Learn the fundamental concepts of parametric equations. This stack covers the definition of a parameter, how to plot points, and understanding the orientation or direction of the curve.
What is the general form of a set of parametric equations?
A set of parametric equations represents a curve using a third variable, called a parameter (often \(t\)). The general form is \(x = f(t)\) and \(y = g(t)\), where \(x\) and \(y\) coordinates are expressed as functions of \(t\).
In parametric equations, what is the 'parameter'?
The parameter, commonly denoted by \(t\) or \(θ\), is an independent variable upon which \(x\) and \(y\) depend. It often represents time, but can also represent an angle or another quantity.
What is the 'orientation' of a parametric curve?
The orientation is the direction in which the curve is traced as the parameter \(t\) increases. It is typically indicated by arrows drawn on the curve.
What is the first step to sketch the graph of a parametric curve?
The first step is to create a table of values. Choose several values for the parameter \(t\) within its given interval, calculate the corresponding \(x\) and \(y\) coordinates, and list them as \((x, y)\) points.
Given the parametric equations \(x = t + 2\) and \(y = t^2\), find the coordinates of the point when \(t = 3\).
Substitute \(t = 3\) into both equations. \(x = 3 + 2 = 5\) and \(y = 3^2 = 9\). The point is \((5, 9)\).
Given \(x = 4\cos(t)\) and \(y = 4\sin(t)\), find the coordinates of the point when \(t = \frac{\pi}{2}\).
Substitute \(t = \frac{\pi}{2}\). \(x = 4\cos(\frac{\pi}{2}) = 4(0) = 0\). \(y = 4\sin(\frac{\pi}{2}) = 4(1) = 4\). The point is \((0, 4)\).
Find the starting and ending points for the curve defined by \(x = 2t - 1\) and \(y = 5 - t\) for \(0 \le t \le 4\).
Starting point (at \(t=0\)): \(x=2(0)-1 = -1\), \(y=5-0=5\), so \((-1, 5)\).
Ending point (at \(t=4\)): \(x=2(4)-1 = 7\), \(y=5-4=1\), so \((7, 1)\).
What does the interval of the parameter, such as \(a \le t \le b\), define for a parametric curve?
The interval for the parameter \(t\) defines a specific segment of the curve. It specifies the starting point (at \(t=a\)) and the ending point (at \(t=b\)), creating a finite arc rather than an infinite curve.
Can two different sets of parametric equations represent the exact same graph?
Yes. This is called reparameterization. For example, \(x=t, y=t^2\) for \(t \in \mathbb{R}\) and \(x=t^3, y=t^6\) for \(t \in \mathbb{R}\) both trace the same parabola \(y=x^2\). However, the speed at which the curve is traced can differ.
Plot the point for \(t = -1\) for the curve given by \(x = t^3 + 1\) and \(y = t^2\).
For \(t = -1\): \(x = (-1)^3 + 1 = -1 + 1 = 0\) and \(y = (-1)^2 = 1\). The point is \((0, 1)\).
If a curve is defined by \(x = 5 - t\) and \(y = t + 2\), what is the direction of motion as \(t\) increases?
As \(t\) increases, \(x\) decreases (because of the \(-t\)) and \(y\) increases. This means the curve is traced from right to left, and upwards.
Does every rectangular equation \(y = f(x)\) have a parametric representation?
Yes. A simple parametric representation can always be created by letting \(x = t\). Then the corresponding \(y\) equation becomes \(y = f(t)\). For example, \(y = x^2 + 2x\) can be parameterized as \(x=t\) and \(y = t^2 + 2t\).
