Before diving into the practice questions, take a moment to think about what you already know about this topic.
- Know: the Cartesian plane has an x-axis (horizontal) and a y-axis (vertical) meeting at the origin (0, 0).
- Understand: any point is described by an ordered pair (x, y) — x tells how far right/left from the origin, y tells how far up/down.
- Can Do: plot points in all four quadrants, identify which quadrant a point is in from its coordinate signs, and find reflections of points in the axes.
The Cartesian plane is a flat grid formed by two number lines:
- The x-axis runs horizontally (left–right).
- The y-axis runs vertically (up–down).
- They cross at the origin (0, 0).
Every point is written as an ordered pair $(x, y)$. The first number (x) tells you how far to move right (positive) or left (negative) from the origin. The second number (y) tells you how far to move up (positive) or down (negative).
Examples: $(3, 5)$: go 3 right, then 5 up. $(-2, 4)$: go 2 left, then 4 up. $(0, -3)$: stay on the y-axis, go 3 down.
The axes divide the plane into four quadrants, numbered anti-clockwise starting from the top-right:
- Quadrant I (top-right): $x > 0$, $y > 0$ — both positive, e.g. $(3, 4)$.
- Quadrant II (top-left): $x < 0$, $y > 0$ — x negative, y positive, e.g. $(-2, 5)$.
- Quadrant III (bottom-left): $x < 0$, $y < 0$ — both negative, e.g. $(-1, -3)$.
- Quadrant IV (bottom-right): $x > 0$, $y < 0$ — x positive, y negative, e.g. $(4, -2)$.
Sign rule summary: look at the signs of $(x, y)$: $(+,+)$ → QI, $(-,+)$ → QII, $(-,-)$ → QIII, $(+,-)$ → QIV.
Memory tip: start at the top-right and count anti-clockwise: I, II, III, IV.
Note: points on an axis (where $x = 0$ or $y = 0$) are NOT in any quadrant.
Reflecting a point creates a mirror image across an axis:
- Reflect in the x-axis — flip the sign of $y$, keep $x$.
$(3, 2) \rightarrow (3, -2)$ - Reflect in the y-axis — flip the sign of $x$, keep $y$.
$(3, 2) \rightarrow (-3, 2)$ - Reflect through the origin (both axes) — flip both signs.
$(3, 2) \rightarrow (-3, -2)$
Quick rule: the axis you reflect in is the one that stays the same. Reflecting in the x-axis leaves the x-coordinate unchanged; reflecting in the y-axis leaves the y-coordinate unchanged.
Task: For each point, identify $x$ and $y$, then state which quadrant it is in.
Points: $(2, 3)$, $(-4, 1)$, $(-3, -2)$, $(1, -4)$.
- $(2, 3)$: $x = 2$ (positive), $y = 3$ (positive) → both positive → Quadrant I.
- $(-4, 1)$: $x = -4$ (negative), $y = 1$ (positive) → x negative, y positive → Quadrant II.
- $(-3, -2)$: $x = -3$ (negative), $y = -2$ (negative) → both negative → Quadrant III.
- $(1, -4)$: $x = 1$ (positive), $y = -4$ (negative) → x positive, y negative → Quadrant IV.
Answer: QI, QII, QIII, QIV — one point in each quadrant.
- Ordered pair is $(x, y)$ — x always first, y always second. Writing $(y, x)$ gives a completely different point.
- The origin is (0, 0), not (1, 1). It is the starting point, not the first positive tick.
- Quadrant II: x is negative but y is positive — don't mix up the signs. The point is to the left of the y-axis but above the x-axis.
Quick Check · 5 questions
Q1. Write the coordinates of the points described:
Q2. For each point, state which quadrant it is in (or if it is on an axis):
Q3. A triangle has vertices at $(1, 1)$, $(4, 1)$, and $(1, 5)$. What type of triangle is it? Find its area.
Extension Problems
A square has one corner at $(2, 3)$ and side length 4. If another corner is at $(6, 3)$, find the ot...
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