Porisma

Worked solution

Find all axial intersections of a parametric curve

Find the coordinates of the intersections of the curve

x=t−4t2,y=t3−4t,t∈Rx=t-4t^2,\qquad y=t^3-4t,\qquad t\in\mathbb R

with the x- and y-axes. Sketch the curve.

CJC · Discussion · Question 1

Answer

(0,0)(0,0), (0,−63/64)(0,-63/64), (−18,0)(-18,0) and (−14,0)(-14,0); see the sketch.

Find the y-axis intersections

x=0x=0 gives t(1−4t)=0t(1-4t)=0, so t=0t=0 or 1/41/4. Then y=0y=0 or 1/64−1=−63/641/64-1=-63/64. The points are (0,0)(0,0) and (0,−63/64)(0,-63/64).

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Find the x-axis intersections

y=0y=0 gives t(t2−4)=0t(t^2-4)=0, so t=0,−2,2t=0,-2,2. The corresponding x-values are 0,−18,−140,-18,-14. Thus the points are (0,0),(−18,0),(−14,0)(0,0),(-18,0),(-14,0).

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Sketch both traced branches

The reconstructed curve uses the given parametric equations and labels all four distinct axial intersections. Its rightmost x-value is 1/161/16 at t=1/8t=1/8; the two crossings of the y-axis are close together. A coarse calculator window can hide that small rightward turn.

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Compare your sketch

Expand diagram

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Exact parametric plot x=t-4t²,y=t³-4t with x-axis intersections -18,-14,0 and y-axis intersections 0,-63/64. A separately scaled near-origin inset shows the narrow rightward turn and both y-intercepts. Main graph is clipped to a finite window; t ranges over all real numbers.
Exact parametric plot x=t-4t²,y=t³-4t with x-axis intersections -18,-14,0 and y-axis intersections 0,-63/64. A separately scaled near-origin inset shows the narrow rightward turn and both y-intercepts. Main graph is clipped to a finite window; t ranges over all real numbers.
Exact parametric plot x=t-4t²,y=t³-4t with x-axis intersections -18,-14,0 and y-axis intersections 0,-63/64. A separately scaled near-origin inset shows the narrow rightward turn and both y-intercepts. Main graph is clipped to a finite window; t ranges over all real numbers.

Common mistakes

Several parameter values can give different intersections with the same axis.

Solution notes

Source calculator sketch independently redrawn from the printed equations; its near-origin scale is explained rather than treating both y-intercepts as one.

Read the hint