---
title: "A particle moves in the \\(xy\\)-plane starting from the origin at time \\(t = 0\\). The position of the particle as a function of time is given by \\(x(t) = \\alpha t^2\\) and \\(y(t) = \\beta t – \\gamma t^2\\), where \\(\\alpha\\), \\(\\beta\\), and \\(\\gamma\\) are positive constants. Which of the following graphs best represents the trajectory of the particle in the \\(xy\\)-plane for \\(t \\ge 0\\)?"
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url: "https://nerd-notes.com/ubq/124035/"
date_modified: "2026-09-28T13:30:14+00:00"
---

# A particle moves in the \(xy\)-plane starting from the origin at time \(t = 0\). The position of the particle as a function of time is given by \(x(t) = \alpha t^2\) and \(y(t) = \beta t – \gamma t^2\), where \(\alpha\), \(\beta\), and \(\gamma\) are positive constants. Which of the following graphs best represents the trajectory of the particle in the \(xy\)-plane for \(t \ge 0\)?

A particle moves in the \(xy\)-plane starting from the origin at time \(t = 0\). The position of the particle as a function of time is given by \(x(t) = \alpha t^2\) and \(y(t) = \beta t - \gamma t^2\), where \(\alpha\), \(\beta\), and \(\gamma\) are positive constants. Which of the following graphs best represents the trajectory of the particle in the \(xy\)-plane for \(t \ge 0\)?

- **A.** Graph A
- **B.** Graph B
- **C.** Graph C
- **D.** Graph D

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124035/*