Eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = √t , y = t + 1; −∞ < t < ∞

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Identify the given parametric equations: \(x = \sqrt{t}\) and \(y = t + 1\), with the parameter \(t\) ranging over all real numbers.

Since \(x = \sqrt{t}\), note that \(x\) is defined only for \(t \geq 0\) because the square root of a negative number is not a real number. This restricts the domain of \(t\) to \(t \geq 0\).

Express \(t\) in terms of \(x\) by squaring both sides of the equation \(x = \sqrt{t}\), which gives \(t = x^2\).

Substitute \(t = x^2\) into the equation for \(y\): \(y = t + 1\) becomes \(y = x^2 + 1\).

The Cartesian equation of the curve is \(y = x^2 + 1\) with \(x \geq 0\). To graph, plot this parabola starting from \(x=0\) (where \(y=1\)) and moving rightward, using arrows to indicate the direction as \(t\) increases.

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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 description of more complex curves and motions.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to express y directly in terms of x, removing t. This is done by solving one equation for t and substituting into the other, which helps in identifying the Cartesian form of the curve.

Orientation of Parametric Curves

Orientation indicates the direction in which the curve is traced as the parameter increases. Using arrows on the graph shows this direction, which is important for understanding the behavior and properties of the curve over the parameter's domain.