4.5 Graphs of Sine and Cosine Functions Page in PreCalc book

Basic Sine and Cosine Curves In this section we will study techniques for _______________ the ______________ of _______________ and ______________ functions. The graph of the _____________ function is actually a ______________ _______________. The picture to the right shows _________ ____________ of the sine curve. This is called one _____________ of the sine curve. This wave pattern continues _____________ in ___________ directions. sketching graph curve cosine sine one cycle period forever both

Basic Sine and Cosine Curves The graph of _______________ is shown to the left. Remembering back from Section 4.2, we know the domain (all the ______ values) of sine and cosine is ____________ _____________ ________________. And the range (all the ______ values) for sine and cosine is the interval ___________, each having a ___________________ of _______________. cosine x allreal numbers y [ -1, 1 ] period 2π

Amplitude and Period Right now we will study the graphic effects of the constants of Right now we will study the graphic effects of the constants of The equations we will look at are of the form: a sin (bx ± c) + da cos (bx ± c) + d _____________________ is the coefficient of the sine or cosine function. It represents ____________ the total _______________ of the function. To find it, you take the _________________ of ______________. _____________________ is the coefficient of the sine or cosine function. It represents ____________ the total _______________ of the function. To find it, you take the _________________ of ______________. BADCBADC A or Amplitude half height Aabsolute value

Ex. 1 Finding Amplitude a. y = -3 cos 4x c. y = 4 sin (2x) – 1d. y = -3 cos (x) + 1 Page 568 Problems 1-6 find amplitude

Amplitude and Period Period describes the _____________________ stretch of the curve. To find the period of the function _________________ _________________ by _______________. _____________ is the ________________ of _________. horizontal divide 2π b Which coefficient x

Ex. 2 Finding Period Page 568 Problems now find period a. y = -3 cos 4x c. y = 4 sin (2x) – 1d. y = -3 cos (x) + 1

Trigonometric Translations Horizontal Translations To the RIGHT “c” unitsTo the LEFT “c” units

Trigonometric Translations Vertical Translations UP “d” units, d > 0DOWN “d” units, d < 0

Trigonometric Translations Flip (or Invert) the Graph This will invert (or flip) the graph over the x-axis.

Ex. 3 Translation Page 568 Problems 17 – 30 a. y = -3 cos (4x – 2) c. y = 4 sin (2x) – 1d. y = -3 cos (x) + 1

Graphing Sine and Cosine Both sine and cosine have five key points that are used when graphing. You must memorize them SineCosine x0π/2π3π/22π2π y0100 x0π/2π3π/22π2π y1001

Graphing Sine and Cosine It is important to remember that amplitude changes the _________________, which are the ________ values. Also, period changes the _________ values. Normal Sine and Cosine Curves have: Amplitude = 1 Period = 2π height y x

y = sin x x0π/2π3π/22π2π y0100

y = cos x x0π/2π3π/22π2π y1001

Ex. 4 Graphing Equations 1.y = sin x x0π/2π3π/22π2π y0100 x0π/2π3π/22π2π y x0π/2π3π/22π2π y3 -5

Ex. 4 Graphing Equations 2.y = 2 – 3 cos x Page 568 problems 31 – 42 x0π/2π3π/22π2π y1001 x0π/2π3π/22π2π y-3030 x0π/2π3π/22π2π y252

Phase Shifting When there is a shift to the left or right, you must adjust your ___________________ and ___________________ values (_________values). Starting ValueEnding Value starting ending x

Ex. 5 Graphing Equations

Page problems

It’s a review of what we have been doing. Please do all problems. Thank you. Please pick up papers in the red bin and on the desk.

**To graph changes, you must do it in the order of BADC y = 2 sin (x – π/2) - 1 B: To graph divide period into 4 equal parts. A: To graph multiply all critical points by A D: To graph move each critical point up or down D units C: To graph move critical points left or right C units

Writing Sine and Cosine Equations A: Amplitude B: used to find period Things to Remember C: Phase Shift D: Vertical Shift A is negative if the graph is ________ To find B set up equation: __________ + C if the graph moves to the _______ - C if the graph moves to the _________ +D if the graph moves ______ -D if the graph moves ________

Ex. 6 Writing Equ’n of Sine and Cosine 1.Write the sine function for A = -2, period = π, C= right 2, D = down 1 2.Write the cosine function for A=3, period =4π, C=left 1, D= up 4 Page 569 problems 61 – 66

20 10 The number of hours after midnight

Review Time Add these problems to your notes paper to help you review!

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)