Unit 4 · Topic 4.4 · about 30 minutes · 7 parts · 13 scored checks, 2 of them multiple choice

AP Precalculus 4.4: Parametrically Defined Circles and Lines

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Not assessed on the AP Exam. Unit 4 is part of the course, but nothing in it is on the exam.

Two graphs side by side. Left: the unit circle traced by x(t) = cos t and y(t) = sin t, with arrows pointing counterclockwise. A square marks the start at (1, 0), labeled t = 0 and t = 2 pi, and dots mark (0, 1) at t = pi/2, (-1, 0) at t = pi and (0, -1) at t = 3 pi/2. Right: a line segment from (1, 2) to (9, 8) traced by x(t) = 1 + 2t and y(t) = 2 + 1.5t for t from 0 to 4. Dots mark the point at t = 0, 1, 2, 3 and 4, evenly spaced, each 2 units right and 1.5 units up from the one before.

Part 1 Think first Swap the coin's coordinates

Make a guess before the lesson starts. A wrong guess still helps the right answer stick.

Predict first

A game designer makes a coin fly in a circle. She sets its position to $x = \cos t$ and $y = \sin t$ and lets the time $t$ run from 0 to $2 π$ . The coin starts at $(1, 0)$ and goes once around counterclockwise. Then she swaps the two, so $x = \sin t$ and $y = \cos t$ . What does the coin do now?

In short. A parametric function can move a point around a circle, built from $(\cos t, \sin t)$ on the unit circle, or along a line segment, built from an initial position and constant rates of change.

$(x (t), y (t)) = (\cos t, \sin t)$ with domain $0 \leq t \leq 2 π$ makes one complete counterclockwise revolution around the unit circle, starting and ending at $(1, 0)$ .
Transformations of it model any circular path: $(h + r \cos t, k + r \sin t)$ has center $(h, k)$ and radius $r$ , and changing the $t$ inside sets the start, the direction and the number of trips.
From $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ in time $T$ : $x (t) = x_{1} + \frac{x_{2} - x_{1}}{T} t$ and $y (t) = y_{1} + \frac{y_{2} - y_{1}}{T} t$ , for $0 \leq t \leq T$ .

Part 2 Learn it Circles, then segments

2a. One trip around the unit circle

You already know the point at angle $t$ on the unit circle: $(\cos t, \sin t)$ . Now let $t$ be time, and let the point move.

At $t = 0$ the point $(x (t), y (t)) = (\cos t, \sin t)$ is at $(1, 0)$ . At $t = \frac{π}{2}$ it is at the top, $(0, 1)$ . Then $(- 1, 0)$ at $t = π$ , $(0, - 1)$ at $t = \frac{3 π}{2}$ , and back to $(1, 0)$ at $t = 2 π$ . The angle keeps growing, so the point turns counterclockwise, the way angles open in standard position.

With domain $0 \leq t \leq 2 π$ , that is one complete counterclockwise revolution around the unit circle, centered at the origin, starting and ending at $(1, 0)$ . To find where the point is at any time, use the unit circle values you already have. At $t = \frac{π}{6}$ it is at $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ .

The domain matters as much as the formula. Stop at $t = π$ and the point covers only the top half. Let $t$ run to $4 π$ and it goes round twice.

Quick check · Match

A point moves along $(x (t), y (t)) = (\cos t, \sin t)$ for $0 \leq t \leq 2 π$ . Tap a time, then where the point is at that time.

2b. Any circle, from the unit circle

Multiply both coordinates by $r$ and the point $(r \cos t, r \sin t)$ stays $r$ from the origin, so it goes around the circle of radius $r$ . Add $h$ and $k$ to move the center to $(h, k)$ :

(x (t), y (t)) = (h + r \cos t, k + r \sin t), 0 \leq t \leq 2 π

That is still one counterclockwise trip. It starts at $(h + r, k)$ , the rightmost point, because $\cos 0 = 1$ and $\sin 0 = 0$ .

