Parametric curves, domains and direction
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| parameter/pəˈræmɪtə/ | tham số |
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.
The method
- 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.
For x=t², y=t, why does eliminating t as y=√x lose half the curve?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Squaring erases sign: with -2≤t≤2, y ranges over negative and positive values while x=y². Writing y=√x silently keeps only the upper branch; the domain and branch must travel with the elimination.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
x=2×3+1=7.
Where it goes wrong
- 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.
Not true: Eliminating a parameter preserves the original domain and direction automatically. Elimination can lose the domain and direction, so state the restrictions from the parameter separately.
For y=t²−2, find y at t=3.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
y=3²−2=7.
Elimination can lose the domain and direction, so state the restrictions from the parameter separately.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Elimination can lose the domain and direction, so state the restrictions from the parameter separately.
Applying it
- 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.
For x=t²,y=t with t≤0, find y at x=4.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Since t≤0 and t²=4, choose t=−2; hence y=−2.
Match each parametric feature to what it preserves.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A Cartesian equation alone records neither the allowed interval nor the direction; both come from the parameter range.
In your course
- AQA A-level Mathematics (7357), specification section C.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Parameter: A variable that determines both coordinates of a point.
- For x=f(t), y=g(t), express t from one equation if possible and substitute into the other.
- Watch for: writing y=√x from x=t²,y=t loses the part where t is negative.