Chapter 8 - Section 8.4 - Plane Curves and Parametric Equations - 8.4 Exercises - Page 617: 24

(a) See the graph. (b) $x^2=y+1,~~x\geq1,~~y\geq0$

(a) $t=0$: $x=sec~0=1$ $y=tan^20=0$ Point: $(1,0)$ $t=\frac{\pi}{4}$: $x=sec~\frac{\pi}{4}=\sqrt 2$ $y=tan^2\frac{\pi}{4}=1$ Point: $(\sqrt 2,1)$ When $t→\frac{\pi}{2}$, $x→+∞$ and $y→+∞$. So, $x\geq1$ and $y\geq0$ (b) $x=sec~t$ $x^2=sec^2t$ $y=tan^2t$ $sec^2t=tan^2t+1$ $x^2=y+1$