Eliminate the parameter t to find a Cartesian equation in the form z = f(y) for: [z(t) = 3t² y(t) = 3 + 4t The resulting equation can be written as a =
Eliminate the parameter t to find a Cartesian equation in the form z = f(y) for: [z(t) = 3t² y(t) = 3 + 4t The resulting equation can be written as a =
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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![### Eliminate the Parameter to Find a Cartesian Equation
To find a Cartesian equation in the form \( x = f(y) \), we need to eliminate the parameter \( t \) from the given parametric equations.
Given:
\[
\begin{cases}
x(t) = 3t^2 \\
y(t) = 3 + 4t
\end{cases}
\]
#### Steps to Eliminate the Parameter:
1. **Solve the \( y(t) \) equation for \( t \):**
\[
y = 3 + 4t \implies t = \frac{y - 3}{4}
\]
2. **Substitute \( t \) back into the \( x(t) \) equation:**
\[
x = 3t^2 = 3\left(\frac{y - 3}{4}\right)^2
\]
3. **Simplify the equation:**
\[
x = 3 \left(\frac{y - 3}{4}\right)^2 = 3 \cdot \frac{(y - 3)^2}{16} = \frac{3(y - 3)^2}{16}
\]
Thus, the resulting Cartesian equation can be written as:
\[
x = \frac{3(y - 3)^2}{16}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f2aa9e5-f479-433e-a47a-cf563a6cc086%2F21e638eb-a14e-4227-bdba-ecd5362196bd%2F1olp8t_processed.png&w=3840&q=75)
Transcribed Image Text:### Eliminate the Parameter to Find a Cartesian Equation
To find a Cartesian equation in the form \( x = f(y) \), we need to eliminate the parameter \( t \) from the given parametric equations.
Given:
\[
\begin{cases}
x(t) = 3t^2 \\
y(t) = 3 + 4t
\end{cases}
\]
#### Steps to Eliminate the Parameter:
1. **Solve the \( y(t) \) equation for \( t \):**
\[
y = 3 + 4t \implies t = \frac{y - 3}{4}
\]
2. **Substitute \( t \) back into the \( x(t) \) equation:**
\[
x = 3t^2 = 3\left(\frac{y - 3}{4}\right)^2
\]
3. **Simplify the equation:**
\[
x = 3 \left(\frac{y - 3}{4}\right)^2 = 3 \cdot \frac{(y - 3)^2}{16} = \frac{3(y - 3)^2}{16}
\]
Thus, the resulting Cartesian equation can be written as:
\[
x = \frac{3(y - 3)^2}{16}
\]
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