In Example \(\PageIndex{1}\), if we let \(t\) vary over all real numbers, we'd obtain the entire parabola. When a curve lies in a plane (such as the Cartesian plane), it is often referred to as a plane curve. Let a curve \(C\) be defined by the parametric equations \(x=t^3-12t+17\) and \(y=t^2-4t+8\). The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. \end{array} What role to the "parameters" lambda and mu have in the parametric equation of the plane? Sketch the graph of the parametric equations \(x=t^2+t\), \(y=t^2-t\). One may have recognized this earlier by manipulating the equation for \(y\): This is graphed in Figure 9.22 (b). Sketch the graph of the parametric equations \(x=\cos^2t\), \(y=\cos t+1\) for \(t\) in \([0,\pi]\). Figure 9.28: Graphing the curve in Example 9.2.9; note it is not smooth at \((1,4)\). Below you can experiment with entering different vectors to explore different planes. This gives \[\cos t = \frac{x-3}{4} \quad \text{and}\quad \sin t=\frac{y-1}{2}.\] The parametric equations limit \(x\) to values in \((0,1]\), thus to produce the same graph we should limit the domain of \(y=1-x\) to the same. &= \frac{1/x-1}{1/x-1+1} \\ We often use the letter \(t\) as the parameter as we often regard \(t\) as representing time. Thus our parametric equations for the shifted graph are \(x=t^2+t+3\), \(y=t^2-t-2\). 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. That is, it has traveled horizontally 64ft and is at a height of 128ft, as shown in Figure 9.24. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. &= \left(\frac1x-1\right)\cdot x \\ Gregory Hartman (Virginia Military Institute). Figure 9.26 plots the parametric equations, demonstrating that the graph is indeed of an ellipse with a horizontal major axis and center at \((3,1)\). The following example demonstrates this. In order to shift the graph to the right 3 units, we need to increase the \(x\)-value by 3 for every point. We demonstrate this once more. Planes. Converting from rectangular to parametric can be very simple: given \(y=f(x)\), the parametric equations \(x=t\), \(y=f(t)\) produce the same graph. 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 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. The user is often prompted to give a \(t\) minimum, a \(t\) maximum, and a "\(t\)-step'' or "\(\Delta t\).'' The orientation shown in Figure 9.21 shows the orientation on \([0,\pi]\), but this orientation is reversed on \([\pi,2\pi]\). The next example demonstrates how such graphs can arrive at the same point more than once. This leads us to a definition. It is sometimes useful to rewrite equations in rectangular form (i.e., \(y=f(x)\)) into parametric form, and vice--versa. y^{\prime}=0 \Rightarrow 2t-4 = 0 \Rightarrow t=2 Find new parametric equations that shift this graph to the right 3 places and down 2. (Experiment with. While the parabola is the same, the curves are different. Example \(\PageIndex{4}\): Graphs that cross themselves. We see at \(t=2\) both \(x^\prime\) and \(y^{\prime}\) are 0; thus \(C\) is not smooth at \(t=2\), corresponding to the point \((1,4)\). Figure 9.21: A table of values of the parametric equations in Example 9.2.2 along with a sketch of their graph. Can we determine concavity? These parametric equations make certain determinations about the object's location easy: 2 seconds into the flight the object is at the point \(\big(x(2),y(2)\big) = \big(64,128\big)\). As an example, given \(y=x^2\), the parametric equations \(x=t\), \(y=t^2\) produce the familiar parabola. We explore these concepts and more in the next section. Their graphs are far more diverse than the graphs of functions produced by "\(y=f(x)\)" functions. The point P belongs to the plane π if the vector is coplanar with the vectors and. We can quickly verify that \(y^{\prime\prime\prime}=-32\)ft/s\(^2\), the acceleration due to gravity, and that the projectile reaches its maximum at \(t=3\), halfway along its path. Let \(f\) and \(g\) be continuous functions on an interval \(I\). If a curve is not smooth at \(t=t_0\), it means that \(x^\prime(t_0)=y^{\prime}(t_0)=0\) as defined. What role to the "parameters" lambda and mu have in the parametric equation of the plane? A curve is a graph along with the parametric equations that define it. Each point has been labeled with its corresponding \(t\)-value. That is, the \(x\)-values are the same precisely when the \(y\)-values are the same. Legal. Examples will help us understand the concepts introduced in the definition. Solution \[y = \frac{t^2}{t^2+1} = 1-\frac{1}{t^2+1} = 1-x.\] A particle traveling along the parabola according to the given parametric equations comes to rest at \(t=0\), though no sharp point is created.\\.