91Ó°ÊÓ

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(x-1)^{2}-(y-2)^{2}=3$$

If there is a \(60^{\circ}\) angle from the positive \(x\) -axis to the positive \(x^{\prime}\) -axis, explain how to obtain the rotation formulas for \(x\) and \(y\)

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y+1)^{2}=-8 x$$

In Exercises \(41-43\), eliminate the parameter. Write the resulting equation in standard form. A circle: \(x=h+r \cos t, y=k+r \sin t\)

Use the polar mode of a graphing utility with angle measure in radians . Unless otherwise indicated, use \(\theta \min =0, \theta \max =2 \pi,\) and \(\theta\) step \(=\frac{\pi}{48} .\) If you are not satisfied with the quality of the graph, experiment with smaller values for \(\theta\) step. Identify the conic that each polar equation represents. Then use a graphing utility to graph the equation. $$r=\frac{12}{4+5 \sin \theta}$$

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y-1)^{2}=-8 x$$

Graph each ellipse and give the location of its foci. $$\frac{(x-3)^{2}}{9}+\frac{(y+1)^{2}}{16}=1$$

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(y-2)^{2}-(x+3)^{2}=5$$

Use the polar mode of a graphing utility with angle measure in radians . Unless otherwise indicated, use \(\theta \min =0, \theta \max =2 \pi,\) and \(\theta\) step \(=\frac{\pi}{48} .\) If you are not satisfied with the quality of the graph, experiment with smaller values for \(\theta\) step. Identify the conic that each polar equation represents. Then use a graphing utility to graph the equation. $$r=\frac{18}{6-6 \cos \theta}$$

In Exercises \(41-43\), eliminate the parameter. Write the resulting equation in standard form. An ellipse: \(x=h+a \cos t, y=k+b \sin t\)