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Lesson 14.1: Graphs of the Sine and Cosine Trig FunctionsLearning Goals:How do the sine and cosine curves relate to the unit circle?How can we find the domain and range of a sine or cosine curve?What is the amplitude of a sine or cosine curve?The sine and cosine functions can be easily graphed by considering their values at the quadrantal angles, those that are integer multiples of 90° or π2 radians. Due to considerations from physics and calculus, most trigonometric graphing is done with the input angle in units of radians, not degrees.24574504572000Graphing the sine function: By using the unit circle, fill out the table below for selected quadrantal angles.-360°-270°-180°-90°0°90°180°270°360°Radians-2π-3π2-π-π20π2π3π22πSinθ010-1010-10sinθ is an odd function because it is reflected over the origin! 013398500466725023368000Graphing the cosine function: By using the unit circle, fill out the table below for selected quadrantal angles.-360°-270°-180°-90°0°90°180°270°360°Radians-2π-3π2-π-π20π2π3π22πCosθ10-1010-101cosθ is an even function because it is reflected over they-axis!032385000The domain and range of the sine and cosine functions are the same. State them below in interval notation:Domain: D=(-∞, ∞)Range: R=[-1, 1]Critical values for the sine curve:Critical values for the cosine curve:0, 1, 0, -1, 01, 0, -1, 0, 1Now we would like to explore the effect of changing the coefficient of the trigonometric function. In essence we would like to look at the graphs of functions of the forms:y=Asinx and y=Acos(x)299085053022500The grid below shows the graph of y=cos?(x). Use your graphing calculator to sketch and label each of the following equations. Be sure your calculator is in RADIAN MODE.y=cosx 1, 0, -1, 0, 1 y=3cosx 3, 0, -3, 0, 3 y=-2cosx-2, 0, 2, 0, -2 308673551625500The basic sine function is graphed below. Without the use of your calculator, sketch each of the following sine curves on the axes below.y=sinx 0, 1, 0, -1, 0 y=2sinx 0, 2, 0, -2, 0 y=-3sinx 0, -3, 0, 3, 0 Amplitude: height of the wave from x-axis! Cannot be negative, so amp =A1. Write the equation of the following graphs:a. y=2sinxb. y=-cosx 16287750002. Graph fθ=-3sinθ in the interval -π≤θ≤π452437517081500Key Points: sinθ are 0, 1, 0, -1, 0a=-3 0, -3, 0, 3, 0 1343025278765003. Graph one cycle of the function fθ=2cosθ5019675345503500Key Points: cosθ are 1, 0, -1, 0, 1a=2 one cycle2, 0, -2, 0, 2 Homework 14.1: Graphs of the Sine and Cosine Trig Functions2667000214630001. On the grid below, sketch the graphs of each of the following equations based on the basic sine function.y=sinx y=3sinx y=-5sinx 2667000445770002. On the grid below, sketch the graphs of each of the following equations based on the basic cosine function.y=cosx y=4cosx y=-3cosx 3. Which of the following represents the range of the trigonometric function y=7sin?(x)?(1) (-7, 7) (2) [-7, 7] (3) [0, 7) (4) (-7, 7]2943225257175004. Which of the following equations describes the graph shown below?(1) y=3cosx(2) y=-3cosx(3) y=3sinx(4) y=-3sin?(x) 5. Which of the following equations represents the periodic curve shown below?26479502095500(1) y=4cosx(2) y=-4cosx(3) y=4sinx(4) y=-4sin?(x)6. Which of the following lines when drawn would not intersect the graph of y=6sin?(x)?(1) x=8 (2) x=3 (3) y=-4 (4) y=9Graphing TangentLearning Goals:How does the tangent curve relate to the unit circle?How do we graph tangent curves with changes to its properties?1. On your plot, sketch the graph of y=tan?(x) by using the following table below. By using the unit circle, fill out the table below for selected quadrantal angles.-360°-270°-180°-90°0°90°180°270°360°Radians-2π-3π2-π-π20π2π3π22πSinθ010-1010-10Cosθ10-1010-101Tanθsinθcosθ0error0error0error0error0Think about the values of tanπ4 and tan-π4 in order to help you graph y=tan?