In the graph of the parametric equations { x = t^2+ t and y = t^2 -t , x is
A) greater than or equal to zero
B) greater than or equal to -0.25
What is your answer and why?
The book says it’s B, but I keep getting A. My graph is different from that sketched in the book. My calculator is CASIO fx-9860GII, mode: radian.
@BethanyD B is correct.
Note that t doesn’t necessarily have to be nonnegative. If t = -1/2, then x = (-1/2)^2 - 1/2 = -1/4. In particular the minimum value of x = t^2 + t (over real t) occurs at t = -1/2, with minimum value -1/4.
@MITer I discovered my mistake - for the graph, my calculator was set to show values of T greater than or equal to zero only, so the full domain wasn’t shown, but now it is. Thank you so much! i now see a rotated parabola with the minimum you mentioned.
I have another question: when graphs “kiss” the x-axis, does the multiplicity have to be 2 or any other even number? and if yes, is it possible to distinguish between mult. = 2 and mult. = 4 only from the graph?
@BethanyD You’d have to specify what you mean by “kiss” the x-axis - I’m guessing what you mean is whether there is a differentiable function tangent to the x-axis at some x = c whose derivative around c changes sign (otherwise y = x^3) would be a counterexample).
For polynomial functions, yes this is true.
You can always tell from the graph given arbitrary precision but it might not work in all cases where precision is needed - for example, it would be very hard for the naked eye to distinguish the graphs of y = x^1000 and y = x^1002 apart.