5.6 Transformations of the Sine and Cosine Graphs

Wed Nov 12

Do NowUse the sine and cosine values of 0, pi/2, pi, 3pi/2 and 2pi to sketch the graph of f(x) = sin x and g(x) = cos x

Review of graphs

• Y = sin x

• Period of 2pi• Amplitude of 1• Goes through (0, 0)

Review of Graphs

• Y = cos x

• Period of 2pi• Amplitude of 1• Goes through (0, 1)

Transformations

• We are interested in the graphs of functions in the form

The Constant A

• Recall that coefficients of functions result in a vertical shift / shrink

• The constant A affects the amplitude of sine and cosine. The amplitude = A

• If A is negative, the graph is upside down

Ex

• Graph the following• 1)

• 2)

• 3)

The Constant B

• Recall that coefficients of X result in a horizontal stretch / shrink

• The constant B affects the period

• The period of these graphs is

Ex

• Sketch a graph of the following• 1)

• 2)

The Constant C

• The constant C, like in previous functions, results in a horizontal shift C units right / left

• This is also known as a phase shift

• Ex: sin (x – 4) is a shift to the right 4 units

• Cos (x + pi) is a shift to the left pi units

The Constant D

• The constant D results in a vertical shift D units up / down

• Ex: y = sin x + 1 shifts up 1 unit

• Ex: y = cos x – 4 shifts down 4 units

• Notice no parenthesis

Combined Transformations

• When working with multiple transformations, we want to rewrite the functions

• This helps you see the phase shift

How to graph

• 1) determine the period, amplitude, and shifts• 2) graph and shift the period, and split it into 4

regions• 3) plot a point in between each region,

including the amplitude and shifts in your calculations

• 4) connect the points in the correct sine or cosine wave

Ex

• Sketch a graph of

Ex

• Sketch a graph of

Closure

• Graph

• HW: p.523 #1-25 odds

5.6 Transformations of Sine and Cosine cont’dThurs Nov 13

• Do Now• Graph the following• 1) y = sin(1/2 x)]

• 2) y = - 2cos( 2x )

HW Review: p.523 #1-25 odds

Review of Sine and Cosine

• Recall the transformations

• A affects the amplitude• B affects the period• C/B affects the phase shift• D affects the vertical shift

Ex

• Graph

Matching

• On p.522

Closure

• What kind of transformations can affect the sine and cosine graphs? How do we determine what transformations occur?

• HW: p.523 #27-43 odds

5.6 Addition and Multiplication of OrdinatesFri Nov 14

• Do Now• Graph

HW Review: p.523 #27-43

Graphs of Sums: Additions of Ordinates

• When graphing a sum of 2 trigonometric functions, we use a strategy called addition of ordinates

Properties of sums

• The period of a sum will be the least common multiple of every period

• Graph each important point by adding the y-values of each trig function

ex

• Graph y = 2sin x + sin 2x

Damped Oscillation: Multiplication of Ordinates

• We’ll just graph these

Finding zeros (review)

• To find zeros of a function,• 1) Graph function• 2) 2nd -> calc -> zeros• 3) Left bound – pick a point slightly left of the

zero you want• 4) Right bound – pick a point slight right of the

zero you want• 5) Guess – hit enter

ex

• Solvethe zeros ofon the interval [-12,12]

closure

• What is addition of ordinates? How do we graph these functions?

• HW: p.524 #45-73 odds

5.6 Other Trig TransformationsMon Nov 17

• Do Now• Graph y = csc x and y = tan x on your calculator

HW Review: p.524 #45-73 odds

Review: f(x) = tan x and cot x

• The period of tangent and cotangent is pi

• Each period is separated by vertical asymptotes

• Amplitude does not affect the graph drastically

Basic graphs

• Y = tan x y = cot x

Review: f(x) = csc x and sec x

• The period of csc x and sec x is 2pi

• Vertical asymptotes occur every half period

• The amplitude represents how close to the center each curve gets

Basic Graphs

• Y = csc x y = sec x

Transformations

• Transformations affect these 4 graphs the same way

Ex

• Sketch the graph of

Ex

• Sketch the graph of

Closure

• Graph

• HW: p. 525 #89-97 odds• CH 5 Test soon

5.6 ReviewTues Nov 18

• Do Now• Sketch the graph of

HW Review p.525 #89-97

Transformations Review

• Basic graphs

• Transformations

• Period, Amplitude, Phase shift, Vertical shift

Closure

• What are some identifying properties of trigonometric functions and their graphs?

• HW: p.529 #1-83 odds skip 53 55 due Thursday

• SGO Assessment Wed Nov 19• Ch 5 Test Fri Nov 21