**Problem Statement:**Write the following as a function of $ x $ and $ y $, eliminating the parameter $ t $. Then graph the function and indicate the orientation of the curve.\[x = (t + 4)^2, \quad y = t - 1\]**Solution Steps:**1. **Express $ t $ in terms of $ y $:** From the equation $ y = t - 1 $, solve for $ t $: \[ t = y + 1 \]2. **Substitute $ t $ into the equation for $ x $:** Substitute $ t = y + 1 $ into the equation $ x = (t + 4)^2 $: \[ x = ((y + 1) + 4)^2 = (y + 5)^2 \]3. **Final form of the function:** The equation of the curve in terms of $ x $ and $ y $ is: \[ x = (y + 5)^2 \]4. **Graphing the Function:** - The equation $ x = (y + 5)^2 $ represents a parabola that opens to the right. - The vertex of this parabola is at $ (0, -5) $. - The parabola will have a vertical line of symmetry along $ y = -5 $.**Orientation of the Curve:**- As $ t $ increases from negative infinity to positive infinity, $ y = t - 1 $ will also increase from negative infinity.- Consequently, as $ y $ increases, $ x = (y + 5)^2 $ increases, indicating the curve moves to the right. In summary, the solution has transformed the parametric equations into the Cartesian form, and the resulting parabola opens to the right with a vertex at $ (0, -5) $. The orientation indicates movement from left to right as $ t $ increases.

Answered: Write the following as a function of x…

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

**Problem Statement:**

Write the following as a function of $ x $ and $ y $, eliminating the parameter $ t $. Then graph the function and indicate the orientation of the curve.

\[
x = (t + 4)^2, \quad y = t - 1
\]

**Solution Steps:**

1. **Express $ t $ in terms of $ y $:**

From the equation $ y = t - 1 $, solve for $ t $:
\[
t = y + 1
\]

2. **Substitute $ t $ into the equation for $ x $:**

Substitute $ t = y + 1 $ into the equation $ x = (t + 4)^2 $:
\[
x = ((y + 1) + 4)^2 = (y + 5)^2
\]

3. **Final form of the function:**

The equation of the curve in terms of $ x $ and $ y $ is:
\[
x = (y + 5)^2
\]

4. **Graphing the Function:**

- The equation $ x = (y + 5)^2 $ represents a parabola that opens to the right.
- The vertex of this parabola is at $ (0, -5) $.
- The parabola will have a vertical line of symmetry along $ y = -5 $.

**Orientation of the Curve:**

- As $ t $ increases from negative infinity to positive infinity, $ y = t - 1 $ will also increase from negative infinity.
- Consequently, as $ y $ increases, $ x = (y + 5)^2 $ increases, indicating the curve moves to the right.

In summary, the solution has transformed the parametric equations into the Cartesian form, and the resulting parabola opens to the right with a vertex at $ (0, -5) $. The orientation indicates movement from left to right as $ t $ increases.

Expert Solution

Step 1

Definition used -

Converting from parametric to cartesian form-

There are two equations for x and y in terms of t, we solve for t from one equation and substitute it in other equation so t will eliminate and we get one equation in x and y. That is called the cartesian equation.

Step 2

Given -

$x = {(t + 4)}^{2}, y = t - 1$

$t = y + 1$

Substituting the value of t in first equation, We got -

$x = {(y + 1 + 4)}^{2}$

$x = {(y + 5)}^{2}$

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