AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/4/4-4)
Unit 4 · Topic 4.4
4.4 Parametrically Defined Circles and Lines
The parametric function (cos t, sin t) for 0 ≤ t ≤ 2π traces the unit circle once counterclockwise, starting at (1, 0). Transforming it gives any circular path. A line segment can be traced by starting at one endpoint and changing x and y at constant rates.
Key terms
- unit circle parametrization
- radius and center
- line segment
- initial point
The unit circle
x(t) = cos t, y(t) = sin t for 0 ≤ t ≤ 2π traces the unit circle exactly once. It starts and ends at (1, 0) and goes counterclockwise, because t plays the role of the angle θ from unit 3.
You can check that every point is on the circle: x² + y² = cos² t + sin² t = 1.
Any circle
Transformations of (cos t, sin t) give every circular path:
- Radius r: multiply both by r, giving (r cos t, r sin t).
- Center (h, k): add h to x and k to y, giving (h + r cos t, k + r sin t).
- Clockwise: replace t with −t, giving (cos t, −sin t).
- Faster: replace t with bt. (cos 2t, sin 2t) goes around once for 0 ≤ t ≤ π.
- Different start: shift t. (cos(t + π/2), sin(t + π/2)) starts at the top, (0, 1).
Checking a circle and drawing arcs
To check a circle parametrization, substitute it into (x − h)² + (y − k)² = r². For (h + r cos t, k + r sin t), the left side becomes r² cos² t + r² sin² t = r².
For only part of a circle, restrict t. The upper half of the unit circle, traced from (1, 0) to (−1, 0), is (cos t, sin t) for 0 ≤ t ≤ π.
Line segments
To go from (x₁, y₁) to (x₂, y₂), start at the first point and change x and y at constant rates: x(t) = x₁ + (x₂ − x₁)t and y(t) = y₁ + (y₂ − y₁)t for 0 ≤ t ≤ 1.
At t = 0 you're at (x₁, y₁), and at t = 1 you're at (x₂, y₂). Halfway, at t = 1/2, you're at the midpoint.
If the trip should take a different amount of time, say T, use rates (x₂ − x₁)/T and (y₂ − y₁)/T and let t run from 0 to T. The path is the same segment; only the timing changes.
To trace the same segment in the other direction, start at (x₂, y₂) instead: x(t) = x₂ + (x₁ − x₂)t and y(t) = y₂ + (y₁ − y₂)t.
Rates on a line
On a parametrized line, x and y each change at a constant rate per unit of t. The ratio of those rates is the slope of the line, which matches topic 4.3: the secant slope between any two points is the same.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A circle with a given center
Write a parametric function for the circle with center (3, −2) and radius 5, traced once counterclockwise starting at its rightmost point.
Show the solutionHide the solution
- Step 1: Start from (cos t, sin t), which begins at its rightmost point (1, 0) and goes counterclockwise.
- Step 2: Scale by 5 and shift by (3, −2): x(t) = 3 + 5 cos t, y(t) = −2 + 5 sin t.
- Step 3: Check the start: t = 0 gives (8, −2), which is 5 units right of the center.
Answer: x(t) = 3 + 5 cos t, y(t) = −2 + 5 sin t, for 0 ≤ t ≤ 2π.
- Example 2
A segment over a set time
A particle moves along the segment from (−1, 4) to (5, 1) at constant rates, taking 3 seconds. Write a parametric function and find where it is at t = 1.5.
Show the solutionHide the solution
- Step 1: Change in x: 5 − (−1) = 6 over 3 seconds, so x changes at 2 units per second.
- Step 2: Change in y: 1 − 4 = −3 over 3 seconds, so y changes at −1 unit per second.
- Step 3: x(t) = −1 + 2t, y(t) = 4 − t, for 0 ≤ t ≤ 3. Check: t = 3 gives (5, 1).
- Step 4: At t = 1.5: (−1 + 3, 4 − 1.5) = (2, 2.5), the midpoint.
Answer: x(t) = −1 + 2t, y(t) = 4 − t, 0 ≤ t ≤ 3; at t = 1.5 it's at (2, 2.5).
- Example 3
Trap: clockwise from the top
Parametrize the unit circle traced once clockwise, starting at (0, 1).
Show the solutionHide the solution
- Step 1: Try x(t) = sin t, y(t) = cos t. At t = 0 this gives (0, 1).
- Step 2: At t = π/2 it gives (1, 0). Going from the top to the right side is a clockwise move.
- Step 3: A common slip is (cos t, sin t) with a shifted t, which still goes counterclockwise. Check two points to confirm the direction.
Answer: x(t) = sin t, y(t) = cos t for 0 ≤ t ≤ 2π.
Common mistakes
- Getting the direction wrong. Always check where the particle is at two nearby times.
- Adding the center to the wrong coordinate: h goes with x and k goes with y.
- Writing a segment as x₁ + x₂t instead of x₁ + (x₂ − x₁)t.
On the exam
- Unit 4 is not on the AP Precalculus Exam, so you'll see this topic on class tests rather than in May.
- Expect to write parametric equations for a circle or segment with a given start, direction and timing, and to check them by plugging in t-values.
Connected topics
Videos
Check yourself
3 questions on 4.4 Parametrically Defined Circles and Lines. Pick an answer to see if you got it, and why.
Which of the following parametric functions traces the circle of radius 4 centered at the origin exactly once, clockwise, starting at the point (0, 4), for 0 ≤ t ≤ 2π?
Which of the following parametric functions traces the line segment from (2, −1) to (8, 3) for 0 ≤ t ≤ 1?
Which of the following describes the curve x(t) = 2 + 3 cos t, y(t) = −1 + 3 sin t for 0 ≤ t ≤ 2π?
0 of 3 answered