Eliminate the parameter $t$ to find a Cartesian equation in the form $x=f(y)$ for:
$\left\{\begin{array}{l}
x(t)=-2 t^2 \\
y(t)=1+3 t
\end{array}\right.$
The resulting equation can be written as $x=$
Answer (2)
Solve for t in terms of y : t = 3 y − 1 .
Substitute the expression for t into the equation for x : x = − 2 ( 3 y − 1 ) 2 .
Simplify the equation: x = − 9 2 ( y − 1 ) 2 .
The Cartesian equation is: x = − 9 2 ( y − 1 ) 2 .
Explanation
Problem Analysis We are given the parametric equations x ( t ) = − 2 t 2 and y ( t ) = 1 + 3 t . Our goal is to eliminate the parameter t and express x as a function of y , i.e., x = f ( y ) .
Solving for t First, we solve the equation y = 1 + 3 t for t in terms of y . Subtracting 1 from both sides gives y − 1 = 3 t . Dividing both sides by 3, we get t = 3 y − 1 .
Substituting t into x(t) Next, we substitute this expression for t into the equation x = − 2 t 2 . This gives x = − 2 ( 3 y − 1 ) 2 .
Simplifying the Equation Finally, we simplify the equation to express x as a function of y . We have x = − 2 ( 9 ( y − 1 ) 2 ) = − 9 2 ( y − 1 ) 2 . Thus, the Cartesian equation is x = − 9 2 ( y − 1 ) 2 .
Final Answer The resulting equation is x = − 9 2 ( y − 1 ) 2 .
Examples
Understanding parametric equations and how to eliminate the parameter is useful in many real-world applications, such as computer graphics and physics. For example, when modeling the trajectory of a projectile, the position can be described by parametric equations with time as the parameter. Eliminating the parameter allows us to find the path of the projectile in terms of x and y coordinates, which can be useful for predicting its landing point. Another example is in computer-aided design (CAD), where parametric equations are used to define curves and surfaces. Eliminating the parameter can help in rendering these shapes on a screen.
To eliminate the parameter from the given parametric equations, we solved for t in terms of y and substituted this expression into the equation for x . The final Cartesian equation obtained is x = − 9 2 ( y − 1 ) 2 .
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