For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.

x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)

검증된 단계별 안내

Step 1: Understand the parametric equations given: \(x = t + 2\) and \(y = t - 4\), where \(t\) is a parameter that can take any real value from \(-\infty\) to \(\infty\).

Step 2: To graph the curve, recognize that as \(t\) varies, the point \((x, y)\) moves along the curve defined by these equations. Plot several points by choosing values of \(t\), calculating corresponding \(x\) and \(y\), and then sketch the curve through these points.

Step 3: To find a rectangular equation (an equation involving only \(x\) and \(y\)), eliminate the parameter \(t\) from the system. From the first equation, express \(t\) in terms of \(x\): \(t = x - 2\).

Step 4: Substitute \(t = x - 2\) into the second equation: \(y = (x - 2) - 4\).

Step 5: Simplify the expression to get the rectangular equation: \(y = x - 6\). This equation represents the same curve as the parametric equations but in terms of \(x\) and \(y\) only.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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, often denoted as t. Understanding how x and y depend on t allows you to describe and analyze curves that may not be functions in the traditional sense.

Eliminating the Parameter to Find Rectangular Equations

To convert parametric equations into a rectangular (Cartesian) equation, you solve one equation for the parameter and substitute into the other. This process removes the parameter t, yielding a direct relationship between x and y.

Graphing Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) for various values of t. Recognizing the shape and direction of the curve helps in visualizing the relationship between x and y and verifying the rectangular equation.

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)