordinate 🧮

Use the point $P = (-7, 2)$ .

1) What is the $x$ -coordinate of $P$ ? 2) What is the $y$ -coordinate of ? From the origin, how many units do you move to reach ?

Putting It Together

You now know the parts of the plane and how to read an address:

Two axes ( $x$ across, $y$ up) crossing at the origin $(0,0)$ .
Every point is an ordered pair $(x, y)$ — across first, up second.

Part 2: Plotting and Reading Points

📍 The Coordinate Plane and Graphing

Part 2 of 5 — Plotting and Reading Points

🔑 The Skill: Plotting is following directions. Start at the origin, walk left/right by the $x$ -value, then up/down by the $y$ -value, and mark the spot.

How to Plot a Point

To plot the point $(x, y)$ :

Start at the origin $(0,$ .

Part 3: The Four Quadrants & Points on the Axes

📍 The Coordinate Plane and Graphing

Part 3 of 5 — The Four Quadrants & Points on the Axes

🔑 The Map: The two axes split the plane into four regions called quadrants. Knowing the sign pattern of each quadrant lets you locate a point before you even plot it.

The Four Quadrants

The quadrants are numbered with Roman numerals, starting in the top right and going counter-clockwise.

Quadrant	Location	$x$ sign	$y$ sign	Example point
I	top right

Part 4: Distance, Reflections & Graphing a Table

📍 The Coordinate Plane and Graphing

Part 4 of 5 — Distance, Reflections & Graphing a Table

🔑 Putting the Plane to Work: Now that you can place points, you can measure between them, flip them across the axes, and graph a rule from a table.

Distance Along a Grid Line

When two points share the same $x$ (a vertical line) or the same $y$ (a horizontal line), the distance between them is just the gap between the other coordinates. Count the squares — or subtract.

Horizontal distance (same $y$ )

$\text{distance} =$ difference of the .

Part 5: Mixed Practice & Mastery Check

📍 The Coordinate Plane and Graphing

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read and write ordered pairs, (2) plot and read points, (3) name quadrants and axis points, and (4) measure distance, reflect points, and graph a table. Let's tie it all together.

Quick Reference

Goal	Key move
Read an address	$(x, y)$ = across first, up second
Find the quadrant	$(+,+)$ =I, =II, =III, =IV

💡 Memory tip: " $x$ is a cross" — the $x$ -axis goes a-cross the page. Whatever is left runs up and down.

The first number is the $x$ -coordinate — how far to move left/right from the origin.
The second number is the $y$ -coordinate — how far to move up/down.

⚠️ Order matters! $(3, 5)$ and $(5, 3)$ are different points. $(3, 5)$ means 3 right, 5 up; $(5, 3)$ means 5 right, 3 up.

💡 Memory tip: Go in the front door ( $x$ , across) before you go up the stairs ( $y$ ). Across, then up — always in alphabetical order, $x$ before $y$ .

🔑 Takeaway: A point is a set of directions from the origin. In Part 2 you'll follow those directions to actually plot points on a grid.

Move horizontally by the

x

-value: right if positive, left if negative.

Move vertically by the

y

-value: up if positive, down if negative.

Mark a dot where you land.

Worked Example: Plot $(4, 3)$

Start at $(0,0)$ . The $x$ -value is $+4$ , so move 4 right. The $y$ -value is $+3$ , so move 3 up. The dot lands 4 to the right and 3 above the origin.

Worked Example: Plot $(-2, 5)$

Start at $(0,0)$ . The $x$ -value is $-2$ , so move 2 left. The $y$ -value is $+5$ , so move 5 up.

💡 Always read the $x$ -value first — even though it feels natural to go up first, the across move comes first.

Concept Check 🎯

Reading a Plotted Point

Reading is plotting in reverse. Given a dot on the grid, ask two questions:

How far left or right of the origin is it? That number (with sign) is the $x$ -coordinate.
How far up or down is it? That number (with sign) is the $y$ -coordinate.

Worked Example

A dot sits 3 units to the right and 4 units below the origin.

Right $3$ → $x = 3$
Below $4$ → $y = -4$

So the point is $(3, -4)$ .

⚠️ Watch the sign on "down" and "left." Down and left are negative; up and right are positive. Forgetting a sign is the #1 plotting error.

Find Each Address 🔽

A point lies 5 units left of the origin and 2 units down.

Plotting Moves 🧮

Answer each as a signed number (use a minus sign for left or down).

1) To plot $(-8, 0)$ , how many units do you move horizontally? (Negative = left.) 2) To plot $(0, -5)$ , how many units do you move vertically? (Negative = down.) 3) A dot is 9 units right and 4 units up. What is its $x$ -coordinate?

🔑 The pattern: Start at Quadrant I (both positive) and spin counter-clockwise. The signs go $(+,+) \to (-,+) \to (-,-) \to (+,-)$ .

💡 Memory tip: " $x$ tells the column, $y$ tells the row." If $x$ is negative you're on the left; if $y$ is negative you're on the bottom.

Concept Check 🎯

Points on the Axes (the Special Cases)

If a point lies on an axis, it is not in any quadrant. There's a simple rule based on which coordinate is zero:

If…	The point lies on…
$y = 0$ , e.g. $(7, 0)$	the $x$ -axis
$x = 0$ , e.g. $(0, -4)$	the $y$ -axis
$x = 0$ and $y = 0$	the origin $(0,0)$

Why? A $y$ -coordinate of $0$ means "don't move up or down," so you stay on the horizontal $x$ -axis. An $x$ -coordinate of $0$ means "don't move left or right," so you stay on the vertical $y$ -axis.

