91Ó°ÊÓ

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=-3 x^{2}+4 x-1$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. \(x=e^{t}, \quad y=t, \quad\) any real number \(t\)

Sketch the graph of the equation and label the vertices. $$r=\frac{15}{3-2 \cos \theta}$$

Identify the conic whose equation is given and find its graph. If it is an ellipse, list its center, vertices, and foci. If it is a hyperbola, list its center, vertices, foci, and asymptotes. $$\frac{(x+3)^{2}}{1}-\frac{(y-2)^{2}}{4}=1$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$y=-3 x^{2}+4 x+5$$

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$$

Convert the rectangular coordinates to polar coordinates. $$(2,4)$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=2 e^{t}, \quad y=1-e^{t}, \quad t \geq 0$$

Sketch the graph of the equation and label the vertices. $$r=\frac{32}{3+5 \sin \theta}$$

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$$