AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/4/4-1)
Unit 4 · Topic 4.1
4.1 Parametric Functions
A parametric function describes a curve by giving both x and y as functions of a third variable, the parameter t. You build a table of points from t-values and connect them in order of increasing t, which also shows the direction the curve is traced.
Key terms
- parameter
- parametric equations
- domain of t
- orientation
What a parametric function is
Usually y depends directly on x. In a parametric function, both x and y depend on a separate independent variable, called the parameter, usually t. You get two equations, x = x(t) and y = y(t), called parametric equations.
Together they form one parametric function: f(t) = (x(t), y(t)). Here x and y are the names of two functions, and each value of t produces one point in the plane.
Think of t as time. At each moment, x(t) tells you how far left or right a moving point is, and y(t) tells you how high it is.
Making a table
Choose several values of t in the domain. For each one, compute x(t) and y(t). Each row of the table gives one point (x, y).
Pick t-values close enough together to show the shape, including any endpoints of the domain and any t where x or y turns around.
If t is time in seconds and x and y are in meters, each row of the table tells you where an object is at that second. Two different rows can even give the same point, if the path passes through it twice. That's fine: the value of t tells you which visit you mean.
Sketching the graph
Plot the points and connect them in order of increasing t, not in order of x. Add arrows showing the direction of increasing t; this is called the orientation of the curve.
The domain of t is often restricted, such as 0 ≤ t ≤ 5. Then the graph has a starting point, at the smallest t, and an ending point, at the largest t. Mark both.
A parametric curve doesn't have to pass the vertical line test. It can loop, cross itself, or double back, because the points are ordered by t, not by x.
Eliminating the parameter
Sometimes you can find a single equation in x and y by solving one parametric equation for t and substituting into the other. That tells you the shape of the path, such as a line or a parabola.
The equation in x and y loses information, though: it doesn't show direction, speed, or where the motion starts and stops. The parametric form keeps all of that.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Table and sketch
Make a table for x(t) = t² − 1, y(t) = 2t on −2 ≤ t ≤ 2, and describe the graph.
Show the solutionHide the solution
- Step 1: t = −2: (3, −4). t = −1: (0, −2). t = 0: (−1, 0). t = 1: (0, 2). t = 2: (3, 4).
- Step 2: Connect in order of t. The curve starts at (3, −4), moves left and up to (−1, 0), then right and up to (3, 4), where it ends.
- Step 3: Eliminating t: t = y/2, so x = y²/4 − 1. That's a parabola opening to the right with vertex (−1, 0).
Answer: Part of a right-opening parabola with vertex (−1, 0), traced from (3, −4) up to (3, 4).
- Example 2
Eliminating the parameter with a restricted domain
Describe the path x(t) = 3 − t, y(t) = t² for 0 ≤ t ≤ 3.
Show the solutionHide the solution
- Step 1: Start: t = 0 gives (3, 0). End: t = 3 gives (0, 9).
- Step 2: Solve x = 3 − t for t: t = 3 − x. Substitute: y = (3 − x)².
- Step 3: As t goes from 0 to 3, x goes from 3 down to 0, so only the part with 0 ≤ x ≤ 3 is traced, moving left and up.
Answer: The piece of y = (3 − x)² with 0 ≤ x ≤ 3, traced from (3, 0) to (0, 9).
- Example 3
Trap: connecting points in x-order
For x(t) = t³ − 3t, y(t) = t on −2 ≤ t ≤ 2, a student sorts the points by x before connecting them. Why is that wrong?
Show the solutionHide the solution
- Step 1: The points in t-order are (−2, −2), (2, −1), (0, 0), (−2, 1), (2, 2).
- Step 2: In t-order the curve goes right, then back left through the origin, then right again: an S shape lying on its side, x = y³ − 3y.
- Step 3: Sorting by x would join (−2, −2) to (−2, 1) and (2, −1) to (2, 2), a completely different picture.
Answer: The points must be connected in order of increasing t. The real curve is a sideways S, x = y³ − 3y, traced from bottom to top.
Common mistakes
- Connecting points in order of x instead of t.
- Leaving off the direction arrows, or the start and end points when the domain of t is restricted.
- Assuming the graph must pass the vertical line test. Parametric curves can loop and cross themselves.
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 build a table, sketch the curve with its orientation, and sometimes eliminate the parameter. Parametric curves return in AP Calculus BC.
Connected topics
Videos
Check yourself
3 questions on 4.1 Parametric Functions. Pick an answer to see if you got it, and why.
A curve is defined by x(t) = t² − 2t and y(t) = t + 1 for −1 ≤ t ≤ 3. Which of the following gives the point where the curve starts (at t = −1) and the point where it ends (at t = 3)?
A curve is defined parametrically by x(t) = 2t − 1 and y(t) = t² for all real t. Which of the following points is on the curve?
A particle moves in the xy-plane so that its position at time t is (x(t), y(t)), where x(t) = t² − 4t and y(t) = t³ − 3t, for 0 ≤ t ≤ 3.
What is the slope of the line through the particle's positions at t = 0 and t = 3?
0 of 3 answered