4.4 Parametrically Defined Circles and Lines

Parametric circles and lines let you describe motion in the plane using a single parameter $t$. A counterclockwise trip around the unit circle is $(x(t), y(t)) = (\cos t, \sin t)$ for $0 \le t \le 2\pi$, and you can shift, scale, and adjust this to make any circle.

Why This Matters for the AP Precalculus Exam

Unit 4 topics, including this one, are not assessed on the AP Precalculus exam. The exam covers Units 1, 2, and 3. Still, this topic is worth learning if your class includes it, because it builds the kind of component-based thinking used in calculus and in science fields where you analyze horizontal and vertical motion separately.

This topic strengthens skills you will use throughout AP Precalculus: recognizing the same curve across multiple representations, choosing a parametrization that fits a situation, and explaining why a setup produces the motion you want. Using technology to set viewing windows and parameter restrictions for parametric graphs is also good practice for the kind of calculator work this course expects.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Key Takeaways

• A full counterclockwise revolution around the unit circle starting and ending at $(1,0)$ is $(x(t), y(t)) = (\cos t, \sin t)$ with domain $0 \le t \le 2\pi$.
• Transforming $(\cos t, \sin t)$ by shifting the center and scaling the radius models any circular path in the plane.
• For a circle centered at $(h, k)$ with radius $r$, use $(x(t), y(t)) = (h + r\cos t, k + r\sin t)$.
• A line segment from $(x_1, y_1)$ to $(x_2, y_2)$ can be parametrized using an initial position and rates of change for $x$ and $y$ with respect to $t$.
• The same curve can be parametrized in more than one way and traversed in different directions.
• Restricting the domain of $t$ controls where the path starts and ends.

Parametric Circles

A complete counterclockwise revolution around the unit circle that starts and ends at $(1,0)$ and is centered at the origin is modeled by

$(x(t), y(t)) = (\cos t, \sin t)$

with domain $0 \le t \le 2\pi$.

The equation $x(t) = \cos t$ gives the x-coordinate of the point as it moves counterclockwise. The value of $\cos t$ ranges from $-1$ to $1$, with $\cos 0 = 1$ and $\cos 2\pi = 1$, which matches the start and end point $(1, 0)$ on the x-axis.

The equation $y(t) = \sin t$ gives the y-coordinate. The value of $\sin t$ ranges from $-1$ to $1$, with $\sin 0 = 0$ and $\sin 2\pi = 0$, which also matches the point $(1, 0)$.

Setting the domain to $0 \le t \le 2\pi$ guarantees the point traces exactly one complete revolution, starting at $(1, 0)$ and returning to $(1, 0)$.

Transformations

The function $(x(t), y(t)) = (\cos t, \sin t)$ is the standard parametrization of the unit circle centered at the origin. By transforming it, you can model any circular path in the plane.

To model a circle centered at $(h, k)$ with radius $r$, use

$(x(t), y(t)) = (h + r\cos t, k + r\sin t)$

This shifts the center from the origin to $(h, k)$ and scales the radius from $1$ to $r$.

You can also shift the starting angle by replacing $t$ with $t + c$:

$(x(t), y(t)) = (\cos(t + c), \sin(t + c))$

where $c$ changes where the point begins on the circle. Combining these transformations lets you model more complex circular paths, including different starting points and centers.

Parametric Lines

A linear path along the line segment from $(x_1, y_1)$ to $(x_2, y_2)$ can be parametrized in many ways. One useful method uses an initial position $(x_1, y_1)$ and rates of change for $x$ and $y$ with respect to $t$.

Start with the direction from the first point to the second:

$(x_2 - x_1,\ y_2 - y_1)$

Then every point on the segment can be written as

$(x_1 + t(x_2 - x_1),\ y_1 + t(y_2 - y_1))$

where $t$ varies from $0$ to $1$. Writing this as separate equations gives

$x = x_1 + t(x_2 - x_1)$

$y = y_1 + t(y_2 - y_1)$

At $t = 0$ you are at $(x_1, y_1)$, and at $t = 1$ you are at $(x_2, y_2)$. Because there are many valid parametrizations, you can choose one that fits the direction, speed, or time range you need.

How to Use This on the AP Precalculus Exam

Unit 4 is not tested on the AP Precalculus exam, so treat the steps below as practice for class assessments and for building calculus-ready reasoning.

Problem Solving

• To write a circle, identify the center $(h, k)$ and radius $r$, then plug into $(h + r\cos t, k + r\sin t)$.
• To write a line segment, pick the starting point, compute the direction $(x_2 - x_1, y_2 - y_1)$, and use $t$ on $[0, 1]$ to move from one endpoint to the other.
• Check direction by testing a few $t$ values in order of increasing $t$ and seeing whether the point moves the way you expect.
• Use the domain of $t$ to set start and end points, since restricting $t$ creates start and end points on the graph.

Common Trap

• Forgetting that $(\cos t, \sin t)$ moves counterclockwise. If you need clockwise motion, you must adjust the parametrization, not just the domain.
• Mixing up the center shift and radius scale. The center adds to the trig term; the radius multiplies it.

Common Misconceptions

• A parametric circle is not just an equation in $x$ and $y$. The parameter $t$ also tells you direction and where the motion starts and ends.
• Changing the domain of $t$ does not change the shape of the circle or line. It changes how much of the path is traced and where it begins and ends.
• The same curve can have many correct parametrizations. Two different sets of parametric equations can draw the same circle or segment while moving at different speeds or in different directions.
• Replacing $t$ with $t + c$ shifts the starting angle, not the size or center of the circle.
• For a line segment, $t$ on $[0, 1]$ keeps you between the endpoints. Values outside that range extend along the same line beyond the segment.

Related AP Precalculus Guides

• 4.10 Matrices
• 4.12 Linear Transformations and Matrices
• 4.6 Conic Sections
• 4.9 Vector-Valued Functions
• 4.13 Matrices as Functions
• 4.5 Implicitly Defined Functions