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Parametric Equations

Study Guide

Key Definition

A parametric equation expresses a set of coordinates $(x, y)$ as functions of a variable $t$.

Important Notes

  • Parametric equations can describe more complex curves than standard equations.
  • $x = f(t)$ and $y = g(t)$ are typical forms for parametric equations.
  • To eliminate the parameter, solve one equation for $t$ and substitute.
  • Parametric equations are useful in physics and engineering for modeling motion.
  • They can represent both closed and open curves.

Mathematical Notation

Standard arithmetic symbols (+, –, ×, ÷, √) and the parameter t are used as usual.Remember to use proper notation when solving problems

Why It Works

Parametric equations allow for the representation of curves that are not functions in the $y = f(x)$ form, enabling modeling of complex paths.

Remember

A parametric curve is defined by $x = f(t)$ and $y = g(t)$.

Quick Reference

Rectangular Equation:$y = x^2 + 5$Parametric Form:$x = t$, $y = t^2 + 5$

Understanding Parametric Equations

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

Parametric equations define x and y in terms of a parameter t. For example, x = t and y = t^2 + 5 traces the parabola y = x^2 + 5 as t varies over real numbers.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Which set of parametric equations represents $y = x^2 + 5$?

A$x = t$, $y = t^2 + 5$B$x = t^2$, $y = t + 5$C$x = t + 5$, $y = t^2$D$x = t - 5$, $y = t^2 + 5$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Car Path

A car travels along a path described by $x = t + 5$ and $y = t^2$ for t in [0, 5]. Eliminate the parameter to find the rectangular equation of the path.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Describe the curve formed by $x = \cos(t)$ and $y = \sin(t)$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Which of the following parametric equations represents a hyperbola?

A$x = \cosh(t)$, $y = \sinh(t)$B$x = \cos(t)$, $y = \sin(t)$C$x = t^2$, $y = t + 1$D$x = t$, $y = t^3$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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