y^{\prime}=0 \Rightarrow 2t-4 = 0 \Rightarrow t=2 As an example, given \(y=x^2\), the parametric equations \(x=t\), \(y=t^2\) produce the familiar parabola. The Pythagorean Theorem can also be used to identify parametric equations for hyperbolas. Below you can experiment with entering different vectors to explore different planes. We sometimes chose the parameter to accurately model physical behavior. This gives us a system of 2 equations with 2 unknowns: \[\begin{array}{c} s^3-5s^2+3s+11 = t^3-5t^2+3t+11 \\ Figure 9.27: A gallery of interesting planar curves. Figure 9.24: Graphing projectile motion in Example 9.2.6. The Pythagorean Theorem gives \(\cos^2t+\sin^2t=1\), so: To shift the graph down by 2 units, we wish to decrease each \(y\)-value by 2, so we subtract 2 from the function defining \(y\): \(y = t^2-t-2\). We explore these concepts and more in the next section. To determine the equation of a plane in 3D space, a point P and a pair of vectors which form a basis (linearly independent vectors) must be known. \cos^2t+\sin^2t &=1 \\ Given a curve defined parametrically, how do we find the slopes of tangent lines? To find the point where the tangent line has a slope of \(-2\), we set \(t=-2\). Our previous experience with cusps taught us that a function was not differentiable at a cusp. Convert the vector equation to the parametric equations. It is not difficult to show that the curves in Examples \(\PageIndex{1}\) and \(\PageIndex{2}\) are portions of the same parabola. Solution The parameters s and t are real numbers. Why does a plane require two non-parallel direction vectors, when a line only required one? Assuming ideal projectile motion, the height, in feet, of the object can be described by \(h(x) = -x^2/64+3x\), where \(x\) is the distance in feet from the initial location. An object is fired from a height of 0ft and lands 6 seconds later, 192ft away. How would you explain the role of "a" in the parametric equation of a plane? Using the methods developed in this section, we again plot points and graph the parametric equations as shown in Figure 9.23. Starting with \(x= 1/(t^2+1)\), solve for \(t\): \( t = \pm\sqrt{1/x-1}\). \left(\frac{x-3}{4}\right)^2 +\left(\frac{y-1}{2}\right)^2 &=1\\ Notice how the vertex is now at \((3,-2)\). Technology Note: Most graphing utilities can graph functions given in parametric form. 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 \(t\). Parametric Equations of the Plane Cartesian Equation of the Plane We have seen graphs with cusps before and determined that such functions are not differentiable at these points. \end{array}\]. Example \(\PageIndex{3}\): Shifting the graph of parametric functions. \end{array} \frac{(x-3)^2}{16}+\frac{(y-1)^2}{4} &=1 Let a curve \(C\) be defined by the parametric equations \(x=t^3-12t+17\) and \(y=t^2-4t+8\). (Experiment with r in the applet to explore.). Example \(\PageIndex{1}\): Plotting parametric functions. It summarizes things in a nifty way. Sketch the graph of the parametric equations \(x=t^2+t\), \(y=t^2-t\). Definition 45 Parametric Equations and Curves Let f and g be continuous functions on an interval I. Why could the equation of a plane be written in infinitely many correct ways? This is graphed in Figure 9.22 (b). A parametrization for a plane can be written as. While the parabola is the same, the curves are different. That is, the \(x\)-values are the same precisely when the \(y\)-values are the same. (Thus \(h(0) = h(192) = 0\)ft.) Find parametric equations \(x=f(t)\), \(y=g(t)\) for the path of the projectile where \(x\) is the horizontal distance the object has traveled at time \(t\) (in seconds) and \(y\) is the height at time \(t\). This leads us to a definition. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. If you like, check out the video below. A curve is a graph along with the parametric equations that define it. Figure 9.28: Graphing the curve in Example 9.2.9; note it is not smooth at \((1,4)\). These values, along with the two arrows along the curve, are used to indicate the orientation of the graph. This can be decidedly more difficult, as some "simple'' looking parametric equations can have very "complicated'' rectangular equations. (Experiment with. In this example, letting \(t\) vary over all real numbers would still produce the same graph; this portion of the parabola would be traced, and re--traced, infinitely. \end{align*}\], Figure 9.25: Graphing parametric and rectangular equations for a graph in Example 9.2.7. 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 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. That is, \(t=a\) corresponds to the point on the graph whose tangent line has slope \(a\). Graphing utilities effectively plot parametric functions just as we've shown here: they plots lots of points. Figure 9.21: A table of values of the parametric equations in Example 9.2.2 along with a sketch of their graph. \[\begin{align*} (Plus, to shift to the right by two, we replace \(x\) with \(x-2\), which is counter--intuitive.) Is there anything you are wondering about parametric form of planes in space? 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). We are familiar with sketching shapes, such as parabolas, by following this basic procedure: The rectangular equation \(y=f(x)\) works well for some shapes like a parabola with a vertical axis of symmetry, but in the previous section we encountered several shapes that could not be sketched in this manner. Thus our parametric equations for the shifted graph are \(x=t^2+t+3\), \(y=t^2-t-2\). A curve \(C\) defined by \(x=f(t)\), \(y=g(t)\) is smooth on an interval \(I\) if \(f^\prime\) and \(g^{\prime}\) are continuous on \(I\) and not simultaneously 0 (except possibly at the endpoints of \(I\)). This, in turn, means that rate of change of \(x\) (and \(y\)) is 0; that is, at that instant, neither \(x\) nor \(y\) is changing. &= \frac{1/x - 1}{1/x} \\ This gives the point \((-1, 1)\). One should be careful to note that a "sharp corner'' does not have to occur when a curve is not smooth. Example \(\PageIndex{7}\): Eliminating the parameter, Find a rectangular equation for the curve described by \[ x= \frac{1}{t^2+1}\quad \text{and}\quad y=\frac{t^2}{t^2+1}.\]. 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. The following example demonstrates one possible alternative. A particle traveling along the parabola according to the given parametric equations comes to rest at \(t=0\), though no sharp point is created.\\. Examples will help us understand the concepts introduced in the definition. The line intersect the xy-plane at the point(-10,2). Example \(\PageIndex{4}\): Graphs that cross themselves. Thus we set \(t=2x\). Planes. We confirm our result by computing \(x(-1) = x(3)=2\) and \(y(-1) = y(3) = 6\). Any point x on the plane is given by s a + t b + c for some value of ( s, t). 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. In order to shift the graph to the right 3 units, we need to increase the \(x\)-value by 3 for every point. It is clear that each is 0 when \(t=0,\ \pi/2,\ \pi,\ldots\). C Parametric Equations of a Plane Let write vector equation of the plane as: (x,y,z) =(x0,y0,z0)+s(ux,uy,uz )+t(vx,vy,vz) or: s t R z z su tv y y su tv x x su tv z z y y x x ∈ ⎪ ⎩ ⎪ ⎨ ⎧ = + + = + + = + +; , 0 0 0 These are the parametric equations of a line.

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