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Module 1 · L10 of 15 ~35 min ⚡ +95 XP available

Parametric Curves & Sketching

A Cartesian equation tells you where a curve lives. But parametric equations also tell you when a point is there, and which way it's travelling. That extra information — direction of tracing, initial and terminal points — transforms a static shape into a dynamic story. In this lesson you'll build the skill of reading that story from a table of values, and telling it back through a well-labelled sketch.

Today's hook — As $t$ increases from $0$, how does the point $(\cos t, \sin t)$ move around the unit circle — clockwise or anticlockwise? And where does it start? Think about this before you read on — your spatial reasoning will sharpen the moment you commit to an answer.

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Recall — your gut answer first

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As $t$ increases from $0$, how does the point $(\cos t, \sin t)$ move around the unit circle? Without working it out algebraically — make a prediction: clockwise or anticlockwise? Where does the point start? Where does it go first?

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The two moves

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Sketching parametric curves comes down to two core skills: build a table of values to find key points, and use the Cartesian equation to confirm the shape. Add direction arrows and label initial/terminal points and you're done.

Every parametric sketch lives on two roads: substitute parameter values to generate key coordinates and plot them, then identify the Cartesian shape to confirm what you're drawing and add direction arrows.

key points + Cartesian shape + direction = sketch

Table of values

Pick convenient $t$ values (often $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$ for trig) and compute $(x, y)$ for each. These become your skeleton points.

Direction of tracing

Add arrows showing how the point moves as $t$ increases. Direction can be non-obvious — always check a few consecutive $t$ values.

Initial & terminal points

If $t$ is bounded, calculate the points at $t_{\min}$ and $t_{\max}$. Mark these clearly — they are often asked for in the exam.

What you'll master

Know

Key facts

The initial point corresponds to the minimum parameter value
The terminal point corresponds to the maximum parameter value
The direction of tracing is determined by how $x$ and $y$ change as $t$ increases

Understand

Concepts

Why a parametric curve can double back on itself
How the parameter range restricts which part of the Cartesian curve is drawn
Why direction information is not available from the Cartesian equation alone

Can do

Skills

Build a table of values from parametric equations
Sketch parametric curves with direction arrows and labelled points
Identify initial and terminal points and describe the curve fully

Key terms

Initial pointThe point on the curve corresponding to the minimum (or starting) parameter value.

Terminal pointThe point on the curve corresponding to the maximum (or ending) parameter value.

Direction of tracingThe path followed by the point $(x, y)$ as the parameter increases through its range.

Table of valuesA systematic list of $(t, x, y)$ triples computed by substituting convenient parameter values.

Sketching strategy

core concept

To sketch a curve given parametrically, follow these five steps:

Find the Cartesian equation (if helpful) to identify the type of curve — line, parabola, circle, ellipse.
Calculate key points by substituting convenient values of $t$ into both equations.
Determine the direction by examining how $x$ and $y$ change as $t$ increases.
Identify any restrictions on $x$ or $y$ that arise from the parameter range.
Mark initial and terminal points if the parameter is bounded, and add direction arrows to the sketch.

Physics connection. In projectile motion, the position of a ball is $x = v_0 \cos\theta \cdot t$ and $y = v_0 \sin\theta \cdot t - \frac{1}{2}gt^2$. Here $t$ is actual time — so the direction of tracing tells you whether the ball is on the way up or coming down. This is information the Cartesian equation $y = x\tan\theta - \frac{gx^2}{2v_0^2\cos^2\theta}$ simply doesn't contain.

Five-step sketching method: (1) find Cartesian form, (2) key points via table, (3) direction, (4) restrictions, (5) label initial/terminal points; Always choose t values that make the trig functions equal to 0, 1 — e.g. multiples of /2

Pause — copy the five-step parametric sketching method into your book: (1) find Cartesian form; (2) table of key points at $t = 0, \pi/2, \pi, 3\pi/2, 2\pi$; (3) mark direction of tracing; (4) note domain restrictions; (5) label initial and terminal points.

Quick check: For $x = t - 1$, $y = 2t + 3$ with $0 \le t \le 3$, what is the initial point (at $t = 0$)?

