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1 Pre-Lesson Assessment Unit 2: Trigonometric Functions Periodic Functions Diagnostic Exam: Page 1 Name: Date: Group: Learning Target: I can determine amplitude, period, frequency, and phase shift, given a graph or equation of a periodic function. 1. The position, S, of a piston in a 6-inch stroke in an engine is given as a function of time, t, in seconds by the formula S=3sin(250πt). What is the amplitude and period of this function? 2. State the amplitude, period, midline, horizontal and vertical phase shifts for the following: a. y = 4 cos(2(x + 5))- 3 b. y= 3 sin(t)+4 c. y = 2 sin(π(x - 2))- 5 d. y =.5 sin(x+6)+2

2 Pre-Lesson Assessment Unit 2: Trigonometric Functions Periodic Functions Diagnostic Exam: Page 2 Learning Target: I can sketch and recognize one cycle of a function of the form y=asinbx or y=acosbx. 3. Sketch one cycle of the graph of y = 2cos( x) below. Make sure to label the axes and scale of your graph. 4. Sketch one cycle of the graph of y =3sin(2x) below. Make sure to label the axes and scale of your graph.

3 Pre-Lesson Assessment Unit 2: Trigonometric Functions Periodic Functions Diagnostic Exam: Page 3 Learning Target: I can sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x). 5. Label the graphs below with the correct equation: y=sec(x), y=csc(x), y=tan(x), or y=cot(x).

4 Pre-Lesson Assessment Unit 2: Trigonometric Functions Periodic Functions Diagnostic Exam: Page 4 Learning Target: I can write the trigonometric function that is represented by a given periodic function. 6. Determine an equation for this graph: Choose from: a. y= cos(2x) b. y=2cos(x) c. y= cos(x) d. y=cos( x) 7. Determine an equation for this graph: Choose from: a. y= 2sin( x) b. y= 2sin(2x) c. y= 2cos( x) d. y= 2cos(2x) 8. Determine an equation for this graph: Choose from: a. y= - cos(2x) b. y= - cos( x) c. y= - sin(2x) d. y= -2cos( x) Images from regentsprep.org

Lesson 1 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 1 Periodic Functions Name: Date: Group: Lesson 1: Periodic Phenomena What does it mean to be periodic?

5 Lesson 1 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 1 Periodic Functions Name: Date: Group: Lesson 1: Periodic Phenomena What does it mean to be periodic? A cycle is: Frequency is: Where do you see Periodic Phenomena? Write a few you like or have seen below. How can we describe periodic phenomena? Recall The Unit Circle: We know: Image from myweb20journal.com

6 Lesson 1 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 2 Graphing the Sine curve: f(x)=sin(x) Degree Radian Approx sin( ) Graphing Cosine curve: f(x)=cos(x) Degree Radian Approx cos( )

7 Lesson 1 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 3 What is the period of the sine and cosine curves and why? Draw y=sin(x) and y=cos(x) together below, make sure to label your functions. Something to experiment before next class: What happens if you put a coefficient in front of the sine or cosine function in y=cos(x) and y=sin(x)? What happens to the curve? What happens if you put a coefficient in front of the x value in the trigonometric functions y=cos(x) and y=sin(x)? What about the curve changes?

Lesson 2 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 4 Lesson 2: Describing Periodic Phenomena: Changing the Period and Frequency of the Sine and Cosine curves

8 Lesson 2 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 4 Lesson 2: Describing Periodic Phenomena: Changing the Period and Frequency of the Sine and Cosine curves Characteristics of y=sin(x): Characteristics of y=cos(x): But thinking back to our periodic phenomena, can they all be modeled by just these two curves? For example, these are sound waves of different instruments and light waves of different stars. What do you notice about the curves? Note findings below: Images from: ykonline.yksd.com, newworldencyclopedia.org

9 Lesson 2 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 5 These models of periodic phenomena have different periods and frequencies. To change the period and frequency of sine and cosine graphs we add and multiply coefficients: Where A, B, C, and D are constants. These two formulas will be important tools for the rest of our unit. Changing B: B determines the and of periodic curves. The period of sine and cosine is: So the period can be determined by: The frequency can be determined by: Practice- What is the period and frequency of the following? Learning Target 1: I can determine amplitude, period, frequency, and phase shift, given a graph or equation of a periodic function. a) y=sin(2x) b) y=cos( x) c) y=cos(4x) d) y=sin( x)

