In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t

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Identify the parametric equations given: \(x = t^{2} + t + 1\) and \(y = 2t\). These describe the coordinates \((x, y)\) in terms of the parameter \(t\).

To sketch the curve, first consider the domain of \(t\). Since \(t\) is a real number parameter, its domain is \((-\infty, \infty)\).

Express \(t\) in terms of \(y\) from the second equation: \(y = 2t \implies t = \frac{y}{2}\). This substitution will help us find the relation between \(x\) and \(y\).

Substitute \(t = \frac{y}{2}\) into the equation for \(x\): \(x = \left(\frac{y}{2}\right)^{2} + \frac{y}{2} + 1 = \frac{y^{2}}{4} + \frac{y}{2} + 1\). This gives the Cartesian form of the curve.

Analyze the domain and range of \(x\) and \(y\): since \(t\) ranges over all real numbers, \(y = 2t\) also ranges over \((-\infty, \infty)\). For \(x\), because it is a quadratic expression in \(t\), determine its minimum value by completing the square or using vertex formula to find the range of \(x\).

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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, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the representation of more complex curves and motions.

Domain and Range in Parametric Form

The domain refers to all possible values of the parameter t for which the parametric equations are defined. The range consists of all possible output values of x and y generated by those t values. Interval notation is used to express these sets clearly.

Sketching Parametric Curves

To sketch a parametric curve, calculate points by substituting values of t into the equations, then plot the corresponding (x, y) points. Understanding how x and y change with t helps visualize the shape and direction of the curve.