Introduction

Suppose we want to plot the graph of y=\sin x, where x is measured in radians.

First, let's create a table of values and include some of the values that we know:

x 0 \dfrac\pi 4 \dfrac\pi 2 \dfrac{3\pi} 4 \pi \dfrac{5\pi} 4 \dfrac{3\pi} 2 \dfrac{7\pi} 4 2\pi
\sin x 0 \dfrac{\sqrt 2}{2} 1 \dfrac{\sqrt 2}{2} 0 -\dfrac{\sqrt 2}{2} -1 -\dfrac{\sqrt 2}{2} 0

Plotting these points, we get the following curve.

From the unit circle, we know that the function y=\sin x repeats itself every 2\pi radians. So we can extend our plot using this periodicity property.

The key characteristics of the graph are:

  • The domain of the function is x\in(-\infty,\infty).

  • The range of the function is y\in[-1,1].

  • The period of the function is 2\pi.

  • Its zeros occur at x=0, \pm \pi, \pm 2\pi,\ldots.

  • Its maximum values occur at \ldots,-\dfrac{3\pi}{2}, \dfrac{\pi}{2},\ldots

  • Its minimum values occur at \ldots,-\dfrac{\pi}{2}, \dfrac{3\pi}{2},\ldots

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Example: Plotting the Graph of the Sine Function

Which of the following plots is the graph of y = \sin x?

EXPLANATION

Let's recall some of the basic properties of y=\sin{x}\mathbin{:}

  • The minimum value is -1, and the maximum value is 1.

  • The period of the function is 2\pi.

  • Its zeros occur at x=0, \pm \pi, \pm 2\pi,\ldots.

  • Its maximum values occur at \ldots,-\dfrac{3\pi}{2}, \dfrac{\pi}{2},\ldots

  • Its minimum values occur at \ldots,-\dfrac{\pi}{2}, \dfrac{3\pi}{2},\ldots

Of the given plots, the only one that displays all of these properties is I.

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Practice: Plotting the Graph of the Sine Function

Which of the following diagrams shows the graph of $y = \sin x?$

a
b
c
d
e
Practice: Plotting the Graph of the Sine Function

Which of the following diagrams shows the graph of $y = \sin x?$

a
b
c
d
e
Example: Identifying Characteristics of the Sine Function

Which of the following is not a zero of y=\sin{x}?

0, \quad -\dfrac{\pi}{2}, \quad 2\pi, \quad -\pi

EXPLANATION

Let's recall the graph of y=\sin{x}.

Note that:

  • The zeros occur at x=0, \pm \pi, \pm 2\pi,\ldots.

Of the given values, the only one that is not a zero of the function is -\dfrac{\pi}{2}.

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Practice: Identifying Characteristics of the Sine Function

Which of the following is a value of $x$ that minimizes the graph of $y=\sin{x}?$

a
 $-\pi$
b
 $\dfrac{\pi}{2}$
c
 $0$
d
 $-\dfrac{\pi}{2}$
e
 $\pi$
Practice: Identifying Characteristics of the Sine Function

Which of the following is a value of $x$ that maximizes the graph of $y=\sin{x}?$

a
 $\dfrac{3\pi}{2}$
b
 $\pi$
c
 $\dfrac{\pi}{2}$
d
 $-\dfrac{\pi}{2}$
e
 $-\pi$
Graphing the Cosine Function

Suppose that we wish to draw the graph of the function y=\cos x, with x in radians.

First, let's write up a table of values, and include some of the specific values of \cos x that we know:

x 0 \dfrac\pi 4 \dfrac\pi 2 \dfrac{3\pi} 4 \pi \dfrac{5\pi} 4 \dfrac{3\pi} 2 \dfrac{7\pi} 4 2\pi
\cos x 1 \dfrac{\sqrt 2}{2} 0 -\dfrac{\sqrt 2}{2} -1 -\dfrac{\sqrt 2}{2} 0 \dfrac{\sqrt 2}{2} 1

Now, plotting these points, we get the following graph:

From the unit circle, we know that y=\cos x repeats every 2\pi radians. Repeating the graph using this periodicity property, we have:

Some key characteristics of the graph are:

  • The domain of the function is x \in (-\infty, \infty ).

  • The range of the function is y\in [-1,1].

  • The period of the function is 2\pi.

  • Its zeros occur at x= \dots ,\pm\dfrac{\pi}{2},\pm\dfrac{3\pi}{2},\dots

  • Its maximum values occur at x=0,\pm2\pi, \pm4\pi,\dots

  • Its minimum values occur at x=\pm\pi,\pm 3\pi,\dots

FLAG
Example: Plotting the Graph of the Cosine Function

Which of the following plots is the graph of y = \cos x?

EXPLANATION

Let's recall some of the basic properties of y=\cos{x}\mathbin{:}

  • The domain of the function is x \in (-\infty, \infty ).

  • The range of the function is y\in [-1,1].

  • The period of the function is 2\pi.

  • Its zeroes occur at x= \dots ,\pm\dfrac{\pi}{2},\pm\dfrac{3\pi}{2},\dots

  • Its maximum values occur at x=0,\pm2\pi, \pm4\pi,\dots

  • Its minimum values occur at x=\pm\pi,\pm 3\pi,\dots

Of the given functions, the only one that has all of these properties is II.

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Practice: Plotting the Graph of the Cosine Function

Which of the following plots is the graph of $y = \cos x?$

a
b
c
d
e
Practice: Plotting the Graph of the Cosine Function

Which of the following plots is the graph of $y = \cos x?$

a
b
c
d
e
Example: Identifying Characteristics of the Cosine Function

Which of the following is a value of x that minimizes the graph of y=\cos{x}? -2\pi , \quad 0, \quad \dfrac{\pi}2, \quad -\pi

EXPLANATION

Let's recall the graph of y=\cos{x}\mathbin{.}

From the graph, we see that the cosine function reaches its minimum value when x=-\pi and x=\pi.

From the given options, the correct answer is x=-\pi.

FLAG
Practice: Identifying Characteristics of the Cosine Function

Which of the following is not a zero of $y=\cos{x}?$

a
 $-\dfrac{\pi}{2}$
b
 $\dfrac{\pi}{2}$
c
 $\dfrac{3\pi}{4}$
d
 $\dfrac{3\pi}{2}$
e
 $-\dfrac{3\pi}{2}$
Practice: Identifying Characteristics of the Cosine Function

Which of the following is a value of $x$ that maximizes the graph of $y=\cos{x}?$

a
 $-\dfrac{3\pi}{2}$
b
 $\dfrac{8\pi}{5}$
c
 $\dfrac{\pi}{2}$
d
 $2\pi$
e
 $-\pi$
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