0. Functions

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?

s = −tan πt

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 $f^{″}\left(x\right)$?

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

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph the function f (x) = sin 2x + cos 3x.

[Technology Exercise]

a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = x + (1/10) sin 30x

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 $t=6$.

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

General Sine Curves

For

f(x) = A sin ((2π/B)(x – C) +D

identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.

y = ½ sin (πx – x) + ½

Identify the amplitude and period of the following functions.

$f\(\left\)(\(\pi\]\right\))=2\(\sin\)2\(\theta\)$

Identify the amplitude and period of the following functions.

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

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

Graph the functions in Exercises 13–22. What is the period of each function?

sin (x/2)

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph the function f (x) = sin³ x.

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph four periods of the function f (x) = −tan 2x.