1 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE 6.3 GRAPHS OF THE SINE AND COSINE Periodic Functions Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period Translations Cominations of Translations Determining a Trigonometric Model Using Curve Fitting Periodic Functions Many things in daily life repeat with a predictale pattern: In warm areas electricity use goes up in the summer and down in the winter, the price of fresh fruit goes down and summer and up in the winter, and attendance at amusement Parks increases in spring and declines in autumn. Because the sine and cosine functions repeat their values in a regular pattern, they are periodic functions. Definition: PERIODIC FUNCTION A periodic function is a function f such that f ( x) = f( x+ np), for every real numer x in the domain of f, every integer n, and some positive real numer p. The smallest possile positive value of p is the period of the function. The sine and the cosine function are periodic with period 2π Graph of the Sine Function 1
2 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE This graph is called a sine wave or sinusoid It has the following properties: The graph is continuous over its entire domain (, ) Its x-intercepts are of the form nπ, where n is an integer Its period is 2π The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin( x) = sin x Graph of the Cosine Function: The cosine function has the following properties: The graph is continuous over its entire domain (, ) π Its x-intercepts are of the form ( 2n + 1) where n is an integer. 2 Its period is 2π The graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, cos(-x) = cos x. Graphing Techniques, Amplitude, and Period The examples that follow show that graphs that are stretched or compressed either vertically, horizontally or oth when compared with the graphs of y = sin x Or y = cos x. Example: Compare the graph of y = cos x with y = 2 cos x. What happens? The amplitude of a periodic function is half the difference etween the maximum and minim values. Thus y = cos x has amplitude = 1 whereas y = 2 cos x has 2
3 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE amplitude = 2. Similarly y = -2 cos x has amplitude = 2 (amplitude is magnitude so we use the asolute value of the coefficient) Definition: Amplitude: The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except with range a, a. The amplitude is a. Homework exercises: 4. Find the graph of y = 2 cos x Homework exercises 5 12. Graph each function over the interval [ 2 π,2π ] Give the amplitude 6. y = 3 sin x 3 8. y = cos x 4 10. y = - sin x 12. y = -3 cos x. Period: Consider the functions y = sin x or y = cos x where >0 is a positive constant. 2π The period changes from 2 π to. Note: use the 5 point summary to graph: Divide an interval (one period) into four equal parts: Find the midpoint of the interval y adding the x-values of the endpoints and dividing y 2 (take the average) Find the midpoints of the two intervals found aove, using the same procedure (this yields the first quarter and third quarter points) 3
4 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE Homework exercises 13 20 Graph each function over a two-period interval. Give the period and amplitude. Summary Guidelines for sketching graphs of sine and cosine functions To graph y = a sin x or y = a cos x, with >0, follow these steps: 1) Find the period, 2 π 2π. Start at 0 on the x-axis, and lay off a distance of. 2) Divide the interval into four equal parts. 3) Evaluate the function for each of the five x-values resulting from step 2. The points will e maximum points, minimum points, and x intercepts. 4) Plot the points found in step 3, and join them with a sinusoidal curve having amplitude a. 5) Draw the graph over additional periods, to the right and to the left, as needed. Translations With trigonometric functions, a horizontal translation is called a phase shift. In the function y = f(x d), the expression x d is called the argument. Homework exercises 23 26 Match each function in Column I with the appropriate description in Column II 24. y = 2 sin (3x 4) 26. y = 2 sin (4x 3) Homework exercises 27 30. Match each function with its graph 28. π y = sin x+ 4 30. y = 1+ sinx (do this one after discussion on vertical translations) 4
5 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE Vertical Translations The graph of a function of the form y = c + f(x) is translated vertically as compared with the graph of y = f(x). The translation is c units up if c > 0 and c units down if c < 0. Cominations of Translations: A function of the form y = c+ asin ( x d) or y = c+ acos ( x d) >0 Can e graphed according to the following guidelines: Further Guidelines for Sketching Graphs of Sine and Cosine Functions Method 1: 1) Find an interval whose length is one period 2 π y solving the three-part inequality 0 x ( d) 2π. 2) Divide the interval into four equal parts. 3) Evaluate the function for each of the five x-values resulting from step 2. The points will e maximum points, minimum points, and points that intersect the line y = c ( middle points of the wave). 4) Plot the points found in step 3, and join them with a sinusoidal curve have amplitude a. 5) Draw the graph over additional periods, to the right and to the left, as needed. Method 2 First graph the asic circular function. The amplitude of the function is a, and the period is 2 π. Then use translations to graph the desired function. The vertical translation is c units up if c > 0 and c units down if c < 0. The horizontal translation (phase shift) is d units to the right if d > 0 and d units to the left if d < 0. Homework exercises 31 36. Find the amplitude, the period, any vertical translation, and any phase shift of the graph of each function: 2 π 32. y = sin x+ 3 2 1 x 34. y= sin + π 2 2 1 36. y=-1+ cos( 2x 3π ) 2 Homework exercises 37 40. Graph each function over a two-period interval: 5
6 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE π 38. y=cos x- 3 40. 3π y = 3sin x 2 Homework exercises 41 44 Graph each function over a one-period interval: 42. y = 3cos( 4x+ π ) 44. 1 3 π y = sin x+ 4 4 8 Homework exercises 45 48. Graph each function over a two-period interval: 46. 2 3 y = 1 sin x 3 4 48. 1 y = 3+ 3sin x 2 Homework exercises 49 52. Graph each function over a one-period interval: 50. y = 4 3cos( x π ) 52. 5 π y = + cos3 x 2 6 Determining a Trigonometric Model Using Curve Fitting A sinusoidal function if often a good approximation of a set of real data points. Homework exercises 55 74 are examples of this. See ook pp. 570 6