10 Lesson 2 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 6 Graphing and sketching one cycle of periodic functions: Learning Target 2: I can sketch and recognize one cycle of a function of the form y=asinbx or y=acosbx. Method 1: Sketching using the graphing calculator Summary of Steps: a) y=sin(2x) b) y=cos( x)

11 Lesson 2 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 7 Method 2: Sketching using the properties of sine and cosine (without a calculator) Summary of Steps: c) y=cos(4x) d) y=sin( x)

12 Lesson 2 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 2: Page 1 Name: Date: Group: Learning Target: I can sketch and recognize one cycle of a function of the form y=asinbx or y=acosbx. 1. Sketch one cycle of the graph of y = 3 sin(x) below. Make sure to label the axes and scale of your graph. 2. Sketch one cycle of the graph of y = 4 sin( x) below. Make sure to label the axes and scale of your graph.

13 Lesson 2 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 2: Page 2 3. Sketch one cycle of the graph of y = cos(x) below. Make sure to label the axes and scale of your graph. 4. Sketch one cycle of the graph of y = 3cos(2x) below. Make sure to label the axes and scale of your graph.

14 Lesson 3 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 8 Lesson 3: Describing Periodic Phenomena: Amplitude, Midline, and Phase Shifts for Sine and Cosine curves Recall these two important formulas for graphing sine and cosine periodic functions: Where A, B, C, and D are constant coefficients. A determines B determines: C determines: D determines: Learning Target 1: I can determine amplitude, period, frequency, and phase shifts, given a graph or equation of a periodic function. Practice: What is the period, frequency, amplitude, midline, vertical and horizontal phase shifts? a. y= 3 sin(x)+4 b. y=2sin(3(x-π))+4 c. y = 2 cos(.5(x - 2π))- 5 d. y=.25sin(x+3π)-6 e. y= 10cos(24x)+100

15 Lesson 3 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 9 f. y= -8cos(3x-3π) g. y= - sin(x+4π)-10 h. y= 6 cos(2x+π)+ Learning Target: I can write the trigonometric function that is represented by a given periodic function. Practice: Choose the equation of the trigonometric function given the periodic graph. It may help to find the period, frequency, amplitude, midline, vertical and horizontal phase shifts of the periodic function to determine the equation. 1. a. y= cos(2x) b. y=2cos(x) c. y= cos(x) d. y=cos( x)

16 Lesson 3 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page a. y=4sin( x) b. y= 4sin(2x) c. y= 4cos( x) d. y= 4cos(2x) 3. a. y=2sin( x)+1 b. y= 2sin(2x)-1 c. y= 2cos( x)-1 d. y= 2cos(2x)+1 4. a. y=sin( x)-3 b. y= sin(2x)+3 c. y= cos(x)+3 d. y= cos(2x)-3

17 Lesson 3 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 11 Review: Coordinates of a point on a circle with radius r is (rcosθ, rsinθ) y=asin(b(x-c))+d and y=acos(b(x-c))+d o is the amplitude o is the frequency o is the period o o C is the horizontal shift y=d is the midline, so D is the vertical shift Steps for Writing Periodic Functions: 1. Decide if sine or cosine by where the function begins a. Sine function begins at its midline and then goes up b. Cosine function begins at its maximum then goes down 2. Identify maximum and minimum 3. Decide if shifted vertically a. D is the midline which is y= b. D is positive: shifted up or D is negative: shifted down 4. Decide on amplitude a. A= maximum-midline OR A = 5. Decide the length of a complete cycle or the period a. Find B, the frequency: the number of completed cycles by 2π 6. Decide if shifted horizontally (C)

18 Lesson 3 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 1: Page 1 Name: Date: Group: Learning Target: I can determine amplitude, period, frequency, and phase shift, given a graph or equation of a periodic function. 1. State the amplitude, period, midline, horizontal and vertical phase shifts for the following: a. y = 2 cos(2(x + 6))- 8 b. y= 4 sin(t)+2 c. y = 3 sin(π(x - 2))- 3 d. y =.5 sin(x+3)+1 e. y = cos (π x+ π5) - 3

19 Lesson 3 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 4: Page 1 Name: Date: Group: Learning Target: I can write the trigonometric function that is represented by a given periodic function. Choose the most appropriate equation for each graph below: 1. a. y= 3sin(2x) b. y= 4cos(x) c. y= 4sin(x) d. y= 3cos( x) 2. a. y= cos( x) - 1 b. y= cos(x) + 1 c. y= cos(x) - 1 d. y= cos( x) a. y= 2cos( x) +2. y= 2cos( x) - 2 c. y= cos(2x) - 2 d. y= cos(2x) + 2

