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
