Exam 4: Graphs of the Circular Functions

Solve the problem. -The chart represents the amount of fuel consumed by a machine used in manufacturing. The machine is turned on at the beginning of the day, takes a certain amount of time to reach its full Power (the point at which it uses the most fuel per hour), runs for a certain number of hours, and is Shut off at the end of the work day. The fuel usage per hour of the machine is represented by a Periodic function. What is the period in hours of this function?

The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph. A) $y = 4 \cos \left( \frac { 1 } { 2 } x \right)$ B) $y = - 4 \cos ( 2 x )$ C) $y = - 4 \cos \left( \frac { 1 } { 2 } x \right)$ D) $y = 4 \sin ( 2 x )$ -The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph.

The function graphed is of the form y = a tan bx or y = a cot bx, where b > 0. Determine the equation of the graph. -

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 5 \pi t$ inches after $t$ seconds. When does the weight first reach its maximum height?

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 3 \cos 7 t$ inches after $t$ seconds. What are the frequency and period of the system?

The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph. -

Solve the problem. -The total sales in dollars of some small businesses fluctuates according to the equation $\mathrm { S } = \mathrm { A } + \mathrm { B } \sin \pi \mathrm { x } / 6$ , where $\mathrm { x }$ is the time in months, with $\mathrm { x } = 1$ corresponding to January, $\mathrm { A } = 5900$ , and $B = 2700$ . Determine the month with the greatest total sales and give the sales in that month.

Find the specified quantity. -Find the amplitude of $y = - 2 \cos \left( 3 x + \frac { \pi } { 4 } \right)$ .

Solve the problem. -A pendulum of length $\mathrm { L }$ , when displaced horizontally and released, oscillates with harmonic motion according to the equation $\mathrm { y } = \mathrm { A } \sin ( ( \sqrt { \mathrm { g } / \mathrm { L } } ) \mathrm { t } + \pi / 2 )$ , where $\mathrm { y }$ is the distance in meters from the rest position $t$ seconds after release, and $g = 9.8 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . Identify the period, amplitude, and phase shift when $\mathrm { A } = 0.39 \mathrm {~m}$ and $\mathrm { L } = 0.49 \mathrm {~m}$ . Round all answers to the nearest hundredth.

Graph the function. - $y=-\frac{3}{5} \cot \left(\frac{1}{2} x-\frac{\pi}{6}\right)$

Match the function with its graph. -1) $y = \sin \left( x - \frac { \pi } { 3 } \right)$ 2) $y = \cos \left( x + \frac { \pi } { 3 } \right)$ 3) $y = \sin \left( x + \frac { \pi } { 3 } \right)$ 4) $y = \cos \left( x - \frac { \pi } { 3 } \right)$ a) b) c) d)

Graph the function. - $y=2-3 \sec \left(x-\frac{\pi}{5}\right)$

The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph. -

Solve the problem. -The minimum length $\mathrm { L }$ of a highway sag curve can be computed by $\mathrm { L } = \frac { \left( \theta _ { 2 } - \theta _ { 1 } \right) \mathrm { S } ^ { 2 } } { 200 ( \mathrm {~h} + \mathrm { S } \tan \alpha ) }$ where $\theta _ { 1 }$ is the downhill grade in degrees $\left( \theta _ { 1 } < 0 ^ { \circ } \right) , \theta _ { 2 }$ is the uphill grade in degrees $\left( \theta _ { 2 } > 0 ^ { \circ } \right)$ , $\mathrm { S }$ is the safe stopping distance for a given speed limit, $h$ is the height of the headlights, and $\alpha$ is the alignment of the headlights in degrees. Compute $L$ for a 55-mph speed limit, where $h = 1.9 \mathrm { ft }$ , $\alpha = 0.7 ^ { \circ } , \theta _ { 1 } = - 5 ^ { \circ } , \theta _ { 2 } = 4 ^ { \circ }$ , and $S = 336 \mathrm { ft }$ . Round your answer to the nearest foot.

Match the function with its graph. 29 -1) $y = - \tan \left( x - \frac { \pi } { 2 } \right)$ 2) $y = \tan \left( x + \frac { \pi } { 2 } \right)$ 3) $y = - \cot \left( x - \frac { \pi } { 2 } \right)$ 4) $y = \cot \left( x + \frac { \pi } { 2 } \right)$ a) b) c) d)

Graph the function over a one-period interval. - $y=4+2 \sin (x-\pi)$

Solve the problem. -The voltage $E$ in an electrical circuit is given by $E = 1.7 \cos 110 \pi t$ , where $t$ is time measured in seconds. Find the frequency of the function (that is, find the number of cycles or periods completed in one second).

Graph the function. - $y=-\frac{1}{2} \csc \left(\frac{1}{2} x-\frac{\pi}{4}\right)$

Graph the function over a one-period interval. - $y=-\frac{1}{3} \cos 4(x-\pi)$

Find the phase shift of the function. - $y = - 2 \sin \left( \frac { 1 } { 4 } x - \frac { \pi } { 4 } \right)$