This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. Thus the astroid is not smooth at these points, corresponding to the cusps seen in the figure. As you do so, consider what you notice and what you wonder. For instance, one can verify that \(x=t^3\), \(y=t^6\) produce the familiar \(y=x^2\) parabola. Example \(\PageIndex{9}\): Determine where a curve is not smooth. The Pythagorean Theorem can also be used to identify parametric equations for hyperbolas. The following example demonstrates one possible alternative. y &= \frac{t^2}{t^2 +1} \\ Consider \(y=x^2\). An object is fired from a height of 0ft and lands 6 seconds later, 192ft away. The portion of the graph defined by the parametric equations is given in a thick line; the graph defined by \(y=1-x\) with unrestricted domain is given in a thin line. We should be careful to limit the domain of the function \(y=1-x\). Taking derivatives, we have: \[x^\prime = -3\cos^2t\sin t\quad \text{and}\quad y^{\prime} = 3\sin^2t\cos t.\]. These examples begin to illustrate the powerful nature of parametric equations. r =(−1,0,2)+s(0,1,−1)+t(1,−2,0); s,t∈R r s t R z s y s t x t x y z s t s t R However, in this parametrization, the curve is not smooth. Plot the graph of the parametric equations \(x=t^2\), \(y=t+1\) for \(t\) in \([-2,2]\). Ex 2. The following example demonstrates this. Watch the recordings here on Youtube! In Example \(\PageIndex{1}\), if we let \(t\) vary over all real numbers, we'd obtain the entire parabola. This conversion is often referred to as "eliminating the parameter,'' as we are looking for a relationship between \(x\) and \(y\) that does not involve the parameter \(t\). The set of all points (x, y) = (f(t), g(t)) in the Cartesian plane, as t varies over I, is the graph of the parametric equations x = f(t) and y = g(t), where t is the parameter. The graph of the parametric equations is given in Figure 9.22 (a). We again start by making a table of values in Figure 9.21(a), then plot the points \((x,y)\) on the Cartesian plane in Figure 9.21(b). The user usually needs to determine the graphing window (i.e, the minimum and maximum \(x\)- and \(y\)-values), along with the values of \(t\) that are to be plotted. The points have been connected with a smooth curve. Let a curve \(C\) be defined by the parametric equations \(x=t^3-12t+17\) and \(y=t^2-4t+8\). As \(y=-x^2/64+3x\), we find \(y= -16t^2+96t\). A plane is determined by a pointP_0 in the plane anda vector n(calledthe normal vector) orthogonal to the plane. Thus parametric equations for the parabola \(y=x^2\) are. The line intersect the xy-plane at the point(-10,2). In this section we introduce a new sketching procedure: Here, \(x\) and \(y\) are found separately but then plotted together. (Plus, to shift to the right by two, we replace \(x\) with \(x-2\), which is counter--intuitive.) The point P belongs to the plane π if the vector is coplanar with the vectors and. Figure 9.20: A table of values of the parametric equations in Example 9.2.1 along with a sketch of their graph. (Experiment with. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Parametric Equations of the Plane Cartesian Equation of the Plane One method is to solve for \(t\) in one equation and then substitute that value in the second. \[y = \frac{t^2}{t^2+1} = 1-\frac{1}{t^2+1} = 1-x.\] How would you explain the role of "a" in the parametric equation of a plane? \[x^\prime = 3t^2-12,\quad y^{\prime} = 2t-4.\], \[\begin{array}{l}x^\prime = 0 \Rightarrow 3t^2-12=0 \Rightarrow t=\pm 2\\ This is graphed in Figure 9.22 (b). A curve is a graph along with the parametric equations that define it. As an example, given \(y=x^2\), the parametric equations \(x=t\), \(y=t^2\) produce the familiar parabola. Often this will be written as, ax+by +cz = d a x + b y + c z = d. where d =ax0 +by0 +cz0 d = a x 0 + b y 0 + c z 0. It is sometimes necessary to convert given parametric equations into rectangular form. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. This is a formal definition of the word curve. The orientation shown in Figure 9.21 shows the orientation on \([0,\pi]\), but this orientation is reversed on \([\pi,2\pi]\). Let \(f\) and \(g\) be continuous functions on an interval \(I\). One might note a feature shared by two of these graphs: "sharp corners,'' or cusps. Graphing utilities effectively plot parametric functions just as we've shown here: they plots lots of points. While the parabola is the same, the curves are different. We are looking for two different values, say, \(s\) and \(t\), where \(x(s) = x(t)\) and \(y(s) = y(t)\). This final equation should look familiar -- it is the equation of an ellipse! We explore these concepts and more in the next section. This is a shortcut that is very specific to this problem; sometimes shortcuts exist and are worth looking for. Since the object travels 192ft in 6s, we deduce that the object is moving horizontally at a rate of 32ft/s, giving the equation \(x=32t\). Sketch the graph of the parametric equations \(x=\cos^2t\), \(y=\cos t+1\) for \(t\) in \([0,\pi]\). &= \left(\frac1x-1\right)\cdot x \\ Thus,x=-1+3t=-10 and y=2. (To plot an ellipse using the above procedure, we need to plot the "top'' and "bottom'' separately.). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Below you can experiment with entering different vectors to explore different planes. We start by computing \(\frac{dy}{dx}\): \(y^{\prime} = 2x\). It is clear that each is 0 when \(t=0,\ \pi/2,\ \pi,\ldots\). Example \(\PageIndex{8}\): Eliminating the parameter, Eliminate the parameter in \(x=4\cos t+3\), \(y= 2\sin t+1\), We should not try to solve for \(t\) in this situation as the resulting algebra/trig would be messy. Because the \(x\)- and \(y\)-values of a graph are determined independently, the graphs of parametric functions often possess features not seen on "\(y=f(x)\)" type graphs. This leads us to a definition. Gregory Hartman (Virginia Military Institute). Figure 9.21: A table of values of the parametric equations in Example 9.2.2 along with a sketch of their graph. Legal. This leads us to a definition. Example \(\PageIndex{7}\): Eliminating the parameter, Find a rectangular equation for the curve described by \[ x= \frac{1}{t^2+1}\quad \text{and}\quad y=\frac{t^2}{t^2+1}.\]. It is sometimes useful to rewrite equations in rectangular form (i.e., \(y=f(x)\)) into parametric form, and vice--versa. We plot the graphs of parametric equations in much the same manner as we plotted graphs of functions like \(y=f(x)\): we make a table of values, plot points, then connect these points with a "reasonable'' looking curve. Figure 9.27: A gallery of interesting planar curves. Convert the vector equation to the parametric equations. The next example demonstrates how such graphs can arrive at the same point more than once. This is called the scalar equation of plane. If you like, check out the video below. As you do so, consider what you notice and what you wonder. This can be substituted into the first equation, revealing that the graph crosses itself at \(t=-1\) and \(t=3\).
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