**
SINE AND COSINE FUNCTIONS**

Example 1:

Given

, do the following:g(x) = 2 sin(2x)a. State the amplitude,

The amplitude is

. This means that the|2|= 2y-values of the peaks areand the2y-values of the valleys are. No larger and smaller2y-values exist.b. State the

EXACTperiod.Since

, the period isb = 2. This means therepresentative pictureisunits in length.c. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic sine function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period.(3) Divide this period into four equal intervals. Each will be of length

./4(4) Create the

representative pictureby using the beginning and ending point of each interval.Note that the

x-values are integer multiples ofand the/4y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(0, 0), (/4, 2), (2/4 = /2, 0), (3/4, 2),.(, 0)(5) Connect the points to form the

representative pictureof the sine function.(6) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

Please note that the peaks and valleys in the graphs of the sine functions are U-shaped!

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!

Example 2:

Given , do the following:

a. State the amplitude.

The amplitude is . This means that the

y-values of the peaks areand they-values of the valleys are . No larger and smallery-values exist.

Please note thatais negative! This means that therepresentative pictureis reflected in thex-axis. Instead of drawing a peak first, we will draw a valley instead!b. State the

EXACTperiod.Since

b =, the period is. This means therepresentative pictureisunits in length.2c. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic sine function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period.2(3) Divide this period into four equal intervals. Each will be of length

.2/4 = 1/2(4) Create the

representative pictureby using the beginning and ending point of each interval.Note that the

x-values are integer multiples ofand the1/2y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(0, 0), (1/2, 1/2), (2/2 = 1, 0), (3/2, 1/2),.(2, 0)

Please note that "a" is negative! This means that therepresentative pictureis reflected in thex-axis. Instead of drawing a peak first, we will draw a valley instead!(5) Connect the points to form the

representative pictureof the sine function.(6) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

Please note that the peaks and valleys in the graphs of the sine functions are U-shaped!

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!

Example 3:

Given , do the following:

a. State the amplitude.

The amplitude is

. This means that the|2|= 2y-values of the peaks areand the2y-values of the valleys are. No larger and smaller2y-values exist.b. State the

EXACTperiod.Since

, the period is . This means theb =representative pictureis4units in length.c. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic cosine function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period4.(3) Divide this period into four equal intervals. Each will be of length

4/4 =.(4) Create the

representative pictureby using the beginning and ending point of each interval.Note that the

x-values are integer multiples ofand they-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(0, 2), (, 0), (2, 2), (3, 0),.(4, 2)(5) Connect the points to form the

representative pictureof the cosine function.(6) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

Please note that the peaks and valleys in the graphs of the cosine functions are U-shaped!

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!

Example 4:

Given , do the following:

a. State the amplitude.

The amplitude is . This means that the

y-values of the peaks areand they-values of the valleys are . No larger and smallery-values exist.

Please note thatais negative! This means that therepresentative waveis reflected in thex-axis. Instead of drawing a peak first, we will draw a valley instead!b. State the

EXACTperiod.Since

, the period isb = /2. This means therepresentative pictureisunits in length.4c. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic cosine function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period.4(3) Divide this period into four equal intervals. Each will be of length

.4/4 = 1(4) Create the

representative pictureby using the beginning and ending point of each interval.Note that the

x-values are integer multiples ofand the1y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(0, 1/2), (1, 0), (2, 1/2), (3, 0),.(4,1/2)

Please note that "a" is negative! This means that therepresentative pictureis reflected in thex-axis. Instead of drawing a peak first, we will draw a valley instead!(5) Connect the points to form the

representative pictureof the cosine function.epresentative pictureon the interval .

Please note that the peaks and valleys in the graphs of the cosine functions are U-shaped!

x-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!

Example 5:

Use Desmos at

https://www.desmos.com/calculatorto draw the graph ofand that of its transformation into the same coordinate system.y = cos(x)Required Graph Characteristics:

- Show the representative picture and a copy of one to the right of it.
- Place numbers on both axes.
- The
x-axis numbers must contain. The beginning/ending point of each interval in each copy of therepresentative picturemust show numbers.- The
y-axis numbers must show the appropriate values for thex-axis numbers used in (3).- Do not make the
y-axis too large. We need to see the peaks and valleys clearly!You can find instructions for Desmos at

http://profstewartmath.com/Math127/A_CONTENTS/desmos.htmSince

, the period is . This means theb =representative pictureisunits in length. This knowledge will allow us to find the required4x-values.Since the amplitude is

|2| = 2, let's restrict they-axis to betweenand3in the Desmos "Graph Settings". One integer more than the amplitude is enough!3NOTE:

Sometimes, we have to try different settings in the Desmos "Graph Settings" window before the required

x-axis numbers show up. Additionally, when we export the image, we need to determine which size best shows all required characteristics.In the graph above, the Desmos "Graph Settings" for the

x-axis are betweenand44and the Step is./2The "Large Square" size was used for Desmos image export. Incidentally, the other settings refused to show all of the

x-axis numbers.