The Coordinate Plane and Graphing

📍 The Coordinate Plane and Graphing

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

Topics in This Part

🔑 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 $(0, 0)$.

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

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:

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

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

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 $P$? 3) From the origin, how many units do you move up to reach $P$?

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.

🔑 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.

📍 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 $(x, y)$:

1. Start at the origin $(0, 0)$.
2. Move horizontally by the $x$-value: right if positive, left if negative.
3. Move vertically by the $y$-value: up if positive, down if negative.
4. 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:

1. How far left or right of the origin is it? That number (with sign) is the $x$-coordinate.
2. 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 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.

🔑 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:

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 zero, then go to the other 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) $(0, 3)$

📍 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 $x$-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:

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: $y = x + 2$

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 $y = 2x - 1$.

📍 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

⚠️ 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.