0. Functions

Given the figure showing four curves labeled A, B, C, and D, which represent the functions $f\left(x\right)=\sin x$, $f^{\prime }\left(x\right)$, $f^{″}\left(x\right)$, and $f^{‴}\left(x\right)$, respectively, which curve corresponds to $f^{″}\left(x\right)$?

Below is a graph of the function $y=\tan\left(bx\right)$. Determine the value of b.

Sketch the function $y=\cos\left(x\right)-1$ on the graph below.

Determine the value of $y=\sin\left(-\frac{\pi}{2}\right)+50$ without using a calculator or the unit circle.

Below is a graph of the function $y=\sec\left(bx-\pi\right)$. Determine the value of b.

Below is a graph of the function $y=\csc\left(bx\right)$. Determine the value of b.

Below is a graph of the function $y=\cot\left(bx+\frac{\pi}{2}\right)$. Determine the value of b.

Identify the amplitude and period of the following functions.

$f\left(\pi\right)=2\sin2\theta$

Identify the amplitude and period of the following functions.

$g\left(\theta\right)=3\cos\left(\frac{\theta}{3}\right)$

Identify the amplitude and period of the following functions.

$p\left(t\right)=2.5\sin\left(\frac12\left(t-3\right)\right)$

Identify the amplitude and period of the following functions.

$q\left(x\right)=3.6\cos\left(\frac{\pi x}{24}\right)$

Identify the amplitude and period of the following functions.

p(t) = 2.5 sin ((1/2)(t-3))

Beginning with the graphs of $y=\sin x$ or $y=\cos x$, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

$q\left(x\right)=3.6\cos\left(\frac{\pi x}{24}\right)+2$

Design a sine function with the given properties.

It has a period of $12$ with a minimum value of $-4$ at $t=0$ and a maximum value of $4$ at $t=6$.

Find a trigonometric function $f$ represented by the graph in the figure. <IMAGE>