Parametric equations of curves · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Curves drawn by a moving dot

Mathematics · Geometry · ages 16-17
Name ______________________   Date ____________

  1. Which point lies on x = 3t, y = t² when t may be any real number?

    • (3, 4)
    • (6, 4)
    • (6, 2)
    • (12, 4)

  2. For x = 2t and y = t², what is the x coordinate when t = -3?

    Answer: ______________

  3. What does the parameter t do in a pair of parametric equations?

    • It sets both coordinates for each allowed t value
    • It sets x while y stays fixed for all allowed t values
    • It sets y while x stays fixed for all allowed t values

  4. Reversing the direction of travel must change the set of points on a curve.

    Circle one:   True   False

  5. For x = 2t, y = t² and 0 <= t <= 2, the equation y = x²/4 without any x restriction describes exactly the traced points.

    Circle one:   True   False

  6. For x = 4 cos t and y = 4 sin t, where 0 <= t <= pi in radians, which Cartesian description gives exactly the traced points?

    • x² + y² = 16, y >= 0
    • x² + y² = 16, y <= 0
    • x² + y² = 16, x >= 0
    • x² + y² = 16, all x and y

  7. For x = t + 2 and y = t², Mira finds t = x - 2, then writes y = x² + 4. Which term did she leave out?

    • -2x
    • 4
    • -4x
    • -4

  8. For x = 5 cos t and y = 5 sin t, where 0 <= t <= 2pi in radians, what does squaring and adding give?

    • x² + y² = 5
    • x² + y² = 250
    • x² + y² = 25

  9. On standard axes, x = 4 cos t and y = -4 sin t for 0 <= t <= 2pi in radians. Where is the point at t = 0, and which way does it travel as t increases?

    • (4, 0), moving anticlockwise
    • (4, 0), moving clockwise
    • (0, -4), moving clockwise

  10. For x = 2t, y = t² and 1 <= t <= 3, eliminating t gives y = x²/4. Which x restriction keeps exactly the traced part?

    • 0 <= x <= 9
    • 1 <= x <= 3
    • 2 <= x <= 6
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Answer key

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

Curves drawn by a moving dot W1-mt_7NC7SCeU2P-s1

  1. (6, 4) · The pair (6,4) uses t = 2 in both equations. The other pairs do not use one t that gives both coordinates.
  2. -6 · Use the x equation: x = 2(-3) = -6.
  3. It sets both coordinates for each allowed t value · Use one allowed value of t in both equations to calculate the point's two coordinates.
  4. False · A circle can be traced clockwise or anticlockwise. Its set of points can stay the same while its parametrisation changes.
  5. False · The parameter only gives 0 <= x <= 4. The unrestricted Cartesian equation also includes points outside that range.
  6. x² + y² = 16, y >= 0 · The identity gives x² + y² = 16. The interval traces the whole upper semicircle, where y >= 0.
  7. -4x · The square is (x - 2)(x - 2) = x² - 4x + 4. Mira omitted the term -4x.
  8. x² + y² = 25 · Squaring gives 25cos²t and 25sin²t. Their sum is 25 because cos²t + sin²t = 1.
  9. (4, 0), moving clockwise · At t = 0 the point is (4,0). At t = pi/2 it is (0,-4), so it moves clockwise from the right side of the circle.
  10. 2 <= x <= 6 · Multiply the bounds on t by 2. Since x = 2t increases with t, the full x range is 2 <= x <= 6.
Worksheet · LightMySky