Direction of tracing

core concept

We just saw the five-step sketching strategy: find Cartesian form, build a key-point table, restrict the domain, and label endpoints. That raises a question: the Cartesian form shows the path but not which way the point moves — how do you add this directional information to the sketch? This card answers it → evaluate $(x,y)$ at a few increasing $t$-values and mark arrows on the curve showing the direction of tracing.

The direction of tracing describes how the point $(x, y)$ moves as $t$ increases. This information is not visible in the Cartesian equation — it is unique to the parametric representation.

To determine direction:

Calculate $(x, y)$ at a few consecutive $t$ values.
Note whether $x$ is increasing or decreasing and whether $y$ is increasing or decreasing.
Mark arrows on your sketch to show this direction.

Key insight: A parametric curve can double back on itself — the same $x$ value can be visited multiple times (with different $y$ or $t$ values). This is impossible to represent in a standard Cartesian function $y = f(x)$.

The unit circle traced anticlockwise from $(1, 0)$ as $t$ increases from $0$ to $2\pi$.

Direction of tracing is determined by how (x, y) changes as t increases; x = t, y = t: starts at (1,0), moves anticlockwise (since is initially increasing)

Pause — copy the direction rule into your book: determine tracing direction by computing $(x,y)$ at two nearby $t$-values; for $x = \cos t$, $y = \sin t$, the point starts at $(1,0)$ and moves anticlockwise as $t$ increases.

Did you get this? True or false: the parametric equations $x = \cos t$, $y = \sin t$ trace the unit circle in an anticlockwise direction as $t$ increases.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · SEMI-CUBIC PARABOLA

Sketch the curve $x = t^2$, $y = t^3$ for $-2 \le t \le 2$, indicating direction.

Build a table of values:

$t$	$x = t^2$	$y = t^3$
$-2$	$4$	$-8$
$-1$	$1$	$-1$
$0$	$0$	$0$
$1$	$1$	$1$
$2$	$4$	$8$

Initial point: $(4, -8)$ at $t = -2$. Terminal point: $(4, 8)$ at $t = 2$. The curve passes through the origin at $t = 0$.

PROBLEM 2 · ELLIPSE

Sketch the curve $x = 2\cos t$, $y = 3\sin t$ for $0 \le t \le 2\pi$, indicating direction.

Cartesian: $\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1$ — an ellipse with $a = 2$ (horizontal), $b = 3$ (vertical).

Use $\cos^2 t + \sin^2 t = 1$ with $\cos t = x/2$ and $\sin t = y/3$.

PROBLEM 3 · LINE SEGMENT

For $x = t - 1$, $y = 2t + 3$ with $0 \le t \le 3$, find the initial and terminal points, the Cartesian equation, and sketch.

Initial point ($t = 0$): $x = -1$, $y = 3$ → $(-1, 3)$. Terminal point ($t = 3$): $x = 2$, $y = 9$ → $(2, 9)$.

Substitute the boundary values of $t$.

Fill the gap: For $x = 3\cos t$, $y = 3\sin t$ ($0 \le t \le \frac{\pi}{2}$), the initial point is and the terminal point is .

Misconceptions to fix · the 3 traps that cost marks

Trap 01

Assuming curves are always traced left to right

Parametric curves can move in any direction — including right to left, or doubling back. Always determine direction by examining how $x$ changes with $t$. For $x = t^2$, as $t$ goes from $-2$ to $0$, $x$ decreases (right to left), then increases again.

Trap 02

Forgetting to label initial and terminal points

Examiners specifically look for labelled initial and terminal points when the parameter is bounded. Stating "starts at $(4, -8)$" without marking it on the sketch costs a mark. Always compute and label both endpoints.

Trap 03

Confusing "anticlockwise" for trig with "clockwise"

$x = a\cos t$, $y = b\sin t$ always traces anticlockwise as $t$ increases. To get a clockwise trace, you need $x = a\cos t$, $y = -b\sin t$ (or $y = b\sin(-t)$). Memorise the standard form.

Did you get this? True or false: for $x = \cos t$, $y = -\sin t$ ($0 \le t \le 2\pi$), the curve is traced in a clockwise direction.