Say a robot mower circles a tree at $(4, 3)$ on a yard map in meters, always 2 meters from the trunk. Its path is $(4 + 2 \cos t, 3 + 2 \sin t)$ , and it starts at $(6, 3)$ .

You can check that against the circle's equation, $(x - 4)^{2} + (y - 3)^{2} = 4$ . Substitute: $(2 \cos t)^{2} + (2 \sin t)^{2} = 4 (\cos^{2} t + \sin^{2} t) = 4$ . Every point of the path is on the circle.

Two things trip people up. The equation shows $r^{2} = 4$ , but the parametric function multiplies by $r = 2$ . And the center keeps its own signs: a tree at $(4, - 3)$ gives $y (t) = - 3 + 2 \sin t$ .

Quick check · Multiple choice

A point goes once counterclockwise around the circle $(x - 3)^{2} + (y + 2)^{2} = 16$ , starting at its rightmost point, for $0 \leq t \leq 2 π$ . Which parametric function models this motion?

2c. Where it starts, which way it goes, how many trips

Changing the $t$ inside cosine and sine changes the trip, not the circle.

Replace $t$ with $- t$ : $(\cos (- t), \sin (- t)) = (\cos t, - \sin t)$ . Same start, $(1, 0)$ , but the point now turns clockwise.
Replace $t$ with $t + c$ : the point starts at angle $c$ instead of 0. $(\cos (t + π), \sin (t + π))$ starts at $(- 1, 0)$ and still turns counterclockwise.
Replace $t$ with $b t$ , for $b > 0$ : the angle grows $b$ times as fast. $(\cos (3 t), \sin (3 t))$ goes around three times for $0 \leq t \leq 2 π$ , and once for $0 \leq t \leq \frac{2 π}{3}$ . The number of trips is $b$ times the length of the interval, divided by $2 π$ .

When a formula fits none of these, test it. Find the point at $t = 0$ , then at a small $t$ such as 0.1, and see which way it moved. Picture a clock: moving right across the top, or down the right side, is clockwise.

Try $(- \cos t, \sin t)$ . At $t = 0$ it is at $(- 1, 0)$ , the leftmost point. At $t = 0.1$ , $y = \sin 0.1$ is just above 0, so the point moved up the left side. On a clock that is clockwise.

Quick check · Sort it

Each card is a parametric function for $0 \leq t \leq 2 π$ , and each one traces a circle. Tap a card, then the way the point goes round.

Counterclockwise

Clockwise

2d. Along a line segment

Straight-line motion needs no trigonometry. You need the initial position and how fast each coordinate changes.

A skateboarder rolls in a straight line at a steady speed from $(1, 2)$ to $(9, 8)$ , in meters, in 4 seconds. Over those 4 seconds $x$ changes by $9 - 1 = 8$ , so the rate of change of $x$ with respect to $t$ is $\frac{8}{4} = 2$ meters per second. $y$ changes by $8 - 2 = 6$ , so its rate is $\frac{6}{4} = 1.5$ . Start at the initial position and add rate times $t$ :

x (t) = 1 + 2 t, y (t) = 2 + 1.5 t, 0 \leq t \leq 4

At $t = 4$ that gives $(9, 8)$ , as it should. The domain does the stopping. The same formulas for $t$ past 4 carry on along the line.

In general, from $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ in time $T$ , the rates are $\frac{x_{2} - x_{1}}{T}$ and $\frac{y_{2} - y_{1}}{T}$ . One segment has many parametrizations. $(1 + 8 t, 2 + 6 t)$ for $0 \leq t \leq 1$ covers it in 1 second. $(9 - 2 t, 8 - 1.5 t)$ for $0 \leq t \leq 4$ covers it backward, from $(9, 8)$ to $(1, 2)$ .

Quick check · Fill the statement

A ball rolls in a straight line at a steady speed from $(- 2, 5)$ to $(6, - 1)$ , in meters, in 4 seconds, with $t$ in seconds. Tap a blank, then tap a number.

In the ball's parametric function, the initial position has $x$ -coordinate . Each second, $x$ changes by meters and $y$ changes by meters, and the domain is $0 \leq t \leq$ .

