In this blog post, we delve into the concept of parametric equations, providing a comprehensive explanation along with a problem-solving approach. Learn how to tackle parametric equations through a real-world example while strengthening your precalculus and algebra skills.
Introduction:
Parametric equations are a powerful mathematical tool used to represent curves and functions in a non-traditional way. In the realm of precalculus and algebra, understanding parametric equations can enhance your problem-solving abilities and expand your mathematical repertoire. In this blog post, we will explore the concept of parametric equations by delving into a real-world problem, explaining the underlying concepts, and offering a step-by-step solution.
Problem Statement:
Imagine a scenario where a particle is moving along a curve in the xy-plane. The particle’s position at any given time can be defined by the parametric equations:
x(t) = 2t^2 + 3t + 1 y(t) = t^3 – 4t
We are tasked with finding the coordinates of the particle at a specific point in time, t = 2, and determining whether it is moving upward or downward.
Understanding Parametric Equations:
Parametric equations provide an alternative way of representing mathematical functions, where both the x and y coordinates are expressed as functions of a third parameter, often denoted as “t.” By varying the value of t, we can trace the path of the particle and study its motion.
Solution:
Step 1: Finding the x-coordinate at t = 2 To determine the x-coordinate of the particle at t = 2, substitute the given value into the x(t) equation: x(2) = 2(2^2) + 3(2) + 1 = 8 + 6 + 1 = 15
Therefore, at t = 2, the x-coordinate of the particle is 15.
Step 2: Finding the y-coordinate at t = 2 Substitute t = 2 into the y(t) equation: y(2) = (2^3) – 4(2) = 8 – 8 = 0
Hence, at t = 2, the y-coordinate of the particle is 0.
Step 3: Determining the direction of motion To determine whether the particle is moving upward or downward at t = 2, we can examine the slope of the curve at that point. The slope can be calculated by taking the derivative of y(t) with respect to x(t):
dy/dx = (dy/dt) / (dx/dt)
Differentiating x(t) and y(t) with respect to t, we get:
dx/dt = 4t + 3 dy/dt = 3t^2 – 4
Substituting t = 2 into the derivatives:
dx/dt = 4(2) + 3 = 11 dy/dt = 3(2^2) – 4 = 8
Now, calculate the slope:
dy/dx = (dy/dt) / (dx/dt) = 8 / 11
Since the slope is positive (8/11 > 0), we can conclude that the particle is moving upward at t = 2.
Conclusion:
Parametric equations are a valuable tool for representing curves and functions in precalculus and algebra. By employing these equations, we can study the behavior of particles in motion, analyze their positions at specific points in time, and determine the direction of movement.