📚 Topic Summary
Eliminating the parameter involves finding a direct relationship between $x$ and $y$ from a set of parametric equations. Parametric equations express $x$ and $y$ in terms of a third variable, often denoted as $t$. The goal is to manipulate these equations to solve for $t$ in one equation and substitute that expression into the other equation, thereby eliminating $t$ and expressing $y$ as a function of $x$ (or vice versa). This process allows us to analyze the curve defined by the parametric equations in a more familiar Cartesian form.
For example, if $x = t + 1$ and $y = t^2$, we can solve for $t$ in the first equation: $t = x - 1$. Substituting this into the second equation gives $y = (x - 1)^2$, which is the equation of a parabola.
🧠 Part A: Vocabulary
Match the term with its correct definition:
- Term: Parameter
- Term: Cartesian Equation
- Term: Parametric Equations
- Term: Elimination
- Term: Variable
- Definition: A symbol representing a value that can change.
- Definition: Expressing $x$ and $y$ in terms of another variable.
- Definition: The process of removing a variable from a set of equations.
- Definition: A symbol representing a constant value in a specific context.
- Definition: An equation relating $x$ and $y$ directly, without a parameter.
✏️ Part B: Fill in the Blanks
Parametric equations use a(n) _______ variable, often denoted as $t$, to express $x$ and $y$. To eliminate the parameter, solve for $t$ in one equation and _______ that expression into the other. This results in a(n) _______ equation relating $x$ and $y$ directly. For instance, if $x = 2t$ and $y = t + 1$, solving for $t$ in the second equation yields $t = _______$. Substituting this into the first equation gives $x = 2(y - 1)$, which simplifies to $x = _______$ .
🤔 Part C: Critical Thinking
Explain why eliminating the parameter can be useful when analyzing the motion of a projectile.