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NSWMaths Extension 1Quick questions

Functions (ME-F1, ME-F2)

Quick questions on Parametric equations: parameter elimination, sketches, and standard curves

12short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is what parametric equations are?
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A parametric curve in the plane is given by two equations,
What is eliminating the parameter?
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To get a Cartesian equation, eliminate tt from the two equations. Strategies:
What is standard parametrisations?
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Line through (x0,y0)(x_0, y_0) with direction (a,b)(a, b):
What is when to use parametric form?
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Parametric form is useful when:
What is direction of motion?
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As tt increases, the point (x(t),y(t))(x(t), y(t)) moves in a definite direction along the curve. For a circle parametrised by t∈[0,2π)t \in [0, 2\pi) with x=cos⁡tx = \cos t, y=sin⁡ty = \sin t, motion is anticlockwise starting at (1,0)(1, 0).
What is circle from parameter?
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The curve is x=3cos⁡tx = 3 \cos t, y=3sin⁡ty = 3 \sin t for t∈[0,2π)t \in [0, 2\pi). Eliminate the parameter.
What is restricted domain?
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The curve is x=t2x = t^2, y=ty = t for t∈[0,2]t \in [0, 2]. Find the Cartesian form and describe.
What is parametrise a line?
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Write a parametrisation for the line through (2,−1)(2, -1) with direction (3,4)(3, 4).
What is standard parabola?
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The parabola y2=8xy^2 = 8 x has 4a=84 a = 8, so a=2a = 2. Parametrise: x=2t2x = 2 t^2, y=4ty = 4 t.
What is squaring without care?
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If you square one parametric equation to eliminate tt, you may introduce extraneous parts (the other branch). Check by substituting back.
What is wrong direction of traversal?
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x=cos⁡tx = \cos t, y=sin⁡ty = \sin t goes anticlockwise. x=cos⁡tx = \cos t, y=−sin⁡ty = -\sin t goes clockwise.
What is mistaking parameter elimination for an identity?
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x=t2x = t^2, y=t3y = t^3 does not give y2=x3y^2 = x^3 cleanly: t=y1/3t = y^{1/3} so x=y2/3x = y^{2/3}, which is the same curve but the algebra has to handle the cube root carefully. :::

All Maths Extension 1Q&A pages