For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.


x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)

Verified step by step guidance

1. Step 1: Understand the parametric equations given: \(x = t + 2\) and \(y = t - 4\), where \(t\) is a parameter that can take any real value from \(-\infty\) to \(\infty\).
2. Step 2: To graph the curve, recognize that as \(t\) varies, the point \((x, y)\) moves along the curve defined by these equations. Plot several points by choosing values of \(t\), calculating corresponding \(x\) and \(y\), and then sketch the curve through these points.
3. Step 3: To find a rectangular equation (an equation involving only \(x\) and \(y\)), eliminate the parameter \(t\) from the system. From the first equation, express \(t\) in terms of \(x\): \(t = x - 2\).
4. Step 4: Substitute \(t = x - 2\) into the second equation: \(y = (x - 2) - 4\).
5. Step 5: Simplify the expression to get the rectangular equation: \(y = x - 6\). This equation represents the same curve as the parametric equations but in terms of \(x\) and \(y\) only.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how x and y depend on t allows you to describe and analyze curves that may not be functions in the traditional sense.

Eliminating the Parameter to Find Rectangular Equations

To convert parametric equations into a rectangular (Cartesian) equation, you solve one equation for the parameter and substitute into the other. This process removes the parameter t, yielding a direct relationship between x and y.

Graphing Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) for various values of t. Recognizing the shape and direction of the curve helps in visualizing the relationship between x and y and verifying the rectangular equation.