20 Lesson 3 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 4: Page 2 4. a. y = sin(x)+3 b. y = sin(3x) c. y = sin(x)-3 d. y = cos(3x) 5. a. y= 3sin(4x) b. y= 2sin(3x) c. y= 3sin(x) d. y= sin(2x) 6. a. y= 2sin(x+2π) -1 b. y= 2sin(x+π)-1 c. y= 2sin (x-π)-1 d. y= 2sin(x-2π)-1 7. a. y= -cos(3x) b. y= -sin(3x) c. y= -3cos(x) d. y= -3sin(x)

21 Lesson 4 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 11 Lesson 4: Other Periodic Phenomena: Recognizing the periodic functions tangent, cotangent, secant, and cosecant. Learning Target 4: I can sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x). Forming Tangent: y=tan(x) Forming Cotangent: y=cot(x)

22 Lesson 4 Assessment Unit 2: Trigonometric Functions Periodic Functions Packet: Page 12 Forming Cosecant: y=csc(x) Forming Secant: y=sec(x)

23 Lesson 4 Assessment Unit 2: Graphing Trigonometric Functions Practice Quiz Learning Target 3: Page 1 Name: Date: Group: Learning Target: I can sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x). Sketch and label at least one cycle of the graphs y=sec(x), y=csc(x), y=tan(x), and y=cot(x) on the axes below. Make sure to label your axes and the scale of each graph.

24 Final Assessment Unit 2: Trigonometric Functions Periodic Functions Quiz: Page 1 Name: Date: Group: Learning Target: I can determine amplitude, period, frequency, and phase shift, given a graph or equation of a periodic function. ( /6) 1. What is one example of periodic phenomena in nature? 2. What can we use to describe periodic phenomena? 3. State the amplitude, frequency, period, midline, horizontal and vertical phase shifts for the following: a. y = 5 cos(x-π)- 3 b. y= sin(t)+4 c. y = 12 sin(2(x - ))- 1 d. y = sin(θ+6π)+2

25 Final Assessment Unit 2: Trigonometric Functions Periodic Functions Quiz: Page 2 Learning Target: I can sketch and recognize one cycle of a function of the form y=asinbx or y=acosbx. ( /2) 4. Sketch the graph of y = 2sin( x) from 0 to 2π below. Make sure to label the axes and scale of your graph. 5. Sketch the graph of y =4cos(2x) from 0 to 2π below. Make sure to label the axes and scale of your graph.

26 Final Assessment Unit 2: Trigonometric Functions Periodic Functions Quiz: Page 3 Learning Target: I can sketch and recognize the graphs of the functions y=sec(x), y=csc(x), y=tan(x), and y=cot(x). ( /4) 6. Label the graphs below with the correct equation: y=sec(x), y=csc(x), y=tan(x), or y=cot(x). a. b. c. d.

27 Final Assessment Unit 2: Trigonometric Functions Periodic Functions Quiz: Page 4 Learning Target: I can write the trigonometric function that is represented by a given periodic function. ( /5) For 7 through 12 select the best choice that models the periodic curve to the right. Choose 5 out of the 6 questions. 7. Determine an equation for this graph: Choose from: a. y= 3cos(2x)+1 b. y=3cos( x)-1 c. y= cos(3x)+1 d. y=3cos( x)+1 8. Determine an equation for this graph: Choose from: a. y= 2cos( x)-2 b. y= 2sin(4x)+2 c. y= 2sin( x)-2 d. y= 2cos(4x)+2 9. Determine an equation for this graph: Choose from: a. y= cos(2x) b. y= cos(x)+2 c. y= sin(x)+2 d. y= 2sin( x)

28 Final Assessment Unit 2: Trigonometric Functions Periodic Functions Quiz: Page Determine an equation for this graph: Choose from: a. y= 3cos(4x) b. y= cos(x)+3 c. y= sin(4x)+3 d. y= 3sin(4x) 11. Determine an equation for this graph: Choose from: a. y= cos(2x) b. y= cos(x)+2 c. y= sin(x)+2 d. y= 2sin( x) 12. Determine an equation for this graph: Choose from: a. y= cos(3x) b. y= 2cos(3x)+2 c. y= sin(2x)+2 d. y= 3sin( x)+4

2.5 Amplitude, Period and Frequency

2.5 Amplitude, Period and Frequency Learning Objectives Calculate the amplitude and period of a sine or cosine curve. Calculate the frequency of a sine or cosine wave. Graph transformations of sine and