Work mode · how are you completing this lesson?

Activities · practice with the ideas

For $x = t - 1$, $y = 2t + 3$ with $0 \le t \le 3$, find the initial and terminal points and state the direction of tracing.

Describe the curve $x = 3\cos t$, $y = 3\sin t$ for $0 \le t \le \pi/2$, including its shape, key points, and direction.

For $x = t^2 - 1$, $y = t$ with $t \ge 0$, find the Cartesian equation and describe the portion traced.

Explain why the parametric curve $x = t^2$, $y = t^3$ for $-2 \le t \le 2$ doubles back on itself, but $y = x^{3/2}$ does not.

What change would you make to $x = 2\cos t$, $y = 3\sin t$ to trace the ellipse in a clockwise direction?

Odd one out: Which of these parametric curves does NOT trace an entire closed loop (i.e. does not return to its starting point)?

Revisit your thinking

Earlier you were asked: as $t$ increases from $0$, how does $(\cos t, \sin t)$ move around the unit circle?

At $t = 0$: point is $(1, 0)$ — the rightmost point. As $t$ increases, $\cos t$ decreases (point moves left) and $\sin t$ increases (point moves up). So the initial motion is anticlockwise. The point completes the full circle at $t = 2\pi$, returning to $(1, 0)$. Knowing the direction of tracing is the key advantage of parametric form over the Cartesian equation $x^2 + y^2 = 1$, which shows the shape but not the motion.

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Multiple choice

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Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 42 marks

Q1. For $x = t - 1$, $y = 2t + 3$ with $0 \le t \le 3$, find the initial and terminal points and sketch the curve indicating its direction. (2 marks)

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ApplyBand 42 marks

Q2. Describe the curve $x = 3\cos t$, $y = 3\sin t$ for $0 \le t \le \pi/2$, including its shape, the key points at each endpoint, and the direction of tracing. (2 marks)

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AnalyseBand 52 marks

Q3. For $x = t^2 - 1$, $y = t$ with $t \ge 0$, find the Cartesian equation, state any restrictions, and sketch the curve. (2 marks)

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Comprehensive answers (click to reveal)

Activity 1: 1. Initial $(-1,3)$, terminal $(2,9)$, direction left-to-right upwards · 2. Quarter-circle of radius 3, centre origin, from $(3,0)$ to $(0,3)$, anticlockwise · 3. $x = y^2 - 1$ ($y \ge 0$) — right branch of a parabola, starting at $(-1, 0)$ when $t=0$ · 4. For $t < 0$, $x = t^2 > 0$ so the curve visits positive $x$ values with negative $y$ (below $x$-axis), then revisits the same $x$ values with positive $y$ after $t=0$. The Cartesian equation $y = x^{3/2}$ only captures $y \ge 0$. · 5. Change $y = 3\sin t$ to $y = -3\sin t$.

Q1 (2 marks): Initial point $(-1, 3)$ at $t=0$; terminal point $(2, 9)$ at $t=3$ [1]. Cartesian: $t = x+1$, so $y = 2(x+1)+3 = 2x+5$; line segment from $(-1,3)$ to $(2,9)$, traced left-to-right [1].

Q2 (2 marks): The curve is the first-quadrant arc of a circle $x^2 + y^2 = 9$, radius 3 [1]. Initial point $(3, 0)$ at $t=0$; terminal point $(0, 3)$ at $t=\pi/2$; traced anticlockwise [1].

Q3 (2 marks): $t = y$ (since $y = t$), so $x = y^2 - 1$ [1]. Restriction $t \ge 0 \Rightarrow y \ge 0$: this is the upper half of the parabola $x = y^2 - 1$, starting at $(-1, 0)$ and moving right and upwards as $t$ increases [1].

Boss battle · The Curve Tracer

earn bronze · silver · gold

Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.

⚔ Enter the arena

Science Jump · platform challenge

Climb platforms by answering parametric curves questions. Lighter alternative to the boss.

Mark lesson as complete

Tick when you've finished the practice and review.

← Lesson 9 · Introduction to Parametric Equations Lesson 11 · Graphical Relationships →

Module overview · Extension 1 · Checkpoint 3 · Module quiz