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: $x=1-t^2$, $y=t-2$

{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 $y(x) = -2 \mp \sqrt{1 - x}$

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 $\hat{y}$, 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}}]}
 ]