Parametric Curves and Orientation

Same path, different journey.

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

What this lesson teaches

The student compares parametrisations of the same curve and describes orientation, speed and restricted domains.

Straightforward practice. Get the method working first.

5 problems

x = −t, y = t². What is x when t = 6?
Answer -6
Why −6.
Two parametrisations give the same rectangular equation. Are they the same motion?
Answer Not necessarily; orientation and domain may differ.
Why The path matches, but the journey may not.
x = t, y = t². What is y when t = −3?
Answer 9
Why Squaring removes the sign.
x = 2t, y = 4t². Eliminating t gives y = xⁿ. What is n?
Answer 2
Why t = x ÷ 2, so y = x².
x = cos t, y = sin t with 0 ≤ t ≤ 90°. What fraction of the circle, as a decimal?
Answer 0.25
Why 90 out of 360.

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 2
Why Subtracting t.
x = t², y = t. What is x when t = −4?
Answer 16
Why (−4)².
How many parametrisations does one curve have? Enter 0 for infinitely many, or the exact count.
Answer 0
Why There is no limit.

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

4 problems

Put the steps for analysing a parametric curve in order.
Answer 1. Build a table of t, x and y. 2. Plot the points in increasing order of t. 3. Draw an arrow showing the orientation. 4. Eliminate t to check the rectangular equation.
Why The table comes before the plotting.
x = 5t, y = 5t. Eliminating t gives y = kx. What is k?
Answer 1
Why They are equal.
The Direction: x = −t, y = 3t. As t increases, does x increase or decrease? Enter 1 for increase, 2 for decrease.
Answer 2
Why It decreases.
The Half Circle: x = cos t, y = sin t with 0 ≤ t ≤ 180°. What fraction of the circle is drawn, as a decimal?
Answer 0.5
Why 0.5.