I'm confused by what my solution manual is telling me.
Part a of a problem I'm doing is: Sketch the plane curve with the given vector equation.
$r(t) = <t-2, t^2+1>, t = -1$
The solution manual basically says:
Since $(x+2)^2 = t^2 = y-1 => y = (x+2)^2 - 1$, the curve is a parabola.
How is $(x+2)^2 = t^2 = y-1?$
Another problem $r(t) = sinti + 2costj, t = pi/4$
I thought I was supposed to sketch it by plugging in values for t, but the solution manual says it's a parabola by doing this:
$x = sint, y = 2cost$ so $x^2 + (y/2)^2 = 1$ and the curve is an ellipse.
How come you are allowed to just square both x and y to get the ellipse?