Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function. \[ f(x)=\frac{4}{x-2} \]

Sketch a graph of \( f \) \( f(x)=2 x-x^{2} \)

1. \( 25 x^{2}+y^{2}-200 x+6 y+384=0 \), centro 2. \( (x+1)^{2}+(y+8)^{2}=100 \), intersecciones con el eje \( x \) 3. \( y^{2}-(x-2)^{2}=1 \), intersecciones con el eje \( y \) 4. \( y^{2}-y+3 x=3 \), pendiente de la recta tangente en \( (1,1) \) 5. \( x=t^{3}, y=4 t^{3} \), nombre de la gráfica rectangular 6. \( x=t^{2}-1, y=t^{3}+t+1 \), intersecciones con el eje \( y \) 7. \( r=-2 \cos \theta \), nombre de la gráfica polar 8. \( r=2+\operatorname{sen} \theta \), nombre de la gráfica polar 19. \( r=\operatorname{sen} 3 \theta, \operatorname{tangentes~a~la~gráfica~en~el~origen~} \) 20. \( r=\frac{1}{2+5 \operatorname{sen} \theta} \), excentricidad 21. \( r=\frac{10}{1-\operatorname{sen} \theta} \), foco yértice 22. \( r=\frac{12}{2+\cos \theta} \), centro

Sketch the segment \( r=\sec \theta \) for \( 0 \leq \theta \leq \frac{\pi}{9} \). Then compute its length in two ways: as an integral in pola coordinates and using trigonometry.

Empic ta aproximacion para C) \( \sqrt{102}+\sqrt[4]{80} \)

Using visual observation, determine whether the graph is symmetric with respect to the (a) \( x \)-axis, (b) \( y \)-axis, or \( (\mathrm{c}) \) the origin. (a) Is the graph symmetric with respect to the \( x \)-axis? Choose the correct answer below. A. No, because for \( \left(\frac{\pi}{4}, 4\right) \), the point \( \left(-\frac{\pi}{4}, 4\right) \) is not on the graph. B. No, because for \( \left(\frac{\pi}{4}, 4\right) \), the point \( \left(\frac{\pi}{4},-4\right) \) is not on the graph. C. Yes, because if \( (a, b) \) is on the graph, then so is \( (-a, b) \). D. Yes, because if \( (a, b) \) is on the graph, then so is \( (a,-b) \). (b) Is the graph symmetric with respect to the \( y \)-axis? Choose the correct answer below. A. Yes, because if \( (a, b) \) is on the graph, then so is \( (-a, b) \). Because for \( \left(\frac{\pi}{4}, 4\right) \), the point \( \left(-\frac{\pi}{4}, 4\right) \) is not on the graph. C. No, because for \( \left(\frac{\pi}{4}, 4\right) \), the point \( \left(\frac{\pi}{4},-4\right) \) is not on the graph. (c) Is the graph symmetric with respect to the origin? Choose the correct answer below. A. Yes, because if \( (a, b) \) is on the graph, then so is \( (-a, b) \). B. Yes, because if \( (a, b) \) is on the graph, then so is \( (-a,-b) \). B C. Yes, because if \( (a, b) \) is on the graph, then so is \( (a,-b) \).