Graphs of Sine and Cosine Functions Trigonometry Section 4.5 Graphs of Sine and Cosine Functions

Exploring the Unit Circle and Basic Sine, Cosine, and Tangent Graphs http://www.walter-fendt.de/m14e/sincostan_e.htm Using the graphing calculator Using a table from 0 to 2π

Sin(x)

Cos(x)

Amplitude The amplitude only applies to the functions _______ and _______. For the function y = a sin(x) and y = a cos(x), the amplitude is _____. If a<0, we have a reflection about the _______. Let’s graph some functions: Example 2

Period of Sine and Cosine Let’s look at some examples and see if we can determine a pattern. Does the amplitude change the period? So, if we ignore the amplitude, consider sin(x) What is the period of sin(2x)? What is the period of sin(3x)? What is the period of sin(4x)? So each time, we divide the period of sin(x) by _____.

Period of Sine and Cosine (cont) Will it work for cos(x) as well? What is the period of cos(2x)? What is the period of cos(3x)? What is the period of cos(4x)? So each time, we divide the period of cos(x) by _____. In general, the period of y = a sin(bx) and y = a cos(bx) is ________. Example 3

Horizontal Shifting THIS IS VERY TRICKY!!! y = a sin(bx-c) and y = a cos(bx-c), the phase (or horizontal) shift is NOT “c units” The phase shift is c/b Let’s take a close look at an example: Sin(x), Sin(x-π), Sin(x- π/2) Sin(2x), Sin(2x-π), Sin(2x- π/2) In other words, the horizontal shift is proportional to the period of the function.

Horizontal Shifting (cont) The phase shift is c/b!!! When negative, we shift to the _______. When positive, we shift to the _______. An easy way to determine a cycle of your graph is to set bx-c=0 and bx-c=2π and solve for x (these are the endpoints). Example 5

Vertical Shifting y = a sin(bx-c) + d, and y = a cos(bx-c) + d d is the vertical shift Up if d is _______ than zero. Down if d is _______ than zero. Example 6

Order of Translations Translations must be done in the following order 1. Horizontal 2. Stretch/Shrink (amplitude in this case) 3. Reflections (you will see in a moment, only x-axis reflection is necessary) 4. Vertical

y = a sin(bx-c) + d Use the period to graph sin(bx). If b<0, use the fact that sine is an odd function to rewrite the equation. (we will have no y-axis reflections, only x-axis) 1. Horizontal—move sin(bx) to the right or left c/b units 2. Stretch—apply the amplitude 3. Reflections: if the coefficient of the sine is negative then reflect about the _______. 4. Vertical—apply the vertical shift.

y = 4 sin(-2x-π/2) + 4 Rewrite the equation as follows: sin [- (2x+ π/2) ]= - sin (2x+ π/2) So in turn y = -4 sin(2x+π/2) + 4 1. Graph sin(2x) shift to the _____ by _____ units. 2. Adjust the amplitude by _____ units. 3. Reflect about the _______. 4. Shift _____ by _____ units.

y = a cos(bx-c) + d Use the period to graph cos(bx). If b<0, use the fact that cosine is an even function to rewrite the equation. (we will have no y-axis reflections, only x-axis) 1. Horizontal—move cos(bx) to the right or left c/b units 2. Stretch—apply the amplitude 3. Reflections: if the coefficient of the cosine is negative then reflect about the _______. 4. Vertical—apply the vertical shift.

y = 2 cos(-2x+π) - 3 Rewrite the equation as follows: cos [- (2x - π)] = cos (2x- π) So in turn y = 2 cos(2x-π) - 3 1. Graph cos(2x) shift to the _____ by _____ units. 2. Adjust the amplitude by _____ units. 3. Reflection? 4. Shift _____ by _____ units.

Homework 1-52