Worked examples — Cartesian plane — axes, quadrants, ordered pairs

This page is a workout. The parent note taught you the ideas; here we hit every kind of problem the Cartesian plane can throw at you — one worked example per scenario, no gaps.

Before we start, three plain-word reminders so every symbol is earned:

The scenario matrix

Here is every case class this topic can hand you. Each of the worked examples below is tagged with the cell it covers, and together they fill the whole table.

#	Case class	What makes it tricky	Example
C1	Point in a "clean" quadrant	reading two non-zero signs	Ex 1
C2	All four quadrants at once	sign flips $(\pm ,\pm )$	Ex 2
C3	Degenerate: point on an axis	one coordinate is $0$ → no quadrant	Ex 3
C4	Degenerate: the origin & zero-distance	both coordinates $0$; distance $=0$	Ex 3, Ex 4
C5	Distance with negative coordinates	signs inside $(\Delta x{)}^2$	Ex 4
C6	Horizontal / vertical special distance	one $\Delta$ is $0$	Ex 5
C7	Real-world word problem	translating words → signs	Ex 6
C8	Exam twist: reflections / unknowns	working backwards from a rule	Ex 7
C9	Limiting / boundary behaviour	quadrant as a point slides across an axis	Ex 8

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Ex 1 — A clean quadrant (cell C1)

Forecast: Guess now — is $P$ up-left, up-right, down-left, or down-right?

1. Start at the origin $O(0,0)$. Why this step? Every address is measured from the centre — that's the meaning of a coordinate.
2. Move to $x=-4$: four units LEFT. Why this step? A negative first coordinate means the opposite of the positive (rightward) direction.
3. From $(-4,0)$ move to $y=3$: three units UP. Why this step? A positive second coordinate means the upward direction.
4. Read the region. Left of the $y$-axis and above the $x$-axis $\Rightarrow$ signs $(-,+)$ $\Rightarrow$ Quadrant II.

Recall Verify

Sign test ::: $x<0$ and $y>0$ is the definition of Quadrant II. Look at the red dot in the figure — it sits in the top-left block. ✓

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Ex 2 — All four quadrants at once (cell C2)

Forecast: Predict the quadrant of each before reading — one lands in each of I, II, III, IV.

1. Read only the two signs of each point. Why this step? The quadrant depends on nothing but the signs; the sizes don't matter.
2. Apply the sign map $(+,+)\to \text{I},(-,+)\to \text{II},(-,-)\to \text{III},(+,-)\to \text{IV}$. Why this step? Starting from I (both positive) and turning counterclockwise, exactly one sign flips at each step.

Point	signs	Quadrant
$A(3,5)$	$(+,+)$	I
$B(-2,4)$	$(-,+)$	II
$C(-3,-2)$	$(-,-)$	III
$D(5,-3)$	$(+,-)$	IV

Recall Verify

Counterclockwise walk ::: I → II → III → IV matches the arrows in the figure, one point sitting in each block. ✓

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Ex 3 — On an axis & at the origin (cells C3, C4)

Forecast: Trap alert — how many of these three are actually inside a quadrant?

1. Check for a zero coordinate. Why this step? A quadrant requires both coordinates non-zero; a single $0$ means the point sits on a boundary line, not in a region.
2. $E(0,-4)$: $x=0$ $\Rightarrow$ on the $y$-axis, below the origin $\Rightarrow$ negative $y$-axis, no quadrant.
3. $F(6,0)$: $y=0$ $\Rightarrow$ on the $x$-axis, right of the origin $\Rightarrow$ positive $x$-axis, no quadrant.
4. $G(0,0)$: both zero $\Rightarrow$ this is the origin, the crossing point of the two axes $\Rightarrow$ no quadrant.

Recall Verify

Count ::: zero of the three lie in a quadrant; all three are boundary points. See the green markers hugging the axes in the figure. ✓

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Ex 4 — Distance with negative coordinates + zero case (cells C5, C4)

Forecast: Will the negative signs make the distance negative? (Distance is a length — can a length be negative?)

1. Write $\Delta x$ and $\Delta y$. Why this step? The distance formula only ever uses the changes in each coordinate. $\Delta x=0-(-3)=3,\Delta y=0-(-4)=4$
2. Square each change. Why this step? Squaring kills the sign — that's exactly why distance can't come out negative. It also sets up Pythagoras. $(\Delta x{)}^2=9,(\Delta y{)}^2=16$
3. Add and take the square root. Why this step? $\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}$ is the hypotenuse of the right triangle whose legs are the two changes. $OH=\sqrt{9+16}=\sqrt{25}=5$
4. Zero case $O$ to $O$: $\Delta x=0,\Delta y=0\Rightarrow \sqrt{0}=0$. Why this step? A point is distance $0$ from itself — the degenerate check that the formula must pass.

