Curve tracing (cartesian and polar)

At a Glance

• Frequency: 2 sub-parts across 2 of 13 years (2022, 2023)
• Priority tier: T3
• Marks (count): 20 (2)
• Average solve time: ~10 min
• Difficulty mix: hard 1, medium 1
• Section: A | Dominant type: computation

Why This Chapter Matters

Curve tracing carries 20 marks — the largest single question in the calculus cluster and twice the weight of a typical Section A item. The examiner asks for a complete systematic analysis: domain, symmetry, intercepts, asymptotes, and a rough sketch. Both past questions involve implicit equations that must be solved for $y^2$, followed by a sign analysis to find the real domain. Mastering the seven-step checklist below and the sign-table method for domain analysis turns this high-stakes item into a routine procedure.

Minimum Theory

Seven-step curve-tracing checklist (Cartesian). For any curve $F(x,y)=0$: (1) Rewrite as $y^2=f(x)$ or $y=f(x)$ if possible. (2) Domain: find where $y^2\ge 0$ via a sign table; boundary points are curve endpoints or asymptotes. (3) Symmetry: $y\to-y$ invariance means $x$-axis symmetry; $x\to-x$ means $y$-axis symmetry; both means origin symmetry. (4) Intercepts: set $x=0$ and $y=0$. (5) Asymptotes: vertical asymptotes where denominator $\to 0$; horizontal asymptotes from $y^2\to c$ as $x\to\pm\infty$; oblique asymptotes from $y/x\to m$. (6) Tangent at special points: at a boundary point $(a,0)$ where $y^2\to 0$, compute $dy/dx$ to determine if the curve ends sharply or with a vertical tangent. (7) Monotonicity: whether $|y|$ increases or decreases as $x$ increases.

Sign table method. To determine the sign of $\frac{P(x)}{Q(x)}$: list all real zeros of $P$ and $Q$ in order, test one value in each interval, and track sign changes. The curve exists precisely where the expression is $\ge 0$.

Vertical tangent criterion. From $y^2=f(x)$, differentiate: $2y\,dy/dx=f'(x)$, so $dy/dx=f'(x)/(2y)$. At a boundary point where $y\to 0^+$ and $f'(x_0)\ne 0$, $dy/dx\to\infty$ — the curve has a vertical tangent at $(x_0,0)$.

Question Archetypes

curve-tracing (2 question(s); 2022, 2023)

Recognition Cues

• “Trace the curve $y^2 Q(x)=P(x)$” — an implicit equation that rearranges to $y^2=P/Q$.
• The question awards 20 marks, signalling that a complete systematic treatment with labelled steps and a sketch is expected.
• The equation involves even powers of $y$ only (so $y=0$ symmetry); the $x$-structure gives a finite real domain.

Solution Template

1. Rewrite as $y^2=f(x)$. Factor numerator and denominator.
2. Sign table. List all zeros of $f$; determine sign of $f(x)$ in each interval; the curve exists where $f\ge 0$.
3. Symmetry. State axis/origin symmetry.
4. Intercepts. Find where $y=0$ (zeroes of numerator) and where $x=0$.
5. Asymptotes. Vertical: where denominator $\to 0$; horizontal: $\lim_{|x|\to\infty}f(x)$.
6. Tangent at boundary points. Compute $dy/dx=f'(x)/(2y)$ at the endpoints; state “vertical tangent” if $dy/dx\to\infty$.
7. Monotonicity. Whether $y$ increases/decreases along each branch.
8. Sketch. Label all key features.

Worked Example

2022 Paper 1, 2022-P1-Q4b (20 marks)

Trace the curve $y^2 x^2=x^2-a^2$ (where $a$ is a real constant).

Step 1 — Rewrite.

$y^2=\frac{x^2-a^2}{x^2}=1-\frac{a^2}{x^2}.$

Step 2 — Domain. $y^2\ge 0\Leftrightarrow 1-a^2/x^2\ge 0\Leftrightarrow x^2\ge a^2\Leftrightarrow|x|\ge|a|$. Curve exists for $x\le -|a|$ and $x\ge|a|$. These are two disjoint branches.

Step 3 — Symmetry. $y^2$ depends only on $x^2$: invariant under both $x\to-x$ and $y\to-y$. Symmetric about both axes (and origin).

Step 4 — Intercepts. At $x=\pm a$: $y^2=0$, so $y=0$. Curve passes through $(\pm a,0)$.

Step 5 — Asymptotes. As $|x|\to\infty$: $y^2\to 1-0=1$, so $y\to\pm 1$. Horizontal asymptotes $y=\pm 1$. No vertical asymptote (curve has no denominator factor tending to zero; the exclusion is a gap, not an asymptote).

Step 6 — Tangent at $(\pm a,0)$.

$\frac{dy}{dx}=\frac{1}{2y}\cdot\frac{2a^2}{x^3}=\frac{a^2}{yx^3}.$

At $x=a^+$, $y\to 0^+$: $dy/dx\to+\infty$. Vertical tangent at $(\pm a,0)$.

Step 7 — Monotonicity. For $x>a$: $y^2=1-a^2/x^2$ increases as $x$ increases (since $a^2/x^2$ decreases). So $|y|$ increases from $0$ at $x=a$ toward $1$ as $x\to\infty$.

