In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other. x = t and y = t² − 4

Verified step by step guidance

1. Identify the parametric equations given: \(x = t\) and \(y = t^{2} - 4\). These describe a curve in the plane where \(t\) is the parameter.
2. Express \(y\) in terms of \(x\) by eliminating the parameter \(t\). Since \(x = t\), substitute \(t\) with \(x\) in the equation for \(y\) to get \(y = x^{2} - 4\).
3. Recognize that the equation \(y = x^{2} - 4\) represents a parabola opening upwards, shifted downward by 4 units along the \(y\)-axis.
4. To graph the curve, plot points by choosing values of \(t\) (or \(x\)), compute corresponding \(y\) values using \(y = t^{2} - 4\), and plot \((x, y)\) pairs on the coordinate plane.
5. Compare this curve to other given parametric curves by analyzing their equations, shapes, and shifts to understand how they differ in position, orientation, or form.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually 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.

Graphing Parametric Curves

To graph parametric curves, compute pairs (x(t), y(t)) for various values of t and plot these points. This method helps visualize the shape and direction of the curve, revealing features like symmetry, intercepts, and turning points.

Comparing Plane Curves

Comparing plane curves involves analyzing differences in shape, orientation, and position. By examining their parametric forms and graphs, one can identify how changes in equations affect the curve’s geometry and behavior.