(x)6667585090One cycle occurs between π2 and -π2 radians. (asymptotes)The length of one cycle (period) is π radians. P=2πBThere are vertical asymptotes at each end of the cycle. The asymptotes occurs at π2 radians and repeats every π radians.The range is -∞, ∞→all reals!.There is no amplitude.The graph is symmetric about the origin, meaning it is an odd function.0One cycle occurs between π2 and -π2 radians. (asymptotes)The length of one cycle (period) is π radians. P=2πBThere are vertical asymptotes at each end of the cycle. The asymptotes occurs at π2 radians and repeats every π radians.The range is -∞, ∞→all reals!.There is no amplitude.The graph is symmetric about the origin, meaning it is an odd function.How do we graph tangent curves with changes to its properties?Each set of axes below shows the graph of fx=tan?(x). Use what you know about function transformations to sketch a graph of y=g(x) for each function g on the interval (0, 2π)2. a. gx=2tan?(x)b. gx=-13tan?(x) c. How does changing the parameter A affect the graph of gx=Atan(x)? Vertical dilation so multiply the y-values by A3. a. gx=tan?x-π2b. gx=tan?(x+π4) c=π2 so move to right π2 c=-π4 so move to left π4 c. How does changing the parameter h affect the graph of gx=tan?(x-h)? Horizontal translation (move graph to the left or right) and it is opposite from what you see!4. a. gx=tanx+1b. gx=tanx+3 c. How does changing the parameter k affect the graph of gx=tanx+k? Vertical translation!5. a. gx=tan?(3x)b. gx=tanx2 multiply x by 13 multiply x by 2 c. How does changing the parameter ω affect the graph of gx=tan?(ωx)?Horizontal dilation so you multiply the x-value by 1ωGraphing Tangent1. Each set of axes below shows the graph of fx=tan?(x). Use what you know about function transformations to sketch a graph of y=g(x) for each function g on the interval (0, 2π).a. gx=-2tan?(x)b. gx=tanx+π2 c. gx=tan2xLesson 14.2: Graphs of the Sine and Cosine Trig Functions – Period and FrequencyLearning Goals:How do we find the period of a sine or cosine curve?How do we find the frequency of a sine or cosine curve?How do we write the equation of a sine or cosine curve given the graph?In this lesson, we will explore graphs of the form:y=AsinBx and y=Acos(Bx)390271041656000Below is the graph of one cycle of y=sinxB=10444500 Below are the graphs of y=sin2x and y=sin12x.B=2 compressed B=12 stretchedAs the value of b changes, what happens to the sine curve? Graph gets stretched or compressed horizontally!Between 0 to 2π, how many cycles of a sine curve appear for each graph? What do you notice about this number compared to value of b?y=sin2x y=sin12xB=2 B=122 cycles from 0 to 2π 12 cycle from 0 to 2π Frequency and Period of Sinusoidal CurvesThe parent graphs for sine and cosine have a period of 2π.By definition, the period of a sinusoidal curve is the horizontal length of one full cycle→must pass through 5 key points!If the value of B changes then the number of cycles from 0 to 2π changes as well.By definition, the frequency of a sinusoidal curve is the number of full cycles from 0 to 2π. B= frequencySince the sine and cosine functions have period 2π, the functions y=Asin(Bx) or y=Acos(Bx) complete one period as Bx varies from 0 to 2π. In other words, to find the period of a sinusoidal curve, simply divide 2π by the value of BPeriod=2πBExercise 1: State the frequency and period of each graph.frequency=2 (2 cycles from 0 to 2π)frequency= 12 (12 cycle from 0 to 2π)Period= 2π2=π length of one cyclePeriod= 2π12=4π length of one cycleExercise 2: State the amplitude, frequency, and period of sinusoidal curve.a. y=3sin?(x)b. y=-cos13xc. y=12cos?(3x)amp=3=3 amp=-1=1 amp=12=12Freq=1 Freq=13 Freq=3 Period=2π1=2π Period=2π13=6π Period=2π3 Exercise 3: Determine the period of fx=8sinπ2xPeriod= 2πB=2ππ2=2π1?2π=4 Exercise 4: State the amplitude, frequency, and period of each sinusoidal curve given below. amp=-2=2 amp=12=12 amp=2=2Freq=1 Freq=2 Freq=12 Period=2π1=2π Period=2π2=π Period=2π12=4π GRAPHING SINUSOIDAL CURVES WITH CHANGES TO FREQUENCY AND PERIOD1. Identify the key points for a SINE or COSINE curve. 0, 1, 0, -1, 0 or 1, 0, -1, 0, 12. Identify all properties of the trig graph.Amplitude, frequency, and period3. Graph one cycle of the wave by using the value of the period and then key points of the curve.4. Use the given interval/directions to determine how many cycles of the function you need.Model Problem 1: Graph two cycles of y=cos2x from 0≤x≤2π022860000amp=1=1 Freq=2 Period=2π2=π 4010025271145001, 0, -1, 0, 1 one cycle-9525040957500Model Problem 2: Graph one cycle of y=3sinθ2 from 0≤θ≤4πamp=3=3 Freq=12 Period=2π12=2π?2=4π 0, 3, 0, -3, 0 one cycle 388620035496500Model Problem 3: On the axes below, graph two cycles of a cosine function with amplitude 2, period π2, and passing through the point (0, -2). Homework 14.2: Graphs of the Sine and Cosine trig functions – Period and Frequency1. For each function, indicate the amplitude, frequency, and period.a. gx=cos?