Each of the following linear system has complex eigenvalues. For eachsystem, (a) find the eigenvalues; (b) determine if the origin is a spiral sink, a spiral source, or a center; (c) determine the natural period and natural frequency of the oscillations; (d) determine the direction of the oscillations in the phase plane; and (e)using HPGSystemSolver, sketch thexy-phase portrait and thex(t)- andy(t)-graphsfor the solutions with the indicated initial conditions:

If you apply the below transformations to the square root parent function, f(x) = vx, what is the equation of the new function? • shift 12 units right. • shift seven units down.

Subscriptions to a popular fashion magazine have gone down by a consistent percentage each year and can be modeled by the function y = 42,000(0.96)t. what does the value 42,000 represent in the function?

Statement reason 1. δabc is similar to δced. given 2. 3. definition of slope 4. slope of slope of definition of slope 5. slope of × slope of multiplying the slopes 6. slope of × slope of substitution property of equality 7. slope of × slope of simplifying the right side the table contains the proof of the relationship between the slopes of two perpendicular lines. what is the reason for statement 2? a. parallel line segments that meet a common perpendicular line are proportional in length. b. the lengths of vertical and horizontal sides in congruent triangles are in a common ratio. c. trigonometric identities determine the lengths of the legs in a right triangle. d. corresponding side lengths in similar triangles are proportional in length.