In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = t² + 3, y = 6 − t³; t = 2

Verified step by step guidance

1. Identify the given parametric equations: \(x = t^{2} + 3\) and \(y = 6 - t^{3}\), and the parameter value \(t = 2\).
2. Substitute the given value of \(t\) into the equation for \(x\): calculate \(x = (2)^{2} + 3\).
3. Substitute the given value of \(t\) into the equation for \(y\): calculate \(y = 6 - (2)^{3}\).
4. Simplify the expressions obtained for \(x\) and \(y\) to find their numerical values.
5. Write the coordinates of the point on the curve as \((x, y)\) using the values found in the previous step.

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.

Substitution of Parameter Values

To find a specific point on a parametric curve, substitute the given parameter value into the parametric equations. This yields the corresponding x and y coordinates, pinpointing the exact location on the curve for that parameter.

Coordinate Plane and Points

The coordinate plane is a two-dimensional space where points are identified by ordered pairs (x, y). Understanding how to interpret these pairs is essential for visualizing and plotting points derived from parametric equations.