M3U7D3 Warm Up Shifted up one Stretched by 3 times 1. How does differ from ? 2. How does differ from ? Shifted up one Stretched by 3 times

HW Check: pp. 10 Interims? Page 10

Page 10

U7D3 Exploring Periodic Data OBJ: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. F-TF.5

Vocabulary Period Period Periodic Function: a repeating pattern of y-values (outputs) at regular intervals. Cycle: One complete pattern is a function. A cycle can occur at any point on the graph of the function Period: the horizontal length of one cycle. Periodic behavior is behavior that repeats over intervals of regularity. Amplitude: half of the distance between the minimum and maximum values of the function. Amplitude Period Period

Exploring Periodic Data Analyze this periodic function. Identify one cycle in two different ways. Then determine the period of the function. Begin at any point on the graph. Trace one complete pattern. The beginning and ending x-values of each cycle determine the period of the function. Each cycle is 7 units long. The period of the function is 7.

Exploring Periodic Data Determine whether each function is or is not periodic. If it is, find the period. a. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. Find points at the beginning and end of one cycle. Subtract the x-values of the points: 2 – 0 = 2. The pattern of the graph repeats every 2 units, so the period is 2.

Exploring Periodic Data (continued) b. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. Find points at the beginning and end of one cycle. Subtract the x-values of the points: 3 – 0 = 3. The pattern of the graph repeats every 3 units, so the period is 3.

Exploring Periodic Data Find the amplitudes of the two functions in Additional Example 2. a. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [2 – (–2)] Substitute. 1 2 = (4) = 2 Subtract within parentheses and simplify. 1 2 The amplitude of the function is 2.

Exploring Periodic Data (continued) b. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [6 – 0] Substitute. 1 2 = (6) = 3 Subtract within parentheses and simplify. 1 2 The amplitude of the function is 3.

Exploring Periodic Data The oscilloscope screen below shows the graph of the alternating current electricity supplied to homes in the United States. Find the period and amplitude. 1 unit on the t-axis = s 1 360

One cycle of the electric current occurs from 0 s to s. 1 60 One cycle of the electric current occurs from 0 s to s. The maximum value of the function is 120, and the minimum is –120. period = – 0  Use the definitions. = Simplify. 1 60 amplitude = [120 – (–120)] = (240) = 120 1 2 The period of the electric current is s. 1 60 The amplitude is 120 volts.

Remember! y = sin x Maximum Maximum intercept intercept intercept 2 Maximum intercept 1 -90º 90º -270º 270º intercept -1 intercept Minimum Minimum -2

y = sin x Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -90º 90º -270º 270º -1 -2 Period: 360°

In other words, how high does it go from its midline? y = sin x Amplitude: half the distance between the maximum and minimum 2 Amplitude = 1 1 -90º 90º -270º 270º -1 -2 In other words, how high does it go from its midline?

y = cos x Maximum Maximum intercept intercept -2 -  2 Minimum 1 intercept intercept -2 -  2 -1 Minimum Minimum -2

y = cos x -2 -  2 Period: 2 Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -2 -  2 -1 -2 Period: 2

How high does it go from its axis? y = cos x How high does it go from its axis? 2 Amplitude = 1 1 -2 -  2 -1 -2

Try it on your calculator! y = sin x y = cos x Try it on your calculator! 2 1 -1 -2

y= sin and y = cos are the mother functions. Changing the equations changes the appearance of the graphs We are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations

Mother Function relative function change? y1 = sin x reflection over x-axis y2 = - sin x y1 = sin x y2 = 4 sin x amplitude = 4 amplitude = y2 = sin x y1 = sin x generalization? y = a sin x amplitude = a

Mother Function relative function change? y1 = sin x y2 = sin 2x Period = 180 or Period = 720 or y1 = sin x y2 = sin x generalization? Period = y = sin bx or

is the horizontal translation Mother Function relative function change? y1 = sin x y2 = sin (x - 45) horizontal translation, 45 degrees to the right. horizontal translation, 60 degrees to the left. y1 = sin x y2 = sin (x + 60) horizontal translation, 30 degrees to the left. y2 = sin (2x + 60) y1 = sin x horizontal translation, 90 degrees to the right. y1 = sin x y2 = sin (3x - 270) generalization? y = sin (bx - c) to the right is the horizontal translation y = sin (bx – (- c)) to the left

‘d’ is the vertical translation Mother Function relative function change? y1 = cos x y2 = 2 + cos x vertical translation, 2 units up. vertical translation, 3 units down. y1 = cos x y2 = -3 + cos x generalization? y = d + cos x ‘d’ is the vertical translation when d is positive, the graph moves up. when d is negative, the graph moves down.

Exploring Periodic Data Determine whether each relation is a function. 1. {(2, 4), (1, 3), (–3, –1), (4, 6)} 2. {(2, 6), (–3, 1), (–2, 2)} 3. {(x, y)| x = 3} 4. {(x, y)| y = 8} 5. {(x, y)| x = y2} 6. {(x, y)| x2 + y2 = 36} 7. {(a, b)| a = b3} 8. {(w, z)| w = z – 36}

Exploring Periodic Data Solutions 1. {(2, 4), (1, 3), (–3, –1), (4, 6)}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 2. {(2, 6), (–3, 1), (–2, 2)}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 3. {(x, y)| x = 3}; no, this is not a function because it is a vertical line and fails the vertical line test. 4. {(x, y)| y = 8}; yes, this is a function because it is a horizontal line and passes the vertical line test.

Exploring Periodic Data 5. {(x, y)| x = y2}; no, this is not a function because an element of the domain is paired with more than one element in the range. Example: 4 = 22 and 4 = (–2)2 6. {(x, y)| x2 + y2 = 36}; no, this is not a function because it is a circle and fails the vertical line test. 7. {(a, b)| a = b3}; yes, this is a function because each element of the domain is paired with exactly one element in the range. 8. {(w, z)| w = z – 36}; yes, this is a function because each element of the domain is paired with exactly one element in the range. Solutions (continued)

You must know how to analyze the equation before you can graph it. The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.

Classwork pp. 11 Homework M3U7D3 pp. 12-13 all