(3x) b. fx=-5sinx2c. hx=10cosπ8x2. For each function, indicate the amplitude and period.3. Which sine function has a period of 8π and an amplitude of 2?(1) fx=2sin?(8x) (2) fx=3sin14x (3) fx=2sin12x (4) fx=2sin14x4. What is the minimum value of f(θ) in the equation fθ=3sin4θ?(1) -1 (2) -2 (3) -3 (4) -45. Which statement is NOT true regarding the graph of the equation y=-2sinπ4x?(1) The amplitude is 2 (2) The range is [-2, 2] (3) The y-intercept is (0,2). (4) The period is 8.6. As angle x increases from π2 to π, the value of sinx will(1) increase from -1 to 0 (2) increase from 0 to 1 (3) decrease from 0 to -1(4) decrease from 1 to 0346773567310007. A radio wave has an amplitude of 3 and a wavelength (period) of π meters. On the accompanying grid, using the interval 0 to 2π, draw a possible sine curve for this wave that passes through the origin.8. On the axes below, graph one cycle 9. On the axes below, graph of a cosine function with amplitude 4, y=-sin12x on the intervalperiod π2, and passing through the 0≤x≤4π. State the amplitude,356235054800500point (0, 4).frequency, and period.0-123761500Lesson 14.3 & 14.4: Vertical Shifts of Sine and Cosine Curves and Phase Shifts, along with Writing Trig EquationsLearning Goals:What is the vertical shift and midline and how do we determine it?How do we graph sine and cosine functions with a translation?What is a phase shift and midline and how do we determine it?How do we write the equation of a trig function from its graph?In this lesson we will explore graphs of the form y=AsinBx+D and y=AcosBx+D.1. Below are the graphs of y=cosx and y=cosx+2What did the value of "d" do to the graph of y=cosx? What type of transformation is this? Moved up 2 units…it is a vertical shift!What would the location of the midline (“new” x-axis) be after this transformation? at y=22. Below are the graphs of y=cosx and y=cosx-1What did the value of "d" do to the graph of y=cosx? What type of transformation is this? Moved down 1 unit…it is a vertical shift!What would the location of the midline (“new” x-axis) be after this transformation? at y=-1SummaryFor curves that have the general form y=AsinBx+D and y=AcosBx+D, the value D represents the translation of a vertical shift (up or down).The value of D also represents the midline of the trigonometric function. It is the horizontal line that the sinusoidal curve rises and falls above and below by a distance of A, the amplitude.If given the graph, the equation of the midline can be found by taking the average of the max and min values of the sinusoidal curve.y=max+min2=3+12=42=2 3419475255905004762532893000Directions: State the range and the equation of the midline for each graph.R=[-5, -1] R=[-6, 4] y=-5+(-1)2=-62=-3 y=4+(-6)2=-22=-1 midline at y=-3midline at y=-1amp=2 amp=5 How to graph sinusoidal curves with vertical shifts:Graph the SINE or COSINE curve without the vertical shift.Identify the vertical shift and use this to move all the key points up or down "D" units.Label your final graph.292100074612500Model Problem 1: Graph and label the function fx=-sinx+3 on the interval -2π≤x≤2π. State the range of the function and the equation of the midline.Vertical shift up 3R=[2, 4] y=2+42=62=3 midline at y=3Period=2π1=2π amp=-1=1 0, 1, 0, -1, 0 amp=-1 0, -1, 0, 1, 0→+3 3, 2, 3, 4,3 Model Problem 2: On the axes below, graph one cycle of a cosine function with amplitude 3, period π2, midline y=-1, and passing through the point (0, 2). State the range of the cosine function.283845011239500Vertical shift down 1R=[2,-4] y=2+42=62=3 midline at y=3Period=π2 amp=3 0, 1, 0, -1, 0→+3 above the midline 3, 0, -3, 0, 3→vertical shift-1→2, -1, -4, -1, 2Directions: State the range of each of the following trigonometric functions.a. y=7sinx+4 b. y=-5cos2x+3c. y=-25sinπ7x-350, 1, 0, -1, 0 R=[-2, 8]R=[-50, -10]a=7 Or ±5 from+3(midline)Or ±25 from-35(midline)d=4 (add 4) 4, 11, 4, -3, 4 R=[-3, 11] Or ±7 from+4(midline)In this lesson we will explore sinusoidal graphs of the form y=AsinBx-C+D.Introduction to Horizontal Shifts (Phase Shifts)The graphs of sine and cosine are the same when sine or cosine is shifted left or right. Such a shift is referred to as a horizontal shift.How could you shift the sine curve below so that it becomes a cosine curve? shift sine to the left by π2 cosx=sinx+π2How could you shift the cosine curve above so that it becomes a sine curve? shift cosine to the right by π2 sinx=cosx-π2Summary of Horizontal Shifts of a Sinusoidal FunctionFor curves that have the general form y=AsinBx-c+D and y=AcosBx-c+D, the value C represents the translation of a horizontal shift (left or right).The graphs of sine and cosine are the same when sine is shifted left by 90° or π2 radians. Such a shifting is referred to as a horizontal shift.sinx+π2=cos?(x) shift sine to the left to create cosinecosx-π2=sin?(x) shift cosine to the right to create sineIn mathematics, a horizontal shift may also be referred to as a phase shift B=1.Remember that with a horizontal shift you must be careful with identifying which way the graph will be translated (moved left or right)!Practice 1: Identify the horizontal shift of the sinusoidal functions given below:a. y=sin?(x-π)b. y=sin?(x+π)c=π so right π c=-π so left πc. y=2cos3x+π6-1d. y=-cosπ2x-3+4c=-π6 so left π6 c=3 so right 3How to Write Trig Equations From Graphs1. Identify if it is a sine or cosine curve: y=AsinBx-c+D and y=AcosBx-c+D. Sine is on midline and Cosine is above/below midline.2. If the max/min of the curve is not equidistant from the x-axis, try to identify the midline. This value goes in for D.3. From the midline, identify the amplitude by looking for the max/min of the curve. This value goes in for A.Although the amplitude must be positive, determine if A will be positive or negative.4. Determine the value of B by looking at either:Period – length of one full cycle P=2πBFrequency – number of full cycles per 2π B=frequencyPractice 1: Write an equation to represent the trigonometric graphs below.01460500No vertical shift!Below the midliney=AcosBx a=-2 starts down first One cycle, so B=1y=-2cosx 4762516129000No shift!Starts on midliney=AsinBx a=-3 starts down first One cycle =π, so 2 cycles up to 2π means B=2y=-3sin2x 0000No shift!Starts on midliney=AsinBx a=2 starts up first One half cycle =2π, so means B=12y=2sin12x 1181107683500No shift!Starts above midliney=AcosBx a=2 starts up first One cycle =2π3, so means B=3y=2cos3x 300037593662500Practice 2: The periodic graph below can be represented by the trigonometric equation y=acosbx+c where a, b, and c are real numbers. State the values of a, b, and c, and write an equation for the graph.Vertical shift!Find the midline first!y=max+min2=8+22=5 (shift)a=-3 below midline, so c=5 B=2πP=2π30=π15 y=-3cosπ15x+5 Practice 3: Find equations of two different functions that can be represented by the graph shown below – one sine and one cosine – using a horizontal translation for one of the equations.0-127000y=-2sinxy=-2cosx-π2 307657551943000Homework 14.3 & 14.4: Vertical Shifts of Sine and Cosine Curves and Phase Shifts, along with Writing Trig Equations1. Graph and label the function fx=-cos12x+1 on the interval -4π≤x≤4π. State the range of the function and the equation of the midline.2914650642620002. On the axes below, graph one cycle of a sine function with the amplitude 2, period 2π, midline y=-3, and passing through the point (0, -3). State the range of the sine function.3057654758825003. For the function below, indicate the amplitude, frequency, period, vertical translation, and equation of the midline. Graph the function together with a graph of the cosine function fx=cos?(x) on the same axes. Graph one full period of each function. hx=4cos2x-1289306037465004. The following graph can be described using an equation of the form y=Acosx+C. Determine the values of A and C. Justify your answers.5. When graphed, the line y=14 would not intersect the graph of which of the following functions?(1) y=5cosx+9 (2) y=-6cosx+10 (3) y=2sinx+15 (4) y=3sinx+206. Which of the following functions has a maximum value of 25?