Parametric curves, domains and direction
| English | 中文 | Pinyin |
|---|---|---|
| parameter/pəˈræmɪtə/ | 参数 | cān shù |
What information disappears when time is eliminated?
- Two coordinates can change together as a third variable changes. Eliminating that variable can hide which part of the curve was traced.
- This lesson studies parameter 参数: A variable that determines both coordinates of a point.
Choose the mathematical structure
- For x=f(t), y=g(t), express t from one equation if possible and substitute into the other. Transfer the parameter interval to the Cartesian curve and mark the direction of increasing t. With trigonometric parameters, use identities and keep sign or branch restrictions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines parameter?
A variable that determines both coordinates of a point.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x=2t+1, y=t²−2 and 0≤t≤3, t=(x−1)/2 gives y=(x−1)²/4−2 with 1≤x≤7. The point moves from (1,−2) to (7,7). For x=t², y=t and −2≤t≤2, the relation is x=y² with −2≤y≤2: both signs of y occur. For x=3cosθ, y=2sinθ and 0≤θ≤π, x²/9+y²/4=1 with y≥0; increasing θ traces the upper half from (3,0) to (−3,0). Conversely, y=x² with −1≤x≤2 can be parametrised by x=t, y=t² and −1≤t≤2.
Parametric curves, domains and direction
For x=f(t), y=g(t), express t from one equation if possible and substitute into the other
Identify which statements preserve the original parameter, curve segment and contextual assumptions.
For x=2t+1, find x at t=3.
x=2×3+1=7.
Test a tempting shortcut
- Writing y=√x from x=t²,y=t loses the part where t is negative. A Cartesian equation alone does not record direction or the original parameter interval. Squaring can erase a sign restriction.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Eliminating a parameter always preserves the original domain and direction automatically. This claim is false. Explain which definition or assumption it violates.
For y=t²−2, find y at t=3.
y=3²−2=7.
Eliminating a parameter always preserves the original domain and direction automatically.
Writing y=√x from x=t²,y=t loses the part where t is negative. A Cartesian equation alone does not record direction or the original parameter interval. Squaring can erase a sign restriction.
Interpret a new situation
- Use a table of parameter values to sketch endpoints and intermediate points. A parameter can revisit an x value or a whole point. Check the original equations and interval when converting back to parametric form.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For x=t²,y=t with t≤0, find y at x=4.
Since t≤0 and t²=4, choose t=−2; hence y=−2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · C. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A variable that determines both coordinates of a point. Choose the relationship, show the method, check its assumptions and interpret the result.