Question

Multiple Choice Which of the following describes the graph of the parametric curve \(x=3 t, y=2 t, t \geq 1 ? \mathrm{E}\) (A) circle (B) parabola (C) line segment (D) line (E) ray

Short answer

The graph of the given parametric curve represents a ray (E).

Step-by-step solution

Understand the Parameters

Firstly, understand the given equations \(x=3t\) and \(y=2t\). These represent the x and y coordinates of every point on the graph as a function of t.

Rewriting Equation

Rewrite the equations to find \(t\) as a function of \(x\) and \(y\). From \(x=3t\), we get \(t=x/3\). From \(y=2t\), we get \(t=y/2\).

Equating Parameters

Set the expressions for \(t\) from step 2 equal to each other. This gives us \(x/3 = y/2\). Then cross-multiply to solve for \(y\), getting \(y = (2/3)x\). This is the equation of a line, also implying that the graph will be a line.

Consider Range of t

Given \(t\geq1\), meaning the graph only contains points where \(t\) is greater than or equal to 1. Therefore, it's not a full line, but rather a ray or line segment starting from a point where \(t=1\). However, it is a ray rather than a line segment because it extends infinitely as \(t\) goes to infinity.