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Copyright © Cengage Learning. All rights reserved.
Polar Coordinates and Parametric Equations Copyright © Cengage Learning. All rights reserved.

8.4 Plane Curves and Parametric Equations
Copyright © Cengage Learning. All rights reserved.

Objectives Plane Curves and Parametric Equations
Eliminating the Parameter Finding Parametric Equations for a Curve Using Graphing Devices to Graph Parametric Curves

Plane Curves and Parametric Equations
In this section we study parametric equations, which are a general method for describing any curve.

Plane Curves and Parametric Equations
We can think of a curve as the path of a point moving in the plane; the x- and y-coordinates of the point are then functions of time. This idea leads to the following definition.

Example 1 – Sketching a Plane Curve
Sketch the curve defined by the parametric equations x = t2 – 3t y = t – 1 Solution: For every value of t, we get a point on the curve. For example, if t = 0, then x = 0 and y = –1, so the corresponding point is (0, –1).

Example 1 – Solution cont’d In Figure 1 we plot the points (x, y) determined by the values of t shown in the following table. Figure 1

Example 1 – Solution cont’d As t increases, a particle whose position is given by the parametric equations moves along the curve in the direction of the arrows.

Plane Curves and Parametric Equations
If we replace t by –t in Example 1, we obtain the parametric equations x = t2 + 3t y = –t – 1 The graph of these parametric equations (see Figure 2) is the same as the curve in Figure 1, but traced out in the opposite direction. x = t2 + 3t, y = –t – 1 Figure 1 Figure 2

Plane Curves and Parametric Equations
On the other hand, if we replace t by 2t in Example 1, we obtain the parametric equations x = 4t2 – 6t y = 2t – 1 The graph of these parametric equations (see Figure 3) is again the same, but is traced out “twice as fast.” x = 4t2 + 6t, y = 2t –1 Figure 3

Plane Curves and Parametric Equations
Thus, a parametrization contains more information than just the shape of the curve; it also indicates how the curve is being traced out.

Eliminating the Parameter

Eliminating the Parameter
Often a curve given by parametric equations can also be represented by a single rectangular equation in x and y. The process of finding this equation is called eliminating the parameter. One way to do this is to solve for t in one equation, then substitute into the other.

Example 2 – Eliminating the Parameter
Eliminate the parameter in the parametric equations of Example 1. Solution: First we solve for t in the simpler equation, then we substitute into the other equation. From the equation y = t – 1, we get t = y + 1.

Example 2 – Solution Substituting into the equation for x, we get
cont’d Substituting into the equation for x, we get x = t2 – 3t = (y + 1)2 – 3(y + 1) = y2 – y – 2 Thus the curve in Example 1 has the rectangular equation x = y2 – y – 2, so it is a parabola.

Finding Parametric Equations for a Curve

Example 5 – Finding Parametric Equations for a Graph
Find parametric equations for the line of slope 3 that passes through the point (2, 6). Solution: Let’s start at the point (2, 6) and move up and to the right along this line. Because the line has slope 3, for every 1 unit we move to the right, we must move up 3 units. In other words, if we increase the x-coordinate by t units, we must correspondingly increase the y-coordinate by 3t units.

Example 5 – Solution This leads to the parametric equations
cont’d This leads to the parametric equations x = 2 + t y = 6 + 3t To confirm that these equations give the desired line, we eliminate the parameter. We solve for t in the first equation and substitute into the second to get y = 6 + 3(x – 2) = 3x

Example 5 – Solution cont’d Thus the slope-intercept form of the equation of this line is y = 3x, which is a line of slope 3 that does pass through (2, 6) as required. The graph is shown in Figure 6. Figure 6

Example 6 – Parametric Equations for the Cycloid
As a circle rolls along a straight line, the curve traced out by a fixed point P on the circumference of the circle is called a cycloid (see Figure 7). If the circle has radius a and rolls along the x-axis, with one position of the point P being at the origin, find parametric equations for the cycloid. Figure 7

Example 6 – Solution Figure 8 shows the circle and the point P after the circle has rolled through an angle  (in radians). The distance d(O, T ) that the circle has rolled must be the same as the length of the arc PT, which, by the arc length formula, is a. Figure 8

Example 6 – Solution cont’d This means that the center of the circle is C(a, a). Let the coordinates of P be (x, y). Then from Figure 8 (which illustrates the case 0 <  <  /2), we see that x = d(O, T) – d(P, Q) = a – a sin  = a( – sin ) y = d(T, C) – d(Q, C) = a – a cos  = a(1 – cos ) so parametric equations for the cycloid are x = a( – sin ) y = a(1 – cos )

Using Graphing Devices to Graph Parametric Curves

Example 7 – Graphing Parametric Curves
Use a graphing device to draw the following parametric curves. Discuss their similarities and differences. (a) x = sin 2t y = 2 cos t (b) x = sin 3t

Example 7 – Solution In both parts (a) and (b) the graph will lie inside the rectangle given by –1  x  1, –2  y  2, since both the sine and the cosine of any number will be between –1 and 1. Thus, we may use the viewing rectangle [–1.5, 1.5] by [–2.5, 2.5].

Example 7 – Solution cont’d (a) Since 2 cos t is periodic with period 2 and since sin 2t has period , letting t vary over the interval 0  t  2 gives us the complete graph, which is shown in Figure 12(a). (a) x = sin 2t, y = 2cos t Figure 12

Example 7 – Solution cont’d (b) Again, letting t take on values between 0 and 2 gives the complete graph shown in Figure 12(b). (b) x = sin 3t, y = 2cos t Figure 12

Example 7 – Solution cont’d Both graphs are closed curves, which means they form loops with the same starting and ending point; also, both graphs cross over themselves. However, the graph in Figure 12(a) has two loops, like a figure eight, whereas the graph in Figure 12(b) has three loops.

Using Graphing Devices to Graph Parametric Curves
The curves graphed in Example 7 are called Lissajous figures. A Lissajous figure is the graph of a pair of parametric equations of the form x = A sin ω1t y = B cos ω2t where A, B, ω1, and ω2 are real constants. Since sin ω1t and cos ω2t are both between –1 and 1, a Lissajous figure will lie inside the rectangle determined by –A  x  A, –B  y  B.

Using Graphing Devices to Graph Parametric Curves
This fact can be used to choose a viewing rectangle when graphing a Lissajous figure, as in Example 7. We know that rectangular coordinates (x, y) and polar coordinates (r, ) are related by the equations x = r cos , y = r sin . Thus we can graph the polar equation r = f() by changing it to parametric form as follows: x = r cos  = f() cos  Since r = f()

Using Graphing Devices to Graph Parametric Curves
y = r sin  = f() sin  Replacing  by the standard parametric variable t, we have the following result.

Example 8 – Parametric Form of a Polar Equation
Consider the polar equation r = , 1    10. (a) Express the equation in parametric form. (b) Draw a graph of the parametric equations from part (a). Solution: (a) The given polar equation is equivalent to the parametric equations x = t cos t y = t sin t

Example 8 – Solution cont’d (b) Since 10  31.42, we use the viewing rectangle [–32, 32] by [– 32, 32], and we let t vary from 1 to 10. The resulting graph shown in Figure 13 is a spiral. (b) x = t cos t, y = t sin t Figure 13

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