In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. _ x = √t, y = t − 1

Verified step by step guidance

1. Start with the given parametric equations: \(x = \sqrt{t}\) and \(y = t - 1\).
2. Express \(t\) in terms of \(x\) from the first equation: since \(x = \sqrt{t}\), then \(t = x^2\) (note that \(x \geq 0\) because square roots are non-negative).
3. Substitute \(t = x^2\) into the second equation to eliminate the parameter: \(y = x^2 - 1\).
4. Recognize that the rectangular equation is \(y = x^2 - 1\), which is a parabola opening upwards, shifted down by 1 unit.
5. To sketch the curve, plot the parabola \(y = x^2 - 1\) for \(x \geq 0\) (since \(x = \sqrt{t}\) implies \(x \geq 0\)), and use arrows pointing in the direction of increasing \(t\) (which corresponds to increasing \(x\) and \(y\) values).

Key Concepts

Parametric Equations and Parameters

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how the parameter t controls the position on the curve is essential for analyzing the curve's shape and orientation.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single rectangular equation in terms of x and y. This process helps to identify the curve's equation in the Cartesian plane, making it easier to analyze and sketch.

Sketching and Orientation of Curves

Sketching the curve requires plotting points that satisfy the rectangular equation and indicating the direction of increasing parameter t with arrows. Orientation shows how the curve is traced as t increases, which is important for understanding the curve's behavior.