Part 3 Watch it done The tip of a minute hand

Read each step, then do the last one yourself.

A clock is drawn on a grid in centimeters with its center at $(2, - 1)$ . The tip of its minute hand is 5 cm from the center. Write a parametric function for the tip over one hour, with $0 \leq t \leq 2 π$ , starting at the top of the clock and moving clockwise.

Step 1 · Start from the unit circle
$(\cos t, \sin t)$ for $0 \leq t \leq 2 π$ goes once around, counterclockwise, from $(1, 0)$ . The minute hand needs two changes: start at the top, and go the other way.
Step 2 · Fix the start and the direction
Replace $t$ with $\frac{π}{2} - t$ : $(\cos (\frac{π}{2} - t), \sin (\frac{π}{2} - t))$ . At $t = 0$ the angle is $\frac{π}{2}$ , so the point is $(0, 1)$ , the top. At $t = 2 π$ the angle is $\frac{π}{2} - 2 π$ , which shares a terminal ray with $\frac{π}{2}$ , so the hour ends at the top too.
Why this step?
Why does replacing $t$ with $\frac{π}{2} - t$ make the point go clockwise?
In $(\cos θ, \sin θ)$ , a growing angle $θ$ turns the point counterclockwise. Here $θ = \frac{π}{2} - t$ , which shrinks as $t$ grows, so the point turns the opposite way.
Step 3 · Stretch and move the circle
Multiply by the radius 5 and add the center's coordinate. For $x$ that gives $x (t) = 2 + 5 \cos (\frac{π}{2} - t)$ , for $0 \leq t \leq 2 π$ . Check $t = 0$ : $x (0) = 2 + 5 \cos \frac{π}{2} = 2$ , the center's $x$ , so the tip starts straight above the center. The $y$ -coordinate is built the same way, from the sine and the center's $y$ .
Your step · Calculate

Finish it. The tip of a minute hand goes once clockwise around a circle with center $(2, - 1)$ and radius 5 cm, starting at the top, and its $x$ -coordinate is $x (t) = 2 + 5 \cos (\frac{π}{2} - t)$ for $0 \leq t \leq 2 π$ . Write $y (t)$ the same way. What is $y (\frac{π}{3})$ ? Give a decimal.

Your answer

By the difference identities, $\cos (\frac{π}{2} - t) = \sin t$ and $\sin (\frac{π}{2} - t) = \cos t$ , so the tip is also $(2 + 5 \sin t, - 1 + 5 \cos t)$ . That is the coin from the start of the lesson: swapping sine and cosine starts at the top and turns clockwise.

Part 4 Try it A drone's straight flight

Warm up on the grapher first. It shows a segment from $(- 2, - 1)$ to $(3, 2)$ , traced for $0 \leq t \leq 1$ , with the path on top and $x$ and $y$ against $t$ underneath. Drag Parameter t from 0 to 1 and watch $x$ grow by 5 and $y$ by 3, each at a steady rate. Then move Interval of t ends at up to 2. The path runs on past $(3, 2)$ along the same line: the formulas do not stop, the domain does. Set Interval of t starts at to $- 1$ and the line runs back past $(- 2, - 1)$ as well.

Lab

Parametric Grapher

The segment is fixed for this warm-up. Move Parameter t and the two interval sliders only.

Open the full Parametric Grapher lab

Now take paper. A delivery drone flies in a straight line at a steady speed from a depot at $(4, 1)$ to a house at $(- 2, 10)$ , with distances in kilometers on a map. The flight takes 3 minutes, and $t$ is the time in minutes since takeoff.

Try it · Calculate

A delivery drone flies in a straight line at a steady speed from a depot at $(4, 1)$ to a house at $(- 2, 10)$ , in kilometers, in 3 minutes, with $t$ in minutes. What is the rate of change of $y$ with respect to $t$ , in kilometers per minute?

