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5.6 Transformations of the Sine and Cosine Graphs Wed Nov 12 Do Now Use 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

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Review of graphs Y = sin x Period of 2pi Amplitude of 1 Goes through (0, 0)

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Review of Graphs Y = cos x Period of 2pi Amplitude of 1 Goes through (0, 1)

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

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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

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Ex Graph the following 1) 2) 3)

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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

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Ex Sketch a graph of the following 1) 2)

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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

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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

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Combined Transformations When working with multiple transformations, we want to rewrite the functions This helps you see the phase shift

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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

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Ex Sketch a graph of

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Ex Sketch a graph of

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Closure Graph HW: p.523 #1-25 odds

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5.6 Transformations of Sine and Cosine cont’d Thurs Nov 13 Do Now Graph the following 1) y = sin(1/2 x)] 2) y = - 2cos( 2x )

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HW Review: p.523 #1-25 odds

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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

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Ex Graph

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Matching On p.522

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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

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5.6 Addition and Multiplication of Ordinates Fri Nov 14 Do Now Graph

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HW Review: p.523 #27-43

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Graphs of Sums: Additions of Ordinates When graphing a sum of 2 trigonometric functions, we use a strategy called addition of ordinates

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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

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ex Graph y = 2sin x + sin 2x

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Damped Oscillation: Multiplication of Ordinates We’ll just graph these

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Finding zeros (review) To find zeros of a function, 1) Graph function 2) 2 nd -> 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

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

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closure What is addition of ordinates? How do we graph these functions? HW: p.524 #45-73 odds

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5.6 Other Trig Transformations Mon Nov 17 Do Now Graph y = csc x and y = tan x on your calculator

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HW Review: p.524 #45-73 odds

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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

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Basic graphs Y = tan xy = cot x

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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

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Basic Graphs Y = csc x y = sec x

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Transformations Transformations affect these 4 graphs the same way

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Ex Sketch the graph of

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Ex Sketch the graph of

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

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5.6 Review Tues Nov 18 Do Now Sketch the graph of

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HW Review p.525 #89-97

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Transformations Review Basic graphs Transformations Period, Amplitude, Phase shift, Vertical shift

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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

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1 Properties of Sine and Cosine Functions MATH 130 Lecture on The Graphs of Trigonometric Functions.

1 Properties of Sine and Cosine Functions MATH 130 Lecture on The Graphs of Trigonometric Functions.

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