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 + 1, y = 5 − t³; t = 2

Verified step by step guidance

1. Identify the given parametric equations: \( x = t^2 + 1 \) and \( y = 5 - t^3 \).
2. Substitute the given value of \( t = 2 \) into the equation for \( x \): \( x = (2)^2 + 1 \).
3. Calculate the expression for \( x \) to find its value.
4. Substitute the given value of \( t = 2 \) into the equation for \( y \): \( y = 5 - (2)^3 \).
5. Calculate the expression for \( y \) to find its value.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically denoted as 't'. In this case, x and y are defined in terms of t, allowing for the representation of curves in a two-dimensional plane. Understanding how to evaluate these equations at specific values of t is crucial for finding points on the curve.

Substitution

Substitution is the process of replacing a variable with a specific value to evaluate an expression. In the context of parametric equations, substituting the given value of t into the equations for x and y allows us to calculate the corresponding coordinates of the point on the curve. This step is essential for determining the exact location on the graph.

Coordinate System

A coordinate system provides a framework for locating points in a plane using pairs of numbers (x, y). In this problem, the Cartesian coordinate system is used, where the x-coordinate is derived from the first parametric equation and the y-coordinate from the second. Understanding how to interpret these coordinates is vital for visualizing the point on the curve.