(1) y=25sinx+12 (2) y=-10cosx+35 (3) y=8cosx+17 (4) y=5sinx+157. Which statement is incorrect for the graph of the function y=-3cosπ3x-4+7?(1) The period is 6. (2) The amplitude is 3. (3) The range is [4, 10]. (4) The midline is y=-4.8. Write an equation to represent the trigonometric graphs below.9. State the amplitude, frequency, period, vertical shift and the horizontal shift of the following function: fx=4cos3πx+2+510. Write the equation of the graph of y=3sinx translated 2 units up and right π units.11. The periodic graph below can be represented by the trigonometric equation y=acosbx+c where a, b, and c are real numbers. State the values of a, b, and c, and write an equation for the graph.17145017145000012. A student attaches one end of a rope to a wall at a fixed point 3 feet above the ground, as shown in the accompanying diagram, and moves the other end of the rope up and down, producing a wave described by the equation y=asinbx+c. The range of the rope’s height above the ground is between 1 and 5 feet. The period of the wave is 4π. State the values of a, b, and c and write an equation that represents this wave.Lesson 14.5: Sinusoidal Regression (Do with next section!)Learning Goals:How do we use our calculator to write a sinusoidal regression equation for a given set of data?How can we explain the parameter of a trig function in terms of its real-life situation?Warm-Up: Answer the following question to prepare for today’s lesson.A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial.Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. y=1018.2839.5969xLocation of Sinusoidal Regression in the CalculatorAll regression equations can be found using the graphing calculator. All types of regressions on the calculator are prepared in a similar manner.When using sine regression, your calculator must be in RADIAN modeYour regression options can be found under STAT→CALC (scroll for more choices) “Iterations” is the number of times the calculator will compute the equation. Any number from 1-16 can be used, but the larger the number the more accurate the equation will be. The default is set at 3. We will always be using 16.Sinusoidal Regression1. The chart below shows the average daily high temperature for each month for Saugerties, NY-228600-63500Write a sine regression equation to model the average daily temperature as a function of the month of the year for Saugerties, NY. Round coefficients to the nearest hundredth. stat→edit→L1,L2y=26.03sin.49x-1.86+58.85 Using your regression equation, what is the predicted average daily high in September, to the nearest degree?y=26.03sin.49(9)-1.86+58.85y=73.36≈73 (close to 74 that is given)2. Suppose that the following table represents the average monthly ambient air temperatures, in degrees Fahrenheit, in some subterranean caverns in southeast Australia for each of the twelve months in a year. We wish to model these data with a trigonometric function. (Notice that the seasons are reversed in the Southern Hemisphere, so January is in summer, and July is in winter.)a) Write a sine regression equation to model the average monthly air temperatures as a function of the month of the year for Australia. Round coefficients to the nearest thousandth. y=7.5815sin.507x+.969+56.733 b) Use your regression equation to predict the average monthly air temperature, to the nearest degree, for January of the following year. x=13 or 13th month y=7.5815sin.50713+.969+56.733 y=64° Turn and Talk: Look at the graph and the picture of the Ferris wheel. Discuss the following questions with your partner.What does the max on the graph tell you about the wheel? Height of the top cart from the groundWhat does the min on the graph tell you about the wheel? Height of the bottom cart from the groundWhat does the midline on the graph tell you about the wheel? Height of the center of the wheel from the groundWhat does the period of 30 tell you about the wheel? 