Question

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

Accepted Answer

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.$$

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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Key Concepts

Parametric Equations

Domain and Range in Parametric Form

Sketching Parametric Curves