Find the equation of the sphere passing through\( P ( - 6,5,4 ) \)and\( Q ( 2 , - 7,5 ) \)with its center at the midpoint of\( P Q \).

The standard equation of the sphere is

(Simplify your answer.)

Identify the intervals over which the function is increasing.

Solve\( 9 \cos ( 2 w ) = 9 \sin ^ { 2 } ( w ) + 3 \)for all solutions\( 0 \leq w < 2 \pi \)

\( w = \)

Give your answers accurate to at least 2 decimal places, as a list separated by commas

Graph the following function.

\( y = - 4 ( x - 1 ) ^ { 2 } - 5\)

Use the Law of Sines to solve for the remaining side(s) and angle(s) of all triangles if

\( \beta = 113 ^ { \circ } , b = 14.25 , c = 11.25 \). Round to two decimal places.

As in the text,\( ( \alpha , a ) , ( \beta , b ) \)and\( ( \gamma , c ) \)are angle-side opposite pairs. If no such triangle exists, enter DNE in each answer box.

Enter the values for an acute triangle solution:

\( \gamma = \square \)degrees

\( \alpha = \square \)degrees

\( a = \square \)degrees

Enter the values for an obtuse triangle solution:

\( \gamma ^ { \prime } = \square \)degrees

\( \alpha ^ { \prime } = \square \)degrees

\( a ^ { \prime } = \square \)degrees