4/29/13. Obj: SWBAT graph periodic functions. Education is Power!

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1 4/9/ Education is Power! Obj: SWBAT graph periodic functions. Bell Ringer: Complete Ferris Wheel assignment HW Requests: -6 Worksheet, Odds # s pg 94 #- Homework: Read pg 79-8, Announcements: Dignity without compromise!

2 Definitions: The period (or wavelength) of f is the length of one complete cycle. Frequency number of cycles in unit interval. The midline is the horizontal line midway between the function s minimum and maximum values. The amplitude is the distance between the function s maximum (or minimum) value and the midline. The phase (or horizontal) shift is the number of units that the start of the cycle is away from being at the midline. We must graph these functions

3 Read page 88 last paragraph Vertical Stretch and Shrink On your calculator. sin x. ½ cos (x). -4 sin(x)

4 Horizontal Stretch and Shrink On your calculator. sin x. sin(x). sin( x ) 4. sin(x) Horizontal Stretch/Shrink y = f(bx) stretch if b < shrink if b > Both cases factor = / b

5 The frequency is the reciprocal of the period. f = b π where T = π b.

6 Periodic Functions Functions that repeat themselves over a particular interval of their domain are periodic functions. The interval is called the period of the function. In the interval there is one complete cycle of the function. To graph a periodic function such as sin x, use the exact values of the angles of 0 0, 45 0, and In particular, keep in mind the quadrantal angles of the unit circle. (0, ) (-, 0) (0, -) (, 0) The points on the unit circle are in the form (cosine, sine). 4..

7 Graphing a Periodic Function Graph y = sin x. Period: p Amplitude: y-intercept: 0 x-intercepts: 0, ±p, ±p,... Domain: all real numbers Range: - y 4..

8 Graphing a Periodic Function Graph y = cos x. Period: p Amplitude: Domain: all real numbers Range: - y y-intercept: x-intercepts:,... p, p 4..4

9 Graphing a Periodic Function Graph y = tan x. Asymptotes: p, p, 5p,..., p p n, n I Period: p Domain: {x x p Range: all real numbers p n, n I, x R} 4..5

10 Determining the Amplitude of y = a sin x Graph y = sin x and y = 0.5sin x. y = sin x y y = sin sin x x y = 0.5sin x 4..6

11 Comparing the Graphs of y = a sin x y = sin x y = sin x y = 0.5sin x Period Amplitude Domain Range p p p 0.5 all real numbers all real numbers all real numbers - y - y -0.5 y 0.5 The amplitude of the graph of y = a sin x is a. When a >, there is a vertical stretch by a factor of a. When 0 < a <, there is a vertical shrink by a factor of a. 4..7

12 Determining the Period for y = sin bx, b > 0 Graph y = sin x and y sin x. y sin x y = sin x y = sin x y = sin x y = sin x 4..8

13 Comparing the Graphs of y = sin bx y = sin x y = sin x y = sin 0.5 x Period Amplitude Domain Range p p 4p all real numbers all real numbers all real numbers - y - y - y The period for y = sin bx is p b, b 0. When b >, there is a horizontal shrink. When 0 < b <, there is a horizontal stretch. 4..9

14 Determining the Period and Amplitude of y = a sin bx Given the function y = sin 4x, determine the period and the amplitude. The period of the function is p. b Therefore, the period is p 4 p. The amplitude of the function is a. Therefore, the amplitude is. y = sin 4x 4..0

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16 A circle with center at (0, 0) and radius is called a unit circle. The equation of this circle would be x y (0,) (-,0) (,0) (0,-) So points on this circle must satisfy this equation.

17 Let's pick a point on the circle. We'll choose a point where the x is /. If the x is /, what is the y value? You can see there x y are two y values. x = / They can be found y by putting / into (0,) the equation for x, and solving for y. y y 4 (-,0) (0,-) (,0), We'll look at a larger version of this and make a right triangle.

18 We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always because it is a radius of the circle. (-,0) (0,), (,0) (0,-)

19 (,0) (0,) (0,-) (-,0) Now let s find the Trigonometric Functions sin cos Notice the sine is just the y value of the unit circle point and the cosine is just the x value. tan,

20 So if I want a trig function for whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and y/x is the tangent., (-,0) (0,) (0,-), (,0), sin tan cos We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.

21 Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? sin 5,,0, , We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions. 90 0, ,,0, These are easy to memorize since they all have the same value with different signs depending on the quadrant.

22 Can you figure out what these angles would be in radians? 7 sin p 4,,0 5 p 4 p 80, 5p 4 5 p 70 0, 90 p p 45 4 The circle is p all the way around so half way is p. The upper half is divided into 4 pieces so each piece is p/4. 0, 7p 4 5 0,,0,

23 Here is the unit circle divided into pieces. Can you figure out how many degrees are in each division? cos0 sin 40,,0, 80, , , We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x. 90 0, 0 00, ,,,0, You'll need to memorize these too but you can see the pattern.

24 Can you figure out what the angles would be in radians?,,0, 80, , , p 0 6 It is still p halfway around the circle and the upper half is divided into 6 pieces so each piece is p/ , 00, ,,,0, We'll see them all put together on the unit circle on the next screen.

25 You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.,

26 The cosine is also periodic with a period of 60 or p. undef Let's label the unit circle with values of the tangent. (Remember this is just y/x) 0 0 undef, We see that they repeat every p so the tangent s period is p.

27 Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen s School Carramar