⚠️ Common trap: $(0, -4)$ has a zero $x$ , so it's on the $y$ -axis — even though the $y$ -axis is named for $y$ . Look at the coordinate that is , then go to the axis.

Quadrant or Axis? 🔽

Classify each point.

Quadrant Numbers 🧮

Enter the quadrant number (1, 2, 3, or 4) for each point. If the point is on an axis, enter 0.

1) $(9, 4)$ 2) $(-1, -8)$ 3) $(0, 3)$

$x$ -coordinates

Example: From $(2, 3)$ to $(9, 3)$ : both have $y = 3$ , so $9 - 2 = 7$ units apart.

Vertical distance (same $x$ )

$\text{distance} =$ difference of the $y$ -coordinates.

Example: From $(-4, 1)$ to $(-4, 6)$ : both have $x = -4$ , so $6 - 1 = 5$ units apart.

💡 Crossing zero? From $(-3, 5)$ to $(4, 5)$ the distance is $4 - (-3) = 7$ . Subtracting a negative adds. You can also just count: 3 squares to the origin, plus 4 more = 7.

⚠️ Distance is always positive. If subtraction gives a negative, take its absolute value: $\lvert x_2 - x_1 \rvert$ .

Measure the Gap 🧮

Each pair of points shares an axis-line. Enter the distance between them (a positive number).

1) $(1, 7)$ and $(1, 2)$ 2) $(-6, 4)$ and $(3, 4)$ 3) $(0, -2)$ and $(0, 5)$

Reflections Across the Axes

A reflection flips a point to the opposite side of an axis, like a mirror image. There are two simple rules:

Reflect across…	What changes	Rule
the $x$ -axis	$y$ flips sign	$(x, y) \to (x, -y)$
the $y$ -axis	$x$ flips sign	$(x, y) \to (-x, y)$

Worked Example

Reflect $(3, 5)$ across the $x$ -axis: only $y$ flips → $(3, -5)$ .

Reflect $(3, 5)$ across the $y$ -axis: only $x$ flips → $(-3, 5)$ .

💡 Why? A mirror on the $x$ -axis (horizontal mirror) flips up/down, which is the $y$ -value. A mirror on the $y$ -axis (vertical mirror) flips left/right, which is the $x$ -value. The axis you flip across is the coordinate that stays the same.

Concept Check 🎯

Graphing a Table of Values

A rule like $y = x + 2$ pairs each input $x$ with an output $y$ . Make a small table, turn each row into an ordered pair, and plot the points.

Worked Example: $y = x + 2$

$x$	$y = x + 2$	Point $(x, y)$
$- 1$

Plot $(-1,1), (0,2), (1,3), (2,4)$ — they fall in a perfectly straight line. Connecting them graphs the rule.

🔑 Big idea: A graph is a picture of a rule. Every point on the line is an $(x, y)$ pair that makes the equation true.

Complete the Table 🔽

Fill in the missing outputs and a point for the rule $y = 2x - 1$ .

⚠️ Top three traps: (1) writing $y$ before $x$ ; (2) wrong sign for "left" or "down"; (3) thinking a point with a zero coordinate is in a quadrant — it's on an axis.

Mixed Practice 🎯

Apply It 🧮

1) For the rule $y = 3x + 1$ , what is $y$ when $x = 4$ ? 2) What is the distance between $(2, -5)$ and $(2, 6)$ ? 3) In which quadrant is $(-8, 7)$ ? (Enter 1, 2, 3, or 4.)

Exit Quiz ✅

Answer all three to finish the lesson.

$x$ -axis	horizontal	right $\rightarrow$	$\leftarrow$ left
$y$ -axis	vertical	up $\uparrow$	$\downarrow$ down

The Coordinate Plane and Graphing

The Coordinate Plane and Graphing - Complete Interactive Lesson

Part 1: Building the Plane: Axes, Origin & Ordered Pairs

📍 The Coordinate Plane and Graphing

Topics in This Part

The Two Number Lines

Ordered Pairs: (x,y)(x, y)(x,y)

Putting It Together

Part 2: Plotting and Reading Points

📍 The Coordinate Plane and Graphing

How to Plot a Point

Part 3: The Four Quadrants & Points on the Axes

📍 The Coordinate Plane and Graphing

The Four Quadrants

Part 4: Distance, Reflections & Graphing a Table

📍 The Coordinate Plane and Graphing

Distance Along a Grid Line

Horizontal distance (same yyy)

Part 5: Mixed Practice & Mastery Check

📍 The Coordinate Plane and Graphing

Quick Reference

Worked Example: Plot (4,3)(4, 3)(4,3)

Worked Example: Plot (−2,5)(-2, 5)(−2,5)

Reading a Plotted Point

Worked Example

Points on the Axes (the Special Cases)

Vertical distance (same xxx)

Reflections Across the Axes

Worked Example

Graphing a Table of Values

Worked Example: y=x+2y = x + 2y=x+2

Ordered Pairs: $(x, y)$

Horizontal distance (same $y$ )

Worked Example: Plot $(4, 3)$

Worked Example: Plot $(-2, 5)$

Vertical distance (same $x$ )

Worked Example: $y = x + 2$