Step 8 — Sketch description. Two symmetric “trumpet” pieces, each with upper and lower arcs. Left piece: from $(-\infty, \pm 1)$ narrowing to the point $(-a,0)$ with a vertical tangent. Right piece: mirror image, from $(a,0)$ upward and rightward to the asymptote $y=\pm 1$.

$\boxed{\;\text{Domain: }|x|\ge|a|;\;\text{symm. both axes;}\;\text{intercepts }(\pm a,0);\;\text{horiz. asymptotes }y=\pm 1;\;\text{vertical tangents at }(\pm a,0).\;}$

2023 Paper 1, 2023-P1-Q4b (20 marks)

Trace the curve $y^2(x^2-1)=2x-1$.

Step 1 — Rewrite.

$y^2=\frac{2x-1}{x^2-1}=\frac{2x-1}{(x-1)(x+1)}.$

Step 2 — Sign table. Zeros: $x=1/2$ (numerator), $x=\pm 1$ (denominator).

Domain: $x\in(-1,\tfrac{1}{2}]\cup(1,\infty)$.

Step 3 — Symmetry. Only $y^2$ appears: symmetric about the $x$-axis. No symmetry in $x$.

Step 4 — Intercepts. At $x=1/2$: $y=0$. At $x=0$: $y^2=(-1)/(-1)=1$, so $y=\pm 1$.

Step 5 — Asymptotes.

Vertical: as $x\to-1^+$: $(x+1)\to 0^+$, $(x-1)\to-2$, $2x-1\to-3$; so $y^2=(-3)/(0^-\cdot(-2))=(-3)/0^-\to+\infty$. Vertical asymptote $x=-1$. As $x\to 1^+$: $(x-1)\to 0^+$, $2x-1\to 1$; $y^2\to+\infty$. Vertical asymptote $x=1$.

Horizontal (right branch): as $x\to+\infty$: $y^2\sim 2x/x^2=2/x\to 0$. The $x$-axis is a horizontal asymptote for the right branch.

Step 6 — Tangent at $(1/2,0)$.

$\frac{dy}{dx}=\frac{1}{2y}\cdot\frac{d}{dx}\!\left(\frac{2x-1}{x^2-1}\right).$

Derivative of $f(x)=(2x-1)/(x^2-1)$: $f'(x)=\frac{2(x^2-1)-(2x-1)(2x)}{(x^2-1)^2}=\frac{-2x^2+2x+2}{(x^2-1)^2}$.

At $x=1/2$: $f'(1/2)=(-1/2+1+2)/((-3/4)^2)=(5/2)/(9/16)=40/9>0$. As $y\to 0$, $dy/dx\to\infty$. Vertical tangent at $(1/2,0)$.

Step 7 — Monotonicity. Left branch $(-1,1/2]$: $|y|$ decreases from $+\infty$ (at $x=-1^+$) to $0$ at $x=1/2$, giving a closed loop. Right branch $(1,\infty)$: $|y|$ decreases from $+\infty$ (at $x=1^+$) toward $0$ as $x\to\infty$.

Step 8 — Sketch description. Left piece: a closed “beak” between $x=-1$ and $x=1/2$, with $y\to\pm\infty$ as $x\to-1^+$ and pinching to a point at $(1/2,0)$. Right piece: two arcs from $\pm\infty$ at $x=1^+$, asymptotically approaching the $x$-axis.

$\boxed{\;\text{Domain: }(-1,1/2]\cup(1,\infty);\;\text{vert. asymp. }x=\pm 1;\;\text{intercepts }(1/2,0),(0,\pm 1);\;\text{right branch }\to x\text{-axis.}\;}$

Common Traps

• Missing the gap. Always set up the sign table first; many students assume the curve is connected and miss the excluded interval (e.g.\ $(1/2,1)$ in 2023 and $(-|a|,|a|)$ in 2022).
• Confusing vertical asymptotes with excluded domain. In 2022, $|x|<|a|$ is just an excluded region (not an asymptote); in 2023, $x=\pm 1$ are genuine vertical asymptotes because $y\to\infty$ there.
• Forgetting vertical tangents. At any boundary point where $y^2\to 0$ and $f'(x)\ne 0$, the tangent is vertical. Always compute $dy/dx=f'/(2y)$ at such points.
• Sign errors in the table. Tracking the sign of $(x-1)(x+1)$ near $x=-1$: for $x$ slightly above $-1$, $x+1>0$ and $x-1<0$, so the product is negative. Verify each interval independently.
• Horizontal asymptote only on one branch. In 2023, the $x$-axis is an asymptote only for the right branch ($x\to+\infty$); the left branch ends at a vertical asymptote, not at the $x$-axis.

Marks-Aware Writing

Both questions are 20 marks. Full marks require all seven features to be clearly addressed. A marking scheme breakdown: domain/sign analysis (4 marks), symmetry (2 marks), intercepts (2 marks), asymptotes — vertical and horizontal (4 marks), tangents at boundary points (3 marks), monotonicity/shape (2 marks), sketch with all features labelled (3 marks). An answer that correctly finds the domain and asymptotes but skips tangents and monotonicity earns about 12 marks. Always present the sign table explicitly — it shows method and earns marks even if a subsequent conclusion has an error.

Practice Set

• 2016-P1-Q8d (15 m) — — Hint: follow the seven-step checklist; isolate $y^2$ and perform the sign analysis before sketching.