30 minutes to complete one full cycle on the wheel.1524000Parameters of Trig Functions in a real-world scenarioAmplitude: height of the wave from the midlineMidline: horizontal line at the center of the curvePeriod: How long to complete one cycleFrequency: how many full cycle up to 2πMaximum/minimum: highest and lowest points from the midline00Parameters of Trig Functions in a real-world scenarioAmplitude: height of the wave from the midlineMidline: horizontal line at the center of the curvePeriod: How long to complete one cycleFrequency: how many full cycle up to 2πMaximum/minimum: highest and lowest points from the midline3. The graph below represents the height of a Ferris wheel for one rotation.a) Explain how you can identify the radius of the wheel from the graph.Look at the amplitude! r=503067050-34480500b) If the center of the wheel is 55 feet above the ground, how high is the passenger car above the ground when it is at the top of the wheel?55midline+50=105 4. The height of the saddle of a horse above the base of a carousel can be modeled by the equation fx=12sin3π8t+42, where t represents seconds after the ride started.a) How much time does it take for the horse to complete one cycle of motion and return to its starting height? Find the period! B=3π8P=2πB=2π3π8=2π1?83π=163=5.3 seconds b) What is the maximum height and the minimum height of the horse’s saddle above the base of the carousel? Midline + amplitude!y=42 and a=±12 so Maximum =54 & Minimum =30Homework 14.5: Sinusoidal Regression1. The average daily temperature T (in degrees Fahrenheit) in Fairbanks, Alaska, is given in the table. Time t is measured in months, with t=0 representing January 1. Write a trigonometric model that gives T as a function of t, rounding coefficients to the nearest hundredth.2. For any given day, the number of degrees that the average temperature is below 65℉ is called the degree-days for that day. This figure is used to calculate how much is spent on heating. The table below gives the total number T of degree-days for each month in t in Dubuque, Iowa, with t=1 representing January.a) Find a sinusoidal model for the data, rounding coefficients to the nearest thousandth.b) Use this model to predict the number of degree-days when t=15, to the nearest day.Lesson 14.6: Modeling with Trig FunctionsLearning Goals:How can we write a trigonometric function that models cyclical behavior?How can we describe the parameters of a trig function within the context of the problem?Warm-Up: Determine if each of the following functions is a cosine or sine function. Then determine if the leading coefficient is positive or negative. sin and-a cos and +acos and -a sin and+aParameters of Trig Functions in a Real-World ScenarioYou almost always want to use a cosine function to model a real-world scenario because it is easier to locate the maximum point to start the curve than it is to find the point that lies on the midline.Amplitude= max- min2 Midline= max+ min2 Period=the time it takes to complete one full cycle Period= 2πb where b=frequency Minimum=midline-amplitude Maximum=midline+amplitude Recall the equation of a trig function is always in the form:y=AsinBx+C+D and y=AcosBx+C+D47148755969000Example 1: In an amusement park, there is a small Ferris wheel, called a kiddie wheel for toddlers. The points on the circle in the diagram to the right represent the position of the cars on the wheel. The kiddie wheel has four cars, makes one revolution every minute, and has a diameter of 20 feet. The distance from the ground to a car at the lowest point is 5 feet. Assume t=0 corresponds to a time when car 1 is closest to the ground.Makes two cycles and is a wave-like function!373380025654000Period =1 minuteDiameter =20, so r=10 amp=10 Min value =5 and Max value =25Midline: y=25+52=15 Starts at the minimum value.a) Sketch the height function for car 1 with respect to time as the Ferris wheel rotates for two minutes. b) Find a formula for a function that models the height of car 1 with respect to time as the kiddie wheel rotates. Starts below the midline → cosine!a=-10 y=AcosBx+C+DD=15 B=2πP=2π1=2π y=-10cos2πx+15Example 2: Once in motion, a pendulums’ distance varies sinusoidally from 12 feet to 2 feet away from a wall every 12 seconds.a) Sketch the pendulum’s distance D from the wall over a 1-minute interval as a function of time t. Assume t=0 corresponds to a time when the pendulum was furthest from the wall.341947522796500Makes two cycles and is a wave-like function!Period =24 secondsMin