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plotting - Eliminating the Parameter: Transform parametric equation to Cartesian equation and draw arrows along parametric growth



Hey guys I really could use some help on this calc 3 problem. I'm stuck on how to write the code for this problem:



  • a) Eliminate the parameter to find a Cartesian equation for a parametric curve.

  • b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.


Given: x=1−t2, y=t−2


{x == 1 - t^2, y == t - 2}

I'm not sure really where to start. Thanks for the help!



Answer




Start by using Eliminate to remove the parametric variable


Eliminate[{x == 1 - t^2, y == t - 2}, t]


-3 - 4 y - y^2 == x



Now you can Solve to get an expression of the form y(x)=−2∓√1−x


sol = (y /. Solve[-3 - 4 y - y^2 == x, y])



{-2 - Sqrt[1 - x], -2 + Sqrt[1 - x]}



To sketch you will need to know where it crosses the axis


sol /. x -> 0


{-3, -1}



That is, it crosses at {{0, -3}, {0, -1}, {-3, 0}}. With which slopes?


D[sol, x]



{1/(2 Sqrt[1 - x]), -(1/(2 Sqrt[1 - x]))}

Slopes are {1/2, -1/2, -1/4} respectively.


Now the plot with arrows growing in the same direction as ˆy, i.e up, to the right for t<−1 and to the left for t>1


Plot[
 sol,
 {x, -9, 2},
 PlotStyle -> Black

 , Frame -> True
 , Prolog -> {Blue, 
   Arrow[Partition[
     Transpose[{1 - t^2, t - 2} /. t -> Range[-5, 5, 0.5]], 2, 1]]}
 , Epilog -> {PointSize[Large], Red, 
   Point[{{0, -3}, {0, -1}, {-3, 0}}]}
 ]


Mathematica graphics




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