Your answer

Try it · Decide and justify

A delivery drone flies in a straight line at a steady speed from a depot at $(4, 1)$ to a house at $(- 2, 10)$ , in kilometers, in 3 minutes, with $t$ in minutes since takeoff. Does $(x (t), y (t)) = (- 2 + 2 t, 10 - 3 t)$ for $0 \leq t \leq 3$ model this flight?

Which reason supports your answer?

Part 5 Explain it Put it in writing

Nothing you type leaves this page. Write your answer, then check it against the model and each line of the list.

Unit 4 is not on the AP Exam, so this is written practice for the course.

Written practice

On a park map in meters, with east as the positive $x$ direction and north as the positive $y$ direction, a round pond has its center at $(40, 30)$ and a radius of 10. (i) Write a parametric function that models a duck swimming once counterclockwise around the edge of the pond, starting and ending at the easternmost point, $(50, 30)$ , for $0 \leq t \leq 2 π$ . (ii) Describe how to change your function so the duck makes the same trip clockwise, and explain how you know it now goes clockwise. (iii) A footpath runs straight from the gate at $(0, 0)$ to $(50, 30)$ . A jogger covers it at a steady speed in 100 seconds. Write a parametric function for the jogger's position, with its domain, and identify the rates of change of $x$ and $y$ with respect to $t$ .

Your answer stays on this page. It is never sent or saved.

Model answer

(i) $(x (t), y (t)) = (40 + 10 \cos t, 30 + 10 \sin t)$ for $0 \leq t \leq 2 π$ . At $t = 0$ it gives $(50, 30)$ , and it turns counterclockwise like $(\cos t, \sin t)$ . (ii) Replace $t$ with $- t$ , which gives $(40 + 10 \cos t, 30 - 10 \sin t)$ . It still starts at $(50, 30)$ , but a moment after $t = 0$ its $y$ is below 30, so the duck moves south from the easternmost point. On a map with north up, that is clockwise. (iii) $x$ changes by 50 meters and $y$ by 30 meters in 100 seconds, so the rates are 0.5 and 0.3 meters per second. The function is $(x (t), y (t)) = (0.5 t, 0.3 t)$ for $0 \leq t \leq 100$ , which reaches $(50, 30)$ at $t = 100$ .

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes $(40 + 10 \cos t, 30 + 10 \sin t)$ for $0 \leq t \leq 2 π$ , with radius 10, not 100.
Replaces $t$ with $- t$ (or negates the sine term) and keeps the start at $(50, 30)$ .
Explains the direction from where the duck goes just after $t = 0$ : south from the easternmost point, which is clockwise.
Writes $(0.5 t, 0.3 t)$ with domain $0 \leq t \leq 100$ and identifies the rates as 0.5 and 0.3 meters per second.

Part 6 Check yourself Moving platforms

The two case questions use the same platform table. Answer them from it.

A designer scripts two moving platforms for a video game level. Positions are in meters on the level map, and $t$ is the time in seconds. Each platform follows the parametric function in the table over the domain given.

Platform paths on the level map
Platform	Position (x(t), y(t))	Domain
$A$	$(5 + 4 \cos t, 2 + 4 \sin t)$	$0 \leq t \leq 2 π$
$B$	$(5 + 4 \cos (2 t), 2 - 4 \sin (2 t))$	$0 \leq t \leq 2 π$

Case question 1 · Multiple choice

Platform $B$ moves along $(5 + 4 \cos (2 t), 2 - 4 \sin (2 t))$ for $0 \leq t \leq 2 π$ , in meters. Which describes its motion?

Case question 2 · Calculate

Platform $A$ moves along $(5 + 4 \cos t, 2 + 4 \sin t)$ for $0 \leq t \leq 2 π$ , in meters, with $t$ in seconds. At what time $t$ is platform $A$ at its lowest point? Give a decimal to two places.

Your answer

The rest stand on their own and do not use the case.

Check yourself · Spot the error

A student writes a parametric function for a point that moves in a straight line at a steady speed from $(3, - 4)$ to $(- 5, 2)$ in 2 seconds. Tap the first line that goes wrong.