value =2 and Max value =12Midline: y=12+22=7 Starts at the maximum value.Graph for 60 secondsb) Write a sinusoidal function for D, the pendulum’s distance from the wall, as a function of time since it was furthest from the wall. y=AcosBx+C y=5cosπ12x+7C=7 Amp =5p=24 B=2π24=π12 Example 3: The tides in a particular bay can be modeled using a sinusoidal function. The maximum depth of water is 36 feet, the minimum depth is 22 feet and high-tide is hit every 12 hours. Write a cosine function in the form d=AcosBt+C, where t represents the number of hours since high-tide and d represents the depth of water in the bay.Max =36 and Min =224486275168592500Midline =36+222=29 amp=36-222=7d=AcosBt+CPeriod=12 hours y=7cosπ6t+29B=2π12=π6 Example 4: A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the ground, and reaches its highest point at 130 feet above the ground. The amount of time it takes to complete one full rotation is equal to 8 minutes. A person’s vertical position, y, can be modeled as a function of time in minutes since they boarded, t, by the equation y=AcosBt+C. Sketch a graph of a person’s vertical position for one cycle and then determine the values of A, B, and C. Show the work needed to arrive at your answers.Homework 14.6: Modeling with Trig Functions1. The High Roller, a Ferris wheel in Las Vegas, Nevada, opened in March 2014. The 550 foot tall wheel has a diameter of 520 feet. A ride on one of its 28 passenger cars lasts 30 minutes, the time it takes the wheel to complete one full rotation. Riders board the passenger car at the bottom of the wheel. Assume that once the wheel is in motion, it maintains a constant speed for the 30-minute ride and is rotating in a counterclockwise direction.a) Sketch a graph of the height of a passenger car on the High Roller as a function of the time the ride began.b) Write a sinusoidal function H that represents the height of a passenger car t minutes after the ride begins.c) Explain how the parameters of your sinusoidal function relate to the situation.d) If you were on this ride, how high would you be above the ground after 20 minutes?2: Once in motion, a pendulums’ distance varies sinusoidally from 20 feet to 2 feet away from a wall every 16 seconds.a) Sketch the pendulum’s distance D from the wall over a 72 second interval as a function of time t. Assume t=0 corresponds to a time when the pendulum was closest to the wall.b) Write a sinusoidal function for D, the pendulum’s distance from the wall, as a function of time since it was closest to the wall.3. Write an equation for the graph shown below:Lesson 14.7: Modeling with Trig Functions Day 2Learning Goal:How can we use trigonometric functions to model cyclical behavior?Warm-Up:470535053340000The following function represents the height of a Ferris wheel at any given time t minutes. What information can you gather about the Ferris wheel from the function given?ht=10cosπ8t+15Sketch a graph if you need a visual.Amp =10 (Radius =10)Midline: y=15 (center of wheel is 15 feet above the ground)Max=15+10=25 (highest point above ground)Min=15-10=5 (lowest point above ground)Period=2πB=2ππ8=2π1?8π=16 (one cycle in 16 minutes)510540021717000Example 1: In the classic novel Don Quixote, the title character famously battles a windmill. In this problem, you will model what happens when Don Quixote battles a windmill, and the windmill wins. Suppose the center of the windmill is 20 feet off the ground, and the sails are 15 feet long. Don Quixote is caught on a tip of one of the sails. The sails are turning at a rate of one counterclockwise rotation every 60 seconds. a) Model Don Quixote’s height, H, above the ground as a function of time, t in seconds, since he was closest to the ground.Cosine → starts below the midline! y=AcosBx+C5400675914400005419725000Midline: y=20 Max=35 Min=5 Period=60seconds a=-15 y=-15cosπ30x+20c=20 B=2π60=π30 b) After 1 minute and 40 seconds, Don Quixote fell off a sail and straight down to the ground. How far did he fall? Use the equation you wrote to find the height!x=timeseconds=60+40=100seconds y=height y=-15cosπ30?100+20=27.5 Example 2: The tidal data for New Canal Station is shown in the table below. Write a sinusoidal function to model the data