Recall Verify

Units & sign ::: length $=5$ (positive, as any length must be); the famous $3$–$4$–$5$ triple. Zero case gives $0$. ✓

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Ex 5 — Horizontal & vertical special distances (cell C6)

Forecast: $A$ and $B$ share a $y$; $B$ and $C$ share an $x$. Guess: which side is horizontal, which is vertical?

1. $AB$: same $y$ ($1=1$), so $\Delta y=0$. Why this step? When one change is zero the segment is purely horizontal and the formula collapses to just $|\Delta x|$. $AB=\sqrt{(3-(-2){)}^2+0^2}=\sqrt{25}=5$
2. $BC$: same $x$ ($3=3$), so $\Delta x=0$. Why this step? Now the segment is purely vertical, length $|\Delta y|$. $BC=\sqrt{0^2+(5-1{)}^2}=\sqrt{16}=4$
3. $AC$ (the slanted side): $AC=\sqrt{(3-(-2){)}^2+(5-1{)}^2}=\sqrt{25+16}=\sqrt{41}$
4. Pythagoras check: Why this step? A right angle sits at $B$ exactly when the two legs' squares add to the longest side's square. $A{B}^2+B{C}^2=25+16=41=A{C}^2✓$
5. Area: legs are perpendicular, so $\text{Area}=\frac{1}{2}\times 5\times 4=10\text{ square units}$

Recall Verify

Right angle & area ::: $25+16=41$ confirms $\angle B=90^{\circ }$; area $=10$ sq units. The red (horizontal) and mint (vertical) legs in the figure meet at $B$. ✓

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Ex 6 — Real-world word problem (cell C7)

Forecast: East then south — which quadrant does that land in?

1. Turn compass words into signs. Why this step? East = right = $+x$; south = down = $-y$. Words must become signs before any maths. $\text{position}=(+8,-6)=(8,-6)$
2. Name the quadrant. Why this step? $(+,-)$ is right-and-down $\Rightarrow$ Quadrant IV.
3. Straight-line distance to the origin. Why this step? The two legs (east, south) meet at a right angle at the turning point, so the direct route is the hypotenuse. $d=\sqrt{8^2+(-6{)}^2}=\sqrt{64+36}=\sqrt{100}=10\text{ m}$

Recall Verify

Sanity ::: the direct hop ($10$ m) is shorter than flying the two legs ($8+6=14$ m) — a straight line always beats a detour. Units in metres. ✓

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Ex 7 — Exam twist: reflection & an unknown (cell C8)

Forecast: Reflecting across the $x$-axis flips one coordinate — which one?

1. Recall what an $x$-axis reflection does. Why this step? Flipping over the horizontal axis keeps left/right the same but sends up $\leftrightarrow$ down. So $x$ stays, $y$ negates: $(x,y)\to (x,-y)$.
2. Match coordinates. Why this step? $N(4,7)$ reflects to $(4,-7)$. Comparing with $M(a,-7)$ forces $a=4$. $a=4$
3. Quadrants. $N(4,7)$ is $(+,+)\Rightarrow$ Quadrant I; $M(4,-7)$ is $(+,-)\Rightarrow$ Quadrant IV. Why this step? A reflection across the $x$-axis always swaps a point between the quadrants stacked above and below (I$\leftrightarrow$IV, II$\leftrightarrow$III).

Recall Verify

Reflection rule ::: $x$ unchanged ($4=4$), $y$ negated ($7\to -7$). The dashed mirror line in the figure is the $x$-axis, with $M$ and $N$ equal heights apart. ✓

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Ex 8 — Limiting behaviour: sliding across an axis (cell C9)

Forecast: With $x$ stuck at $-1$ (always left), what happens exactly when $y$ passes through $0$?

1. $y=2$ and $y=1$: $x<0,y>0\Rightarrow$ Quadrant II. Why this step? Left and above $\Rightarrow (-,+)$.
2. $y=0$: the point is $(-1,0)$ $\Rightarrow$ on the $x$-axis, no quadrant. Why this step? At the instant of crossing, one coordinate is $0$; a quadrant needs both non-zero — so there is a momentary "gap" between regions.
3. $y=-1$: $x<0,y<0\Rightarrow$ Quadrant III. Why this step? Left and below $\Rightarrow (-,-)$.
4. The limiting insight. Why this step? As $y$ decreases through $0$, the point does not jump from II straight into III — it must touch the boundary axis first. Quadrants are open regions; their edges belong to neither side.

Recall Verify

Boundary crossing ::: II ($y=2,1$) → axis ($y=0$, no quadrant) → III ($y=-1$). The colour of the dot in the figure changes lavender → slate (on-axis) → coral to show the handover. ✓

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Quick self-test

Guess before revealing.

See also: Distance formula, Midpoint formula, Area of triangle, Polar coordinates.