Check yourself · Put it in order

A point moves along $(- 3 \sin t, 3 \cos t)$ for $0 \leq t \leq 2 π$ . Tap the four points in the order the point first reaches them.

Part 7 Keep it From earlier lessons

Ideas you have not used in a while fade. One finds a point on a circle, where this lesson started; the other reads a slope from the rates of change of $x$ and $y$ , as a line segment does.

A groundskeeper maps a soccer field with the origin at the center spot. East is the polar axis, so the positive $x$ -axis points east and the positive $y$ -axis points north. Distances are in meters and angles in radians. Three markers were recorded in different ways.

Markers on the field map
Marker	Recorded as	Coordinate system
P	$(10, \frac{5 π}{6})$	Polar
Q	$(- 8, \frac{π}{4})$	Polar
R	$(- 4 \sqrt{3}, 4)$	Rectangular

From 3.13 Trigonometry and Polar Coordinates · Calculate

A field map has the origin at the center spot and east as the polar axis, in meters. Marker P has polar coordinates $(10, \frac{5 π}{6})$ . How many meters north of the center spot is marker P? That is its $y$ -coordinate.

Your answer

From 4.3 Parametric Functions and Rates of Change · Calculate

A particle moves along $(2^{t}, t^{2} + 2 t)$ . Use the average rates of change of $x$ and $y$ over $1 \leq t \leq 3$ to find the slope of the graph between the points where $t = 1$ and $t = 3$ .

Your answer

In three lines

$(\cos t, \sin t)$ for $0 \leq t \leq 2 π$ goes once counterclockwise around the unit circle, from $(1, 0)$ back to $(1, 0)$ .
$(h + r \cos t, k + r \sin t)$ moves the circle to center $(h, k)$ and radius $r$ ; $- t$ reverses it, $t + c$ moves the start, and $b t$ makes $b$ times as many trips.
A segment from $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ in time $T$ starts at $(x_{1}, y_{1})$ and changes at the rates $\frac{x_{2} - x_{1}}{T}$ and $\frac{y_{2} - y_{1}}{T}$ .

Vocabulary

Parametric function: A function that gives a point $(x (t), y (t))$ in the plane for each value of the parameter $t$ in its domain.
Unit circle: The circle of radius 1 centered at the origin, traced once by $(\cos t, \sin t)$ for $0 \leq t \leq 2 π$ .
Counterclockwise revolution: One full trip around a circle in the direction angles open in standard position, ending where it started.
Line segment: The part of a line between two points, such as $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ ; a parametric function stays on it when its domain stops it at both ends.
Initial position: Where the moving point is when $t$ starts, such as $(x_{1}, y_{1})$ for motion from $(x_{1}, y_{1})$ to $(x_{2}, y_{2})$ .
Rate of change of x with respect to t: How much $x$ changes per unit of $t$ ; along a segment covered at a steady speed in time $T$ it is $\frac{x_{2} - x_{1}}{T}$ .
Rate of change of y with respect to t: How much $y$ changes per unit of $t$ ; along a segment covered at a steady speed in time $T$ it is $\frac{y_{2} - y_{1}}{T}$ .

Course alignment, for teachers

AP Precalculus topic 4.4, Unit 4: Functions Involving Parameters, Vectors, and Matrices.

Skill 1.B: Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context.
Skill 1.C: Construct new functions, using transformations, compositions, inverses, or regressions, that may be useful in modeling contexts, criteria, or data, with and without technology.

This lesson maps all 3 essential knowledge statements of the topic to the part that teaches each one: 3 are checked by a scored question. The map is in the lesson file.

Part 1 Think first Swap the coin's coordinates

Part 2 Learn it Circles, then segments

2a. One trip around the unit circle

2b. Any circle, from the unit circle

2c. Where it starts, which way it goes, how many trips

2d. Along a line segment

Part 3 Watch it done The tip of a minute hand

Part 4 Try it A drone's straight flight

Parametric Grapher

Part 5 Explain it Put it in writing

Part 6 Check yourself Moving platforms

Part 7 Keep it From earlier lessons

Vocabulary

Labs that use this topic