### Parametric Equations Example#### Consider the parametric equations:\[ x = \sqrt{t} \]\[ y = \sqrt{4 - t} \]Complete each of the following (show your work):**(a) Graph the equation of the parametric equation on the interval $0 \leq t \leq 4$. Be sure to indicate the direction in your graph.**- **Instructions**: Insert your graph in the designated area.- **Graph Section**: - *There is a placeholder for a graph below:* - ![Graph Placeholder](data:image/png;base64,iVBORw...)**(b) Eliminate the parameter.**- **Instructions**: Show your work for eliminating the parameter in the designated area.**(c) Write the domain of the parametric equations.**- **Instructions**: In the designated space, write the domain of the parametric equations.---#### Detailed Steps and Explanation:1. **Graphing the Parametric Equations**: - To graph the equations $x = \sqrt{t}$ and $y = \sqrt{4 - t}$, use the interval $0 \leq t \leq 4$. - Plot the points by substituting values for $t$ within the interval and calculating the corresponding $x$ and $y$ values. - Indicate the direction of the graph as $t$ increases from 0 to 4.2. **Eliminating the Parameter**: - Begin by expressing $t$ in terms of $x$: \[ t = x^2 \] - Substitute $t = x^2$ into the equation for $y$: \[ y = \sqrt{4 - t} \Rightarrow y = \sqrt{4 - x^2} \] - Now, the parametric equations are represented as the Cartesian equation $y = \sqrt{4 - x^2}$.3. **Domain of the Parametric Equations**: - The range for $t$ is from 0 to 4. - Since $x = \sqrt{t}$, $x$ will be between $0$ and $2$ ($ \sqrt{4} = 2 $). - Hence, the domain for $x$

You need to start by checking values of x and y at different point only then you will get the idea…

Consider the parametric equations: a = Vt y = V4 -t Complete each of the following (show your work): (a) Graph the equation of the parametric equation on the interval Os ts 4. Be sure to indicate th direction in your graph. (b) Eliminate the parameter. (c) In the designated space, write the domain of the parametric equations. a) Put your graph here. Draw or add images here b) Show your work here for removing the parameter. c) What are the valid x-values for these parametric equations?

Consider the parametric equations: a = Vt y = V4 -t Complete each of the following (show your work): (a) Graph the equation of the parametric equation on the interval Os ts 4. Be sure to indicate th direction in your graph. (b) Eliminate the parameter. (c) In the designated space, write the domain of the parametric equations. a) Put your graph here. Draw or add images here b) Show your work here for removing the parameter. c) What are the valid x-values for these parametric equations?

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Parametric Equations Example

#### Consider the parametric equations:
\[ x = \sqrt{t} \]
\[ y = \sqrt{4 - t} \]

Complete each of the following (show your work):

**(a) Graph the equation of the parametric equation on the interval $0 \leq t \leq 4$. Be sure to indicate the direction in your graph.**
- **Instructions**: Insert your graph in the designated area.
- **Graph Section**:
- *There is a placeholder for a graph below:*
- ![Graph Placeholder](data:image/png;base64,iVBORw...)

**(b) Eliminate the parameter.**
- **Instructions**: Show your work for eliminating the parameter in the designated area.

**(c) Write the domain of the parametric equations.**
- **Instructions**: In the designated space, write the domain of the parametric equations.
---

#### Detailed Steps and Explanation:

1. **Graphing the Parametric Equations**:
- To graph the equations $x = \sqrt{t}$ and $y = \sqrt{4 - t}$, use the interval $0 \leq t \leq 4$.
- Plot the points by substituting values for $t$ within the interval and calculating the corresponding $x$ and $y$ values.
- Indicate the direction of the graph as $t$ increases from 0 to 4.

2. **Eliminating the Parameter**:
- Begin by expressing $t$ in terms of $x$:
\[ t = x^2 \]
- Substitute $t = x^2$ into the equation for $y$:
\[ y = \sqrt{4 - t} \Rightarrow y = \sqrt{4 - x^2} \]
- Now, the parametric equations are represented as the Cartesian equation $y = \sqrt{4 - x^2}$.

3. **Domain of the Parametric Equations**:
- The range for $t$ is from 0 to 4.
- Since $x = \sqrt{t}$, $x$ will be between $0$ and $2$ ($ \sqrt{4} = 2 $).
- Hence, the domain for $x$

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