The given parametric equations are:

$x = 3 \cos(\theta) \quad \text{and} \quad y = 2 \sin(\theta)$ with the parameter range: $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$

Step 1: Eliminate the parameter to find the Cartesian equation.

To eliminate $\theta$, we can use trigonometric identities. Since $x = 3 \cos(\theta)$ and $y = 2 \sin(\theta)$, let's express $\cos(\theta)$ and $\sin(\theta)$ in terms of $x$ and $y$:

$\cos(\theta) = \frac{x}{3} \quad \text{and} \quad \sin(\theta) = \frac{y}{2}$

Now, use the Pythagorean identity: $\cos^2(\theta) + \sin^2(\theta) = 1$

Substitute $\cos(\theta)$ and $\sin(\theta)$ in this identity: $\left(\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1$

Simplifying: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

This is the equation of an ellipse with semi-major axis 3 along the $x$-axis and semi-minor axis 2 along the $y$-axis.

Step 2: Determine the direction of increasing $\theta$.

For $\theta = -\frac{\pi}{2}$ to $\theta = \frac{\pi}{2}$:

• At $\theta = -\frac{\pi}{2}$, $x = 0$ and $y = -2$ (bottom of the ellipse).
• At $\theta = 0$, $x = 3$ and $y = 0$ (rightmost point of the ellipse).
• At $\theta = \frac{\pi}{2}$, $x = 0$ and $y = 2$ (top of the ellipse).

So, the curve starts at the bottom of the ellipse, moves to the right, and then reaches the top. The direction is counterclockwise.

Step 3: Draw the curve.

The curve is an ellipse with axes of 3 and 2, centered at the origin. You should draw arrows in the counterclockwise direction to indicate increasing $\theta$.

Step 4: Determine restrictions on $x$ and $y$.

Given that $\theta$ lies between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, the cosine function (which defines $x$) is non-negative, meaning $x \geq 0$. However, $y$ can take both positive and negative values because the sine function ranges from $-1$ to $1$.

Thus, the region for the curve is restricted to $x \geq 0$, but $y$ can be either positive or negative.

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Would you like more details on any of these steps, or have any other questions?

Here are 5 related questions for further practice:

1. How do you parametrize an ellipse with different center points?
2. What is the effect of changing the range of $\theta$ on the parametric curve?
3. How do you graph parametric equations using a calculator?
4. What is the physical meaning of parametric curves in motion modeling?
5. How do you calculate the arc length of a parametric curve?

Tip: Always check the range of your parameter carefully, as it affects the portion of the curve you are plotting!