Eliminate The Parameter $t$ To Find A Cartesian Equation In The Form $x=f(y)$ For:${ \left{ \begin{array}{l} x(t)=2 T^2 \ y(t)=-8+1 T \end{array} \right. }$The Resulting Equation Can Be Written As $x =

by ADMIN 208 views

Introduction

In parametric equations, the parameter t is used to describe the relationship between the variables x and y. However, in many cases, it is desirable to eliminate the parameter t and express x as a function of y, or vice versa. This process is known as eliminating the parameter. In this article, we will show how to eliminate the parameter t to find a Cartesian equation in the form x = f(y) for the given parametric equations.

The Parametric Equations

The given parametric equations are:

{x(t)=2t2y(t)=−8+1t\left\{ \begin{array}{l} x(t)=2 t^2 \\ y(t)=-8+1 t \end{array} \right.

These equations describe the relationship between the variables x and y in terms of the parameter t.

Eliminating the Parameter t

To eliminate the parameter t, we need to find a way to express x in terms of y. One way to do this is to use the second equation to solve for t in terms of y, and then substitute this expression into the first equation.

Step 1: Solve for t in Terms of y

From the second equation, we can solve for t in terms of y as follows:

y(t)=−8+1t⇒t=8−yy(t)=-8+1 t \Rightarrow t = 8 - y

Step 2: Substitute the Expression for t into the First Equation

Now that we have expressed t in terms of y, we can substitute this expression into the first equation to get:

x(t)=2t2⇒x(8−y)=2(8−y)2x(t)=2 t^2 \Rightarrow x(8-y)=2 (8-y)^2

Step 3: Simplify the Expression

To simplify the expression, we can expand the squared term and combine like terms:

x(8−y)=2(8−y)2⇒x(8−y)=2(64−16y+y2)⇒x(8−y)=128−32y+2y2x(8-y)=2 (8-y)^2 \Rightarrow x(8-y)=2 (64-16 y+y^2) \Rightarrow x(8-y)=128-32 y+2 y^2

Step 4: Express x in Terms of y

Now that we have simplified the expression, we can express x in terms of y as follows:

x=128−32y+2y2x=128-32 y+2 y^2

The Cartesian Equation

The resulting equation is a Cartesian equation in the form x = f(y). This equation describes the relationship between the variables x and y without the need for the parameter t.

Conclusion

In this article, we showed how to eliminate the parameter t to find a Cartesian equation in the form x = f(y) for the given parametric equations. We used the second equation to solve for t in terms of y, and then substituted this expression into the first equation to get the resulting equation. The resulting equation is a Cartesian equation that describes the relationship between the variables x and y without the need for the parameter t.

Example Use Cases

The resulting equation can be used in a variety of applications, such as:

  • Graphing: The equation can be used to graph the relationship between the variables x and y.
  • Optimization: The equation can be used to optimize the relationship between the variables x and y.
  • Modeling: The equation can be used to model real-world phenomena, such as the motion of an object.

Tips and Tricks

When eliminating the parameter t, it is often helpful to use the following tips and tricks:

  • Use the second equation to solve for t in terms of y: This can help to simplify the expression and make it easier to substitute into the first equation.
  • Substitute the expression for t into the first equation: This can help to eliminate the parameter t and get the resulting equation.
  • Simplify the expression: This can help to make the resulting equation easier to read and understand.

Conclusion

Introduction

In our previous article, we showed how to eliminate the parameter t to find a Cartesian equation in the form x = f(y) for the given parametric equations. In this article, we will answer some frequently asked questions about eliminating the parameter t and provide additional tips and tricks to help you master this technique.

Q: What is the purpose of eliminating the parameter t?

A: The purpose of eliminating the parameter t is to express x in terms of y, or vice versa, without the need for the parameter t. This can be useful in a variety of applications, such as graphing, optimization, and modeling.

Q: How do I know if I can eliminate the parameter t?

A: You can eliminate the parameter t if the parametric equations are given in the form:

{x(t)=f(t)y(t)=g(t)\left\{ \begin{array}{l} x(t)=f(t) \\ y(t)=g(t) \end{array} \right.

where f(t) and g(t) are functions of t.

Q: What are the steps to eliminate the parameter t?

A: The steps to eliminate the parameter t are:

  1. Solve for t in terms of y using the second equation.
  2. Substitute the expression for t into the first equation.
  3. Simplify the expression to get the resulting equation.

Q: What if I have a parametric equation with a quadratic term?

A: If you have a parametric equation with a quadratic term, you can use the quadratic formula to solve for t in terms of y. For example:

x(t)=2t2⇒t=±x2x(t)=2 t^2 \Rightarrow t = \pm \sqrt{\frac{x}{2}}

Q: Can I eliminate the parameter t if the parametric equations are given in the form x = f(y, t) and y = g(y, t)?

A: No, you cannot eliminate the parameter t if the parametric equations are given in the form x = f(y, t) and y = g(y, t). In this case, the parameter t is not a function of t alone, but rather a function of both y and t.

Q: What are some common mistakes to avoid when eliminating the parameter t?

A: Some common mistakes to avoid when eliminating the parameter t include:

  • Not solving for t in terms of y correctly.
  • Not substituting the expression for t into the first equation correctly.
  • Not simplifying the expression correctly.

Q: Can I use a calculator or computer software to eliminate the parameter t?

A: Yes, you can use a calculator or computer software to eliminate the parameter t. Many calculators and computer software programs, such as Mathematica and Maple, have built-in functions to eliminate the parameter t.

Q: What are some real-world applications of eliminating the parameter t?

A: Some real-world applications of eliminating the parameter t include:

  • Graphing: Eliminating the parameter t can be used to graph the relationship between the variables x and y.
  • Optimization: Eliminating the parameter t can be used to optimize the relationship between the variables x and y.
  • Modeling: Eliminating the parameter t can be used to model real-world phenomena, such as the motion of an object.

Conclusion

In conclusion, eliminating the parameter t to find a Cartesian equation in the form x = f(y) is a useful technique that can be used in a variety of applications. By following the steps outlined in this article and avoiding common mistakes, you can master this technique and use it to solve a wide range of problems.