Topic 2: Equations of Sinusoidal Functions

Describe the characteristics of a sinusoidal function by analyzing its equation. Match equations in a given set to their corresponding graphs. Interpret the graph of a sinusoidal function that models a situation, and explain the reasoning.

Information Any sinusoidal function can be expressed as where a, b, c, and d are the parameters that affect the characteristics of the graph.

Information Many characteristics of a sinusoidal function can be calculated using the parameters a, b, and d.

Example 1 Determining the characteristics of a sine function based on its equation Consider the function for . a) Describe the graph of the function by stating the amplitude, equation of the midline, range, and period. b) Explain how you could use technology to verify your solutions. Amplitude is 2 and the equation of the midline Is y = 1. 2 Range Period y = 1 2 Graph the equation in Y1 and adjust your window settings. Then double check to make sure your characteristics are accurate.

Note : If no degree symbol is present in an equation, assume that the x is measured in radians. Example 2 Determining the characteristics of a sine function based on its equation Consider the function for . Describe the graph of the function by stating the amplitude, equation of the midline, range, and period. a = 3, b = 2, c = 7, d =0 Amplitude is 3 and the equation of the midline Is y = 0. Range Period 3 y = 0 3

Example 3 Match each graph with the corresponding equation below. Matching equations to graphs Match each graph with the corresponding equation below. a) Which of the above equations represents Graph 1? b) Which of the above equations represents Graph 2? Graph 1 has a period of 360. This means that the b-value is 360 360 =1 . Also Graph 1 has a midline (d-value) of 1 and an amplitude (a-value) of 4. Equation i. Graph 2 has a period of 180. This means that the b-value is 360 180 =2 . Also, graph 2 has a midline (d-value) of 2 and an amplitude (a-value) of 3. Equation iii.

Example 4 Determining the equation of a function given the characteristics Create a sinusoidal function of the form y = asin(bx)+d that has the following characteristics Range: {y|-3≤y≤4,y∈R} Period of 6 One full cycle must occur in 6 radians. 4 6 -3

Example 5 Solving a problem using a sinusoidal function The Far North is called “the Land of the Midnight Sun” for a good reason: during the summer months, in some locations, the sun can be visible for 24 hours a day. The number of hours of daylight in Iqaluit, Nunavut, can be represented by the function where x is the day number in the year. Graph this equation in Y1, making sure that your calculator is in radian mode. Adjust your window settings to reflect the context of the question. Since x represents the day number in the year, and appropriate range of values might be 0 – 400. Since y represents the amount of sunlight in a day, an appropriate range of values might be 0 – 24.

Example 5 a) How many hours of daylight occur in Iqaluit on the shortest day of the year, to the nearest tenth? b) How many daylight hours does Iqaluit get on February 22? (Jan. has 31 days) The shortest day of the year is represented by the minimum. Press 2nd Trace 3: Minimum and follow the calculator steps. There are 4.34 hours of sunlight on the shortest day of the year. February 22nd is the 53rd day of the year (31 in Jan + 22 in Feb). We want to find the number of daylight hours (y) for day 53 (x). Press 2nd Trace 1: Value and enter an x-value of 53. There are 8.77 hours of sunlight on February 22nd.

Example 5 c) What is the period of this sinusoidal function? Explain how the period relates to the context of the problem. d) What is the first day that Iqaluit will have 16 hours of daylight? Since days of sunlight vary according to time of year, the period is equal to the number of days in a year. We want to find the day number (x) for 16 daylight hours (y). Enter the y-value of 16 into Y1. Then find the intersection point by press 2nd Trace 5: Intersect. On day 106, there are 16 hours of sunlight.

Need to Know Any sinusoidal function can be expressed as: where a, b, c, and d are the parameters that affect the characteristics of the graph.

Need to Know Many characteristics of a sinusoidal function can be calculated using the parameters a, b, and d.

You’re ready! Try the homework from this section. Need to Know The expression sinx+c indicates that sine operates only on x. To indicate that sine operates on the expression x+c, parentheses are used: sin(x+c).