11.1B Graphs of Sine and Cosine

Objectives: F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F.BF.3: Identify the effect on the graph of repleacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs…

For the Board: You will be able to graph trigonometric functions.

Application: Sine and cosine functions can be used to model real-world phenomena, such as sound waves. Different sounds create different waves. Sounds are distinguished by their frequency. Frequency is the number of cycles in a given unit of time. It is the reciprocal of the period of a function. Hertz (Hz) is the standard measure of frequency and represents one cycle per second.

Example: A sound has a frequency of 400 Hz. What is the period of the wave? The wave repeats 400 times in 1 second and the period of the function is 1/400 or 0.0025.

White Board Activity: Practice: A sound has a frequency of 800 Hz. What is the period of the wave? The wave repeats 800 times in 1 second and the period of the function is 1/800 or 0.00125.

Open the book to page 756 and read example 3. Example: Use a sine function to graph a sound wave with a period of

0.002 s and an amplitude of 3 cm. Find the frequency in 3 hertz for this sound wave. x y 0.002 = 2/1000 0 0 Frequency = 1000/2 = 500 Hz 0 0.001 0.002 .0005 3 / 0 π/2 π 3π/2 2π .001 0 3π/2 2π -3 .0015 -3

.002 0 White Board Activity: Practice: Use a sine function to graph a sound wave with a period of 0.004 s and an amplitude of 3 cm. Find the frequency in 3 hertz for this sound wave. x y 0.004 = 4/1000 0 0 Frequency = 1000/4 = 250 Hz 0 0.002 0.004 .001 3 / .002 0 3π/2 2π -3 .003 -3 .004 0

Sine and cosine can be translated vertically and horizontally. The horizontal translation of a periodic function is called a phase shift.

General Form of the Sine Function y = a sin b(x – h) + k a is the amplitude: vertical stretch or compression b is used to determine the period: 2π/b h is the phase shift: horizontal translation Since x represents an angle in degrees or radians, h will also be expressed in degrees or radians. k is the vertical translation Since y represents a number, k will also be expressed as a number.

Open the book to page 757 and read example 4. Example: Using f(x) = sin x as a guide, graph x y g(x) = 3 sin (x – π/4). Identify the x-intercepts 0 0 3 and phase shift. π/2 3 x-intercepts: π/4 + n π π 0 O π/2 π 3π/2 2π phase shift: right π/4 3π/2 -3 0 π/2 π 3π/2 2π 3π/2 2π 0 2π -3

White Board Activity: Practice: Using f(x) = cos x as a guide, graph x y 2 g(x) = cos (x – π). Identify the x-intercepts 0 1 and the phase shift. 1 x-intercepts: π/2 + nπ π/2 0 π -1 phase shift: right π 0 π 2π 3π 4π 3π/2 0 -1 2π 1

-2

Open the book to page 758 and read example 5. Example: The number of people, in thousands, employed in a resort town can be modeled by 8

g(x) = 1.5 sin π/6(x + 2) + 5, where x is the month of 6 the year. a. Graph the number of people employed in the 4 town for one complete period. 2π 2π π 2 Period :   x y π 1 6 6 0 0 0 3 6 9 12 2π 6 3 1.5    12 1 π 6 -1 9 -1.5 12 1 Translate the curve up 5. Translate the curve left 2.

b. What is the maximum number of people employed? In the basic function the maximum occurs at 3 and is 1.5. Adjusting for the shift up of 5 and the shift left of 2, the maximum occurs at 1 and is 6.5. 6.5 x 1000 is 6500 people are employed in January.

White Board Activity: Practice: Suppose that a Ferris wheel in feet can be modeled π 40 by H(t) = -16 cos t + 24, where t is the time in 45 30 seconds. a. Graph the height of a cabin for two 20 complete periods. x y π 10 Period : 2π  0 -16 45 22.5 0 0 45 90 135 180 45 45 16  2π   90 -10 π 67.5 0 90 -16

Translate the curve up 24. b. What is the maximum height of a cabin? In the basic function the maximum occurs at 45 and is 16. Adjusting for the shift up of 40, the maximum occurs at 45 seconds and 135 seconds and is 16 + 24 = 40 feet.

Assessment: Question student pairs.

Independent Practice: Text: pg. 759 prob. 7 – 11, 18 – 23, 25 – 28.