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.

Follow the seven step strategy to graph the following rational function. \( f(x)=\frac{6 x}{x-2} \) To graph the function, first determine the symmetry of the graph of f . Choose the correct answer below. y-axis symmetry origin symmetry neither \( y \)-axis symmetry nor origin symmetry What is the \( y \)-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept is (Type an integer or a simplified fraction.)

Use transformations of the absolute value function, \( \mathrm{f}(\mathrm{x})=|\mathrm{x}| \), to graph the function \( g(x)=-|\mathrm{x}-3|-5 \). ... \( \square \) A. Horizontal shift \( \square \) B. Vertical stretch/shrink \( \square \) C. Horizontal stretch/shrink \( \square \) D. Vertical shift \( \square \) E. Reflection about the \( x \)-axis \( \square \) F. Reflection about the \( y \)-axis

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (0,-8) \) and \( (0,8) \); vertices at \( (0,3) \) and \( (0,-3) \)

The exponential model \( \mathrm{A}=474.2 e^{0.006 t} \) describes the population, A , of a country in millions, \( t \) years after 2003 . Use the model to determine the population of the country in 2003 . The population of the country in 2003 was \( \square \) million.

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (-7,0) \) and \( (7,0) \); vertices at \( (3,0) \) and \( (-3,0) \)