Question

What parametric curve is described by the function $$ \gamma(t)=(t-3)+i(2 t+1) $$ \(0 \leq t \leq 2\) ? [Hint: What would you do if you were instead considering the parametric equations \(x=t-3\) and \(y=2 t+1]\)

Short answer

The parametric curve described by the function \(\gamma(t)=(t-3)+i(2 t+1)\) for \(0 \leq t \leq 2\) is the straight line \(y = 2x + 7\) in the Cartesian plane.

Step-by-step solution

Translate into parametric equations

Note that a complex function can be written as \(\gamma(t) = x(t) + i y(t)\), and this corresponds to the parametric equations \(x = x(t), y = y(t)\) in the Cartesian coordinate system. The real part of \(\gamma(t)\) corresponds to \(x(t)\), and the imaginary part corresponds to \(y(t)\). So, from \(\gamma(t)=(t-3)+i(2 t+1)\), we extract \(x(t) = t -3\) and \(y(t) = 2t + 1\).

Identify the parametric curve

The equations \(x(t) = t -3\) and \(y(t) = 2t + 1\) are linear, implying that they represent a straight line in the Cartesian plane. To find the line equation, solve \(x(t)\) equation for \(t\) and substitute into the \(y(t)\) equation to eliminate \(t\).

Find and Analyze the line

Solving \(x(t) = t - 3\) for \(t\) gives us \(x + 3 = t\). Substituting this into \(y(t) = 2t + 1\) gives \(y = 2(x + 3) + 1 = 2x + 6 + 1 = 2x + 7\). Hence, the parametric curve described by this function is the straight line \[y = 2x + 7\] in the Cartesian plane.

Key concepts

Complex Function

When delving into the realm of complex numbers, we often encounter functions that output complex values. A complex function like the one involved in our exercise, \(\gamma(t) = (t-3) + i(2t+1)\), offers a wonderful illustration of how we can decipher mathematical expressions involving the imaginary unit \(i\).

Understanding a complex function begins with recognizing its two parts, namely the real part and the imaginary part. These parts, when mapped respectively on the real and imaginary axes of a complex plane, construct a vivid graphical representation. The function essentially combines two relationships: one for the real component \(x(t)\) and another for the imaginary component \(y(t)\). This duality allows us to navigate between algebraic expressions and geometric interpretations with ease.

Parametric Equations

Parametric equations represent a set of equations that express a set of quantities as explicit functions of one or more independent variables, known as parameters. Unlike standard equations that directly relate \(x\) to \(y\), parametric equations define both \(x\) and \(y\) with respect to a third variable, often \(t\), which is considered a parameter.

As seen in our problem, \(\gamma(t)\) is broken down into \(x(t) = t - 3\) and \(y(t) = 2t + 1\). These equations describe the x and y coordinates of points on the curve as functions of \(t\). Such parametric representation gives us the superpower to trace the curve as \(t\) evolves, offering a clear path through the Cartesian plane, point by point.

Cartesian Coordinate System

The Cartesian coordinate system is a cornerstone of analytical geometry, providing a framework to locate points on a plane using ordered pairs of numbers. Invented by René Descartes, this system uses two intersecting axes, typically labeled as the \(x\)-axis (horizontal) and the \(y\)-axis (vertical), which collectively define the plane.

In our exercise, the Cartesian system plays a crucial role by creating a visual context for the parametric equations. When the values computed from \(x(t)\) and \(y(t)\) are plotted on this system, the result is a concrete graphical representation of the parametric curve. The beauty of Cartesian coordinates lies in their simplicity and the intuitive way they allow us to map complex functions and equations to a visual plane.

Line Equation

A line equation conveys the relationship between the \(x\) and \(y\) coordinates of every point along a straight line within the Cartesian plane. The most familiar form is the slope-intercept form, \(y = mx + b\), highlighting the line’s slope \(m\) and its y-intercept \(b\).

The exercise walks through deriving the line equation from parametric equations. By isolating the parameter \(t\) in the \(x(t)\) equation and substituting it into the \(y(t)\) equation, we eliminate the parameter to find a direct relationship between \(x\) and \(y\). Consequently, we arrive at the equation \(y = 2x + 7\), capturing the essence of the straight line that our parametric equations sketch out in the Cartesian plane.