Section 5.2 Graphs of the Sine and Cosine Functions

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1 Section 5.2 Graphs of the Sine and Cosine Functions We know from previously studying the periodicity of the trigonometric functions that the sine and cosine functions repeat themselves after 2 radians. Hence, their period is 2. This also tells us that we can find the whole number line wrapped around the unit circle. f ( x) sinx f ( x) cosx All sine and cosine graphs will have this general shape called a sinusoid. They are periodic. They can be used to model situations that repeat themselves. Notice that the graphs of sine and cosine are exactly the same except for a horizontal shift. Section 5.2 Graphs of the Sine and Cosine Functions 1

2 A Periodic Function and Its Period A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in the domain of f. The smallest such number p is called the period of f. The graphs of periodic functions display patterns that repeat themselves at regular intervals. Amplitude Let f be a periodic function and let m and M denote, respectively, the minimum and maximum M m values of the function. Then the amplitude of f is the number. 2 In other words the amplitude is half the height. Example 1: Given the following graph, state its height, amplitude and period. Height: Amplitude: M m 2 = Period: Section 5.2 Graphs of the Sine and Cosine Functions 2

3 For the following functions: y Asin( Bx C) D and y Acos( Bx C) D *Amplitude = A (Note: Amplitude is always positive.) If A > 1, then it s a vertical stretch by a factor of A. If 0 < A < 1, then it s a vertical shrink by a factor of A. *If B > 1, then it s a horizontal shrink by a factor of 1/B. *If 0 < B < 1, then it s a horizontal stretch by a factor of 1/B. *Period = 2 B *Translation in horizontal direction (called the phase shift) = C B If this value is negative, this means a phase shift to the right. If this value is positive, this means a phase shift to the left. *If D > 0, then it s a vertical shift upward. *If D < 0 then it s a vertical shift downward. *If the coefficient of the function is negative, this refers to an x-axis reflection. Example 2: State the transformations for 1 1 f( x) cos x Section 5.2 Graphs of the Sine and Cosine Functions 3

4 Graphing Sine f ( x) sinx Since the period of the sine function is 2, we will graph the function on the interval [0, 2 ]. The rest of the graph is made up of repetitions of this portion. Key points in graphing sine functions are obtained by dividing the period into four equal parts. (Assuming no vertical shifting.) One complete cycle of the sine curve includes three x-intercepts, one maximum point and one minimum point. The graph has x-intercepts at the beginning, middle, and end of its full period. The graph of y Asin( Bx C) completes one cycle from x = C B to x = C 2. B B Section 5.2 Graphs of the Sine and Cosine Functions 4

5 Example 3: Sketch the graph of Amplitude: A = 1 f( x) 3sin x 3. 2 Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Domain: Range: Section 5.2 Graphs of the Sine and Cosine Functions 5

6 Example 4: Sketch the graph of Amplitude: A = 1 f( x) sin x 4 8. Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Domain: Range: Section 5.2 Graphs of the Sine and Cosine Functions 6

7 Example 5: Give an equation of the form graph. y Asin( Bx C) D that could represent the given Let s begin by recalling one cycle of the basic sine graph. Then choose one cycle on the graph above. Amplitude: A = M m 2 = Vertical Shift, D: It ll be half-way between the maximum and the minimum values. Use the period to find B: Recall the period formula 2 = B Compare your chosen cycle to the original one cycle of sine. Any phase shift? Any other transformations? Sine Function: Section 5.2 Graphs of the Sine and Cosine Functions 7

8 Graphing Cosine f ( x) cosx Since the period of the cosine function is 2, we will graph the function on the interval [0, 2 ]. The rest of the graph is made up of repetitions of this portion. Key points in graphing cosine functions are obtained by dividing the period into four equal parts. (Assuming no vertical shifting and no x-axis reflection.) One complete cycle of the cosine curve includes two x-intercepts, two maximum points and one minimum point. The graph has x-intercepts at the second and fourth points of its full period. The graph of y Acos( Bx C) completes one cycle from x = C B to x = C 2. B B Section 5.2 Graphs of the Sine and Cosine Functions 8

9 x Example 6: Sketch the graph of f( x) 2cos 3. Amplitude: A = Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Domain: Range: Section 5.2 Graphs of the Sine and Cosine Functions 9

10 Example 7: Sketch the graph of f( x) cos2x 2. Amplitude: A = Period: 2 = B Phase Shift: C B = One cycle begins at the phase shift and ends at: C 2 B B Any other transformations? Domain: Range: Section 5.2 Graphs of the Sine and Cosine Functions 10

11 Example 8: Recall the given graph in Example 5. Now give an equation of the form y Acos( Bx C) D that could represent the given graph. Let s begin by recalling one cycle of the basic cosine graph. Then choose one of these cycles on the graph above. A, D, B are the same as in Example 5: A = 5 D = -2 B = 4 Now compare your chosen cycle to the original one cycle of cosine. Any phase shift? Any other transformations? Cosine Function: Section 5.2 Graphs of the Sine and Cosine Functions 11

12 Example 9: A mass on a smooth table attached to a spring moves in simple harmonic motion described by f ( x) 10cos t, with t measured in seconds and f in centimeters. Find the 6 maximum displacement and the time required for one cycle. 7 Example 10: The function Pt () 10525sin t approximates the blood pressure P, in 4 millimeters of mercury, at time t, of a person at rest. Find the period and the highest blood pressure. Section 5.2 Graphs of the Sine and Cosine Functions 12

13 Example 11: Write a sine function with a positive vertical displacement given that the amplitude is 2, the horizontal shift is 9 2 to the right, the period is 6, and its y-intercept is (0, 6). Section 5.2 Graphs of the Sine and Cosine Functions 13

14 Try this one: Give an equation of the form f (x) = Acos(B x - C) + D which could be used to represent the given graph. (Note: C or D may be zero.) a. f( x) 2cos(3 x ) 1 b. f( x) 2cos(3 x ) c. f( x) 2cos(3 x ) 1 d. f( x) 4cos(3 x ) 1 e. f( x) 4cos(3 x ) 1 Try this one: Give an equation of the form f (x) = Asin(B x - C) + D which could be used to represent the given graph. (Note: C or D may be zero.) a. f ( x) 3sin x b. f ( x) 6sin x 1 c. f ( x) 6sin x d. f ( x) 3sin x 1 e. f ( x) 3sin x Section 5.2 Graphs of the Sine and Cosine Functions 14

Section 5.2 Graphs of the Sine and Cosine Functions

A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in