Parametric Curves and Orientation

Same path, different journey.

12 problems · about 21 minutes · TEKS P.3.C

NameDate

Figure — use these to answer the problems

This parabola is the path of x = t, y = t². Every parametrisation of it draws this same shape.

y = 1x² + 0x + 0

Point on the curve2

Figure 1

Straightforward practice. Get the method working first.

5 problems

x = −t, y = t². What is x when t = 6?
Answer
2
Two parametrisations give the same rectangular equation. Are they the same motion?
- Not necessarily; orientation and domain may differ.
- Yes, always.
x = t, y = t². What is y when t = −3?
Answer
x = 2t, y = 4t². Eliminating t gives y = xⁿ. What is n?
Answer
x = cos t, y = sin t with 0 ≤ t ≤ 90°. What fraction of the circle, as a decimal?
Answer

The same ideas with more to keep track of.

3 problems

x = 3 − t, y = t. As t increases, does x increase or decrease? 1 or 2.
Answer
x = t², y = t. What is x when t = −4?
Answer
How many parametrisations does one curve have? Enter 0 for infinitely many, or the exact count.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the steps for analysing a parametric curve in order.
Write 1 to 4 in the boxes to put these in order.
- Build a table of t, x and y.
- Plot the points in increasing order of t.
- Draw an arrow showing the orientation.
- Eliminate t to check the rectangular equation.
x = 5t, y = 5t. Eliminating t gives y = kx. What is k?
Answer
The Direction
x = −t, y = 3t. As t increases, does x increase or decrease? Enter 1 for increase, 2 for decrease.
Answer
The Half Circle
x = cos t, y = sin t with 0 ≤ t ≤ 180°. What fraction of the circle is drawn, as a decimal?
Answer