Question

Sketching parametric equations: Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.

$x=\tan t$,$y=1+se{c}^{2}t$,$t\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

Short answer

The curve is parabola and required graph is as follows,

Step-by-step solution

Step 1. Given Information.

The equations are$x=\tan t$and$y=1+se{c}^{2}t$such that$t\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

Step 2. Explanation.

Start by eliminating the parameter. Square the equations for x and add 1..

$x^2+1={\tan }^{2}t+1$

Subtract 1 from y.

$y-1=se{c}^{2}t$

Apply trigonometric identity

$$\begin{gathered} se{c}^{2}t=1+{\tan }^{2}t \\ y-1=x^2+1 \end{gathered}$$

Simplify the equation.

$y=x^2+2$

The equation present parabola.

Step 3. Calculation.

Make a table to draw the table as follows,

t	$-\frac{\mathrm{\pi }}{2}$	$-\frac{\mathrm{\pi }}{4}$	0	$\frac{\mathrm{\pi }}{4}$	$\frac{\mathrm{\pi }}{2}$
$x=\tan t$	$\infty$	-1	0	1	$\infty$
$y=1+se{c}^{2}t$	$\infty$	3	2	3	$\infty$

Step 4. Graph.

The required graph is as follows,