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. To graph a parametric curve by hand, a useful first step is to:
- Worked Example: First identify exactly what the question is asking: To graph a parametric curve by hand, a useful first step is to:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A table shows points along the path.
- Ordering by t shows direction.
- Then plot the points in sequence.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: make a table of t, x(t), and y(t)
Practice problems
1. To graph a parametric curve by hand, a useful first step is to:
Choices: make a table of t, x(t), and y(t) · divide every y-value by x · ignore direction · replace t with pi every time
Show solution
- Warm-up: First identify exactly what the question is asking: To graph a parametric curve by hand, a useful first step is to:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A table shows points along the path.
- Ordering by t shows direction.
- Then plot the points in sequence.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: make a table of t, x(t), and y(t)
2. To graph a parametric curve by hand, a useful first step is to: (variation 2)
Choices: make a table of t, x(t), and y(t) · divide every y-value by x · ignore direction · replace t with pi every time
Show solution
- Warm-up: First identify exactly what the question is asking: To graph a parametric curve by hand, a useful first step is to:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A table shows points along the path.
- Ordering by t shows direction.
- Then plot the points in sequence.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: make a table of t, x(t), and y(t)
3. To graph a parametric curve by hand, a useful first step is to: (variation 3)
Choices: make a table of t, x(t), and y(t) · divide every y-value by x · ignore direction · replace t with pi every time
Show solution
- Core Practice: First identify exactly what the question is asking: To graph a parametric curve by hand, a useful first step is to:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A table shows points along the path.
- Ordering by t shows direction.
- Then plot the points in sequence.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: make a table of t, x(t), and y(t)
4. For x = t and y = t + 3, eliminate t and write y in terms of x.
Show solution
- Core Practice: First identify exactly what the question is asking: For x = t and y = t + 3, eliminate t and write y in terms of x.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Since x = t, replace t with x in y = t + 3.
- Then y = x + 3.
- The rectangular equation is linear.
- Check the result by substituting or estimating: the response should match y = x + 3 and make sense in the original problem.
Answer: y = x + 3
5. For x = t and y = t + 3, eliminate t and write y in terms of x. (variation 2)
Show solution
- Core Practice: First identify exactly what the question is asking: For x = t and y = t + 3, eliminate t and write y in terms of x.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Since x = t, replace t with x in y = t + 3.
- Then y = x + 3.
- The rectangular equation is linear.
- Check the result by substituting or estimating: the response should match y = x + 3 and make sense in the original problem.
Answer: y = x + 3
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