8.1 Graphs of the Sine and Cosine Functions

Determine the period of the function $f \left(x\right) = \sin \left(\frac{\pi}{6} x\right) .$

Determine the period of the function $g \left(x\right) = \cos \left(\frac{x}{3}\right) .$

What is the amplitude of the sinusoidal function $f \left(x\right) = −4 \sin \left(x\right) ?$ Is the function stretched or compressed vertically?

What is the amplitude of the sinusoidal function $f \left(x\right) = \frac{1}{2} \sin \left(x\right) ?$ Is the function stretched or compressed vertically?

Determine the direction and magnitude of the phase shift for $f \left(x\right) = \sin \left(x + \frac{\pi}{6}\right) - 2.$

Determine the direction and magnitude of the phase shift for $f \left(x\right) = 3 \cos \left(x - \frac{\pi}{2}\right) .$

Determine the direction and magnitude of the vertical shift for $f \left(x\right) = \cos \left(x\right) - 3.$

Determine the direction and magnitude of the vertical shift for $f \left(x\right) = 3 \sin \left(x\right) + 2.$

Determine the midline, amplitude, period, and phase shift of the function $y = 3 \sin \left(2 x\right) + 1.$

Determine the midline, amplitude, period, and phase shift of the function $y = \frac{1}{2} \cos \left(\frac{x}{3} - \frac{\pi}{3}\right) .$

The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.

Why are the sine and cosine functions called periodic functions?

For the equation $A \cos \left(B x + C\right) + D ,$ what constants affect the range of the function and how do they affect the range?

How can the unit circle be used to construct the graph of $f \left(t\right) = \sin t ?$

Let $f \left(x\right) = \sin x .$

On $\left[\right. 0 , 2 \pi ),$ solve $f \left(x\right) = 0.$

Let $f \left(x\right) = \sin x .$

Evaluate $f \left(\frac{\pi}{2}\right) .$

Let $f \left(x\right) = \sin x .$

On $\left[\right. 0 , 2 \pi ),$ the maximum value(s) of the function occur(s) at what x-value(s)?

Let $f \left(x\right) = \sin x .$

Show that $f \left(− x\right) = - f \left(x\right) .$ This means that $f \left(x\right) = \sin x$ is an odd function and possesses symmetry with respect to ________________.

Let $f \left(x\right) = \cos x .$

On $\left[\right. 0 , 2 \pi ),$ solve $f \left(x\right) = \frac{1}{2} .$

Let $f \left(x\right) = \cos x .$

On $\left[\right. 0 , 2 \pi ),$ find the x-values at which the function has a maximum or minimum value.