for New Canal Station. L=Minimum and H=Maximum min=0.11 max=0.53 60×24=1440+29=1469minper cycleamp=.53-.112=.21 start at the min value!midline=.53+.112=.32 fx=-.21cos2π1469x+.32Example 3: A tsunami is a series of ocean waves that send surges up to 100 feet high onto land. They are caused by underwater earthquakes or explosions. The water level in a tsunami will initially fall below its normal level, then rise an equivalent amount above the normal level, finally returning back to its normal level. Assume the period of this cycle to be 18 minutes. A tsunami is observed from a coastal pier where the normal depth of the water is 10 feet. Following the cycle pattern described above, the tsunami has an amplitude of 20 feet above the normal water depth. The depth of the water will vary sinusoidally with time.291147545275500a) Graph one full cycle of the wave, with time in minutes on the x-axis and water height in feet on the y-axis.Period=18 minutesAmp =20Midline at: y=10Max=10+20=30Min=10-20=-10b) Write a sine function of the form fx=AsinBx+D to model this tsunami.Graph a sine wave…it goes down first!D=10 A=-20 y=-20sinπ9x+10B=2π18=π9 c) Predict the depth of the water 12 minutes after the tsunami first reaches the pier, to the nearest tenth of a foot.Plug x=12 in the equationy=-20sinπ9*12+10=27.3°d) What will be the minimum depth of the water during this cycle, and when will it occur? y=-10Example 4: An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum blood pressure, known as systolic, was 110 and her minimum blood pressure, known as diastolic, was 70. If each heartbeat cycle takes 0.75 seconds, then determine a sinusoidal model, in the form y=Asin(Bt)+C, for her blood pressure as a function of time t in seconds. Show the calculations that lead to your answer.max=110 y=20sin8π3x+90min=70 Period=0.75 seconds→B=2π.75=2π34=2π1?43=8π3 midline: y=110+702=90 Homework 14.7: Modeling with Trig Functions Day 21. Write an equation that could represent this function.2. An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum blood pressure, known as systolic, was 120 and her minimum blood pressure, known as diastolic, was 60. If each heartbeat cycle takes 0.8 seconds, then determine a sinusoidal model, in the form y=Asin(Bt)+C, for her blood pressure as a function of time t in seconds. Show the calculations that lead to your answer.3. Rapidly vibrating objects send pressure waves through the air that are detected by our ears and then interpreted by our brains as sound. Our brains analyze the amplitude and frequency of these pressure waves.A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to vibrate. By adjusting the current, the amplitude and frequency of vibrations can be controlled.The following graph shows the pressure intensity (I) as a function of time (x), in seconds, of the pressure waves emitted by a speaker set to produce a single pure tone.a. Does it seem more natural to use a sine or cosine function to fit this graph? Explain.b. Find the equation of a trigonometric function that fits this graph.4. Evie is on a swing thinking about trigonometry (no seriously). She realizes that her height above the ground is a periodic function of time that can be modeled using h=3cosπ2t+5, where t represents time in seconds. Which of the following is the range of Evie’s height?(1) 2≤h≤8 (2) 4≤h≤8 (3) 3≤h≤5 (4) 2≤h≤55. Below is a table that shows the average high temperatures for Harrison, NY for each month of the year. Write a trigonometric equation that could fit this data, rounding coefficients to the nearest thousandth.6. During one cycle, a sinusoid has a maximum at (4, 12) and a minimum at (12, -2). What is the period of this sinusoid?(1) 8 (2) 8π (3) 16 (4) 16π (5) 327. Graph and label the function294957567881500 fx=2cos12x-2 on the interval -4π≤x≤4π. State the range of the function and the equation of the midline.8. For the function below, indicate the amplitude, frequency, period, vertical translation, and equation of the midline. Graph the function together with a graph of the sine function fx=sin?(x) on the same axes. Graph at least one full period of the function. fx=-3sin2x+1 1047757302500 ................
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