Coordinate Plane & Graphs

The coordinate plane uses a horizontal and a vertical axis to describe any point as a pair of numbers. Like a map that names a column and a row, the point (3, 2) sits 3 steps right and 2 steps up. Turning a position into two numbers lets you draw shapes and graphs precisely. Here, drag the point and watch how its coordinates are decided.

Intuition — locating a position like a treasure map

x-coord3

y-coord2

🗺️ What Are Coordinates?

①On a treasure map, "3 east, 2 north" pins down a location

②That is a coordinate: (3, 2) = (across 3, up 2)

③x-coordinate: distance left/right from the origin

④y-coordinate: distance up/down from the origin

⑤Origin O = (0, 0)

Identifying Quadrants

Quadrant Test

I:(+,+) II:(-,+) III:(-,-) IV:(+,-)

Look at the signs to identify the quadrant instantly

🔍 Quadrant Cheat Sheet

①Quadrant I: x>0, y>0 (top-right)

②Quadrant II: x<0, y>0 (top-left)

③Quadrant III: x<0, y<0 (bottom-left)

④Quadrant IV: x>0, y<0 (bottom-right)

⑤Points on an axis like (0,3) or (2,0) belong to no quadrant!

Reflections — image in a mirror

Reflection over x-axis

(a, b) → (a, -b)

Mirror at the floor → only y flips sign

Reflection over y-axis

(a, b) → (-a, b)

Mirror at the wall → only x flips sign

Reflection through origin

(a, b) → (-a, -b)

180° rotation about the origin → both signs flip

💡 Reflection Mnemonics

①x-axis: floor mirror → flip y only

②y-axis: wall mirror → flip x only

③Origin: flip both → 180° rotation

④Drag the point above to see the reflections move with it

Distance Between Two Points (preview)

B x-coord4

B y-coord-1

Distance Between Two Points

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Hypotenuse of the right triangle (Pythagorean theorem, learned in Grade 9)

🔍 Why Does It Work?

①Connect the two points and form a right triangle

②Horizontal leg = difference in x; vertical leg = difference in y

③Hypotenuse = distance between the points

④Pythagoras: hyp² = horiz² + vert²

Work It Out

Example 1

Which quadrant is the point P(−3, 2) in?

Decide the quadrant from the signs of the x- and y-coordinates.

x = −3 < 0, y = 2 > 0

(−, +) is the second quadrant.

(−, +) ⇒ Quadrant II

▸ Quadrant II

The quadrant is set by the sign combination of x and y.

Example 2

Reflect the point A(2, −5) across the x-axis and find its image.

Reflection across the x-axis flips only the sign of y.

(x, y) → (x, −y)

Substitute.

(2, −5) → (2, 5)

▸ (2, 5)

x-axis reflection flips only y; y-axis reflection flips only x.

Exam Key Points

Coordinate Essentials

Point P(a, b) — (x, y) order matters!

(2, 3) and (3, 2) are completely different points

Grade-7 school exam type

If (a, b) is in Quadrant III, which quadrant is (−a, b) in?

①Quadrant I

②Quadrant II

③Quadrant III

④Quadrant IV

⑤on the x-axis

▸ ④ Quadrant IV

In Quadrant III, a < 0 and b < 0.

a < 0, b < 0

Then −a > 0 and b < 0, so the signs are (+, −).

(−a, b) = (+, −) ⇒ Quadrant IV

🎯 Exam Key Points

①Order is (x, y), not (y, x)

②On x-axis: y=0; on y-axis: x=0

③Identify quadrants from sign combinations

④Check the axis of reflection (x? y? origin?)

⑤Read grid scale carefully — one square may not equal 1

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Linear Equations

Direct & Inverse Proportion

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