# parametric equation

#### kevin11

Hey all, sorry if this is in the wrong section. I'm in a precalc class but we're jumping between trig and this so not really sure anymore. Any help would be appreciated!

a. Eliminate parameter t to find the equation in rectangular form.
Sketch the curve defined by:
$$\displaystyle x = \sec t, y = \tan^2 t, 0\leq t \leq \pi$$

So far I made a chart with 3 columns, labeled: t, x, and y. For the t column I chose the values:
$$\displaystyle \frac {\pi}{3}, \frac {\pi}{4}, \frac {\pi}{6}, \frac {3\pi}{4}, \frac {2\pi}{3}, \frac {5\pi}{6}$$

Am I supposed to convert? If so, how do I convert $$\displaystyle tan^2 t$$?
$$\displaystyle \sec t = \frac {1}{\cos t}?$$

#### skeeter

MHF Helper
Hey all, sorry if this is in the wrong section. I'm in a precalc class but we're jumping between trig and this so not really sure anymore. Any help would be appreciated!

a. Eliminate parameter t to find the equation in rectangular form.
Sketch the curve defined by:
$$\displaystyle x = \sec t, y = \tan^2 t, 0\leq t \leq \pi$$

So far I made a chart with 3 columns, labeled: t, x, and y. For the t column I chose the values:
$$\displaystyle \frac {\pi}{3}, \frac {\pi}{4}, \frac {\pi}{6}, \frac {3\pi}{4}, \frac {2\pi}{3}, \frac {5\pi}{6}$$

Am I supposed to convert? If so, how do I convert $$\displaystyle tan^2 t$$?
$$\displaystyle \sec t = \frac {1}{\cos t}?$$
$$\displaystyle \tan^2{t} = \sec^2{t} - 1$$

$$\displaystyle y = x^2 - 1$$

• kevin11

#### Soroban

MHF Hall of Honor
Hello, kevin11!

(a) Eliminate parameter $$\displaystyle t$$ to find the equation in rectangular form.

(b) Sketch the curve defined by: .$$\displaystyle \begin{Bmatrix}x &=& \sec t \\ y &=& \tan^2\!t\end{Bmatrix}\quad 0\leq t \leq \pi$$
skeeter is absolutely correct!

The equation is: .$$\displaystyle y \:=\:x^2-1\quad\hdots$$ a parabola.
. . It opens upward, vertex (0,-1), x-intercepts (±1, 0)
But there's more . . .

I too made a chart and got these values (some approximate).

. . $$\displaystyle \begin{array}{c|cc} t & x & y \\ \hline \\[-4mm] 0 & 1 & 0 \\ \\[-4mm] \frac{\pi}{6} & 1.15 & 0.33 \\ \\[-4mm] \frac{\pi}{4} & 1.41 & 1 \\ \\[-4mm] \frac{\pi}{3} & 2 & 3 \end{array}$$
L . $$\displaystyle \begin{array}{c|cc} \frac{\pi}{2} & \infty & \infty \\ \\[-4mm] \frac{2\pi}{3} & \text{-}2 & 3 \\ \\[-4mm] \frac{3\pi}{4} & \text{-}1.41 & 1 \\ \\[-4mm] \frac{5\pi}{6} & \text{-}1.15 & 0.33 \\ \\[-4mm] \pi & \text{-}1 & 0 \end{array}$$

So the graph looks like this:

Code:
                  |
*          3+           *
|
*          |          *
*        |        *
- - - - * - - + - - * - - - -
-2    -1     |     1     2
|

• kevin11

#### kevin11

Hey guys, thanks for your help, really helped me out!