Graphing Parametric Curves
A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.
A parametric curve is a path. The parameter controls where the point is and which way it moves. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Trace a parametric path and describe direction
- Use graphing parametric curves in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. A curve is traced by x = t and y = t + 2. Eliminate the parameter to write y in terms of x.
- Worked Example: First identify exactly what the question is asking: A curve is traced by x = t and y = t + 2. Eliminate the parameter to write y in terms of x.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Since x = t, the parameter t is just x itself.
- Substitute t = x into y = t + 2.
- So y = x + 2.
- Check the result by substituting or estimating: the response should match y = x + 2 and make sense in the original problem.
Answer: y = x + 2
Practice problems
1. A curve is traced by x = t and y = t + 2. Eliminate the parameter to write y in terms of x.
Show solution
- Warm-up: First identify exactly what the question is asking: A curve is traced by x = t and y = t + 2. Eliminate the parameter to write y in terms of x.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Since x = t, the parameter t is just x itself.
- Substitute t = x into y = t + 2.
- So y = x + 2.
- Check the result by substituting or estimating: the response should match y = x + 2 and make sense in the original problem.
Answer: y = x + 2
2. What is the recommended first step for graphing a parametric curve by hand?
Choices: Build a table of t, x(t), and y(t) values · Divide every y-value by its x-value · Set t equal to 0 and stop · Erase the parameter before plotting anything
Show solution
- A parametric curve gives x and y separately in terms of t.
- Listing t alongside the x and y it produces gives plottable points.
- You then connect those points in t-order.
Answer: Build a table of t, x(t), and y(t) values
3. For x = t + 5 and y = 2t, find the point on the curve when t = 4.
Show solution
- Core Practice: First identify exactly what the question is asking: For x = t + 5 and y = 2t, find the point on the curve when t = 4.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Put t = 4 into x = t + 5: x = 4 + 5 = 9.
- Put t = 4 into y = 2t: y = 2(4) = 8.
- So the point is (9, 8).
- Check the result by substituting or estimating: the response should match (9, 8) and make sense in the original problem.
Answer: (9, 8)
4. For x = t and y = t + 5, which way does the point move as t increases?
Choices: to the right and upward · to the left and downward · to the right and downward · straight up with no horizontal change
Show solution
- As t increases, x = t increases, so the point moves right.
- At the same time y = t + 5 increases, so it moves up.
- Both coordinates grow with t, so the motion is right-and-up.
Answer: to the right and upward
5. Why is a parametric curve called a path rather than just a shape?
Choices: The parameter records the order and direction the point is traced · Because it can only be a straight line · Because x and y must always be equal · Because it has no equation in x and y
Show solution
- A bare shape shows where points are but not how they are visited.
- The parameter t tells you the order points are reached and which way you travel.
- That extra order/direction information is what makes it a path.
Answer: The parameter records the order and direction the point is traced
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