Question

Transformations Let \(x=(2 \cos t)+h\) and \(y=(2 \sin t)+k\) (a) Writing to Learn Let \(k=0\) and \(h=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi]\) . Describe the role of \(h\) (b) Writing to Learn Let \(h=0\) and \(k=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi] .\) Describe the role of \(k\) (c) Find a parametrization for the circle with radius 5 and center at \((2,-3)\)(d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi-minor axis of length 2 parallel to the \(y\) -axis (d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi minor axis of length 2 parallel to the \(y\) -axis.

Short answer

(a,b) The value of \(h\) and \(k\) translates the graph of the circle horizontally and vertically respectively. For different values of \(h\) and \(K\), the circle moves along the x and y axis respectively. (c) The circle of radius 5 centered at (2, -3) is parametrized by \(x = 5\cos(t) + 2\), \(y = 5\sin(t) - 3\). (d) The ellipse described is parametrized by \(x = -3 + 5cos(t)\), \(y = 4 + 2sin(t)\).

Step-by-step solution

Understanding role of \(h\) & \(k\)

Rewrite the equations as \(x = 2cos(t) + h\) and \(y = 2sin(t) + k\). We can see that \(h\) and \(k\) respectively shift the graphs in the x and y direction by \(h\) and \(k\) units.

Parametrization of the circle

Using the general form of parametric equations for a circle \(x = rcos(t) + h\) and \(y = rsin(t) + k\), where \(r\) is the radius and \(h, k\) the coordinates of the centre. Substituting \(r = 5\), \(h = 2\) and \(k = -3\) one gets \(x = 5cos(t) + 2\) and \(y = 5sin(t) - 3\). These represent the circle of radius 5 and centre (2, -3).

Parametrization of the ellipse

For an ellipse in parametric form, consider the following equations: \(x = h + a\cos(t)\) and \(y = k + b\sin(t)\), where \(a\) and \(b\) are the semi-major and semi-minor axis lengths respectively. By substituting \(h = -3\), \(k = 4\), \(a = 5\), and \(b = 2\) one obtains \(x = -3 + 5\cos(t)\) and \(y = 4 + 2\sin(t)\). This gives a parametrization of the ellipse centred at (-3, 4) with semi major axis of length 5 parallel to the x-axis and semi minor axis of 2 parallel to the y-axis