Finding Cartesian from Parametric Equations

In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.

x = cos t, y = sin 3t

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Step 1: Identify the parametric equations given: \(x = \cos t\) and \(y = \sin 3t\).

Step 2: Understand the behavior of \(x = \cos t\): it oscillates between -1 and 1 with period \(2\pi\).

Step 3: Understand the behavior of \(y = \sin 3t\): it oscillates between -1 and 1 but with a period of \(\frac{2\pi}{3}\), which is three times faster than \(x\).

Step 4: Look for a graph where the \(x\)-values oscillate smoothly between -1 and 1, while the \(y\)-values oscillate more rapidly, creating multiple waves within one period of \(x\). This will create a pattern with three oscillations in \(y\) for every one oscillation in \(x\).

Step 5: Match this behavior to the graphs provided. The graph labeled F shows a wave pattern with multiple oscillations in \(y\) for each oscillation in \(x\), consistent with \(x = \cos t\) and \(y = \sin 3t\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing representation of more complex curves like loops or spirals.

Conversion to Cartesian Form

Converting parametric equations to Cartesian form involves eliminating the parameter t to find a direct relationship between x and y. This helps identify the shape of the curve and compare it to standard graphs, such as circles or ellipses.

Trigonometric Parametric Curves

When parametric equations involve trigonometric functions like sine and cosine, the resulting curves often represent periodic or oscillatory shapes. Understanding how frequencies and amplitudes affect the curve is essential for matching equations to their graphs.

Finding Cartesian from Parametric EquationsIn Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.x = cos t, y = sin 3t