5.6 transformations of the sine and cosine graphs wed nov 12 do now use the sine and cosine values...

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