Curves drawn by a moving dot

Mathematics · Geometry · ages 16-17

Name ______________________ Date ____________

Which point lies on the curve x = 2t, y = t squared?
- (4, 4)
- (2, 2)
- (6, 4)
- (4, 2)
The curve is given by x = 2t and y = t squared. When t = 3, what is the x coordinate?
Answer: ______________
What does the parameter t do?
- It drives x and y separately along the curve
- It always equals y
- It removes the need for tables
Parametric plots need arrows showing which way t increases.
Circle one: True False
The Cartesian equation always traces exactly what the parameter traces.
Circle one: True False
What curve do x = 3 cos t, y = 3 sin t describe?
- A circle of radius 3 centred at the origin
- A circle of radius 9 centred at the origin
- An ellipse through (9, 0) and (0, 9)
- A straight line through the origin
Eliminate t from x = 2t, y = t squared. What is the Cartesian equation?
- y = x squared over 4
- y = 2x squared
- y = x squared
- y = 4x squared
Why do x equals 3 cos t and y equals 3 sin t trace a circle?
- Cosine and sine are always equal
- Squaring and adding gives x squared plus y squared equals 9
- The parameter t equals 3
Where does the circle dot start at t zero, and which way does it go?
- (3, 0), moving anticlockwise
- (0, 3), moving clockwise
- The origin, moving outward
After eliminating t you get a full parabola, but t only ran 0 to 2. What must you add?
- A second parameter at once
- A bigger table with decimals
- The restriction t forces on x or y

Answer key

For grown-ups. Fold this page away before handing over the rest.

(4, 4) · At t = 2 the curve passes through (4, 4), and none of the other points fits any single t value.
6 · Substitute t = 3 into x = 2t to get x = 6.
It drives x and y separately along the curve · Both coordinates hang off t, which acts like time.
True · Direction is part of the curve, not decoration.
False · Elimination can widen the road beyond what t reaches.
A circle of radius 3 centred at the origin · Squaring and adding gives x squared plus y squared = 9, using cos squared plus sin squared = 1, so it is a circle of radius 3.
y = x squared over 4 · From x = 2t we get t = x over 2, and substituting into y = t squared gives y = x squared over 4.
Squaring and adding gives x squared plus y squared equals 9 · The identity shortcut turns the pair into the circle rule.
(3, 0), moving anticlockwise · Cosine starts at 1 and the angle grows anticlockwise from the axis.
The restriction t forces on x or y · State the limits the parameter imposed, or the curve overclaims.