1. *Sketch sine and cosine graphs *Use amplitude and period *Sketch translations of sine and cosine graphs 4.5 Graphs of Sine and Cosine Functions

2. Fill in the table • Plot the points on the graph Sine Function Graph

3. Fill in the table • Plot the points on the graph Cosine Function Graph

4. One period = the intercepts, the maximum points, and the minimum points Key Points

5. Amplitude: of y = a sinx and y = a cosx represent half the distance between the maximum and minimum values of the function and is given by • Amplitude = IaI • Period: Let be be a positive real number. The period of y = a sinbx and y = a cosbx is given by • Period = (2π)/b Amplitude and Period of Sine and Cosine Curves

6. Sketch the graph y = 2sinx by hand on the interval [-π, 4π] Graphing By Hand

7. Sketch the graph of y = cos (x/2) by hand over the interval [-4π, 4π] Graphing Using Period

8. The constant c in the general equations • y = a sin(bx – c) and y = a cos(bx – c) create horizontal translations of the basic sine and cosine curves. • One cycle of the period starts at bx – c = 0 and ends at bx – c = 2π • The number c/b is called a phase shift Translations of Sine and Cosine

9. Sketch the graph y = ½ sin (x – π/3) Horizontal Translation

10. Sketch the graph y = 2 + 3 cos2x Vertical Translations

11. Find the amplitude, period, and phase shift for the sine function whose graph is shown. Then write the equation of the graph. Writing an Equation

12. For a person at rest, the velocity v (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by where t is the time (in seconds) . (inhalation occurs when v> 0 and exhalation occurs when v < 0 ) • Graph on the calculator • Find the time for one full respiratory cycle • Find the number of cycles per minute • The model is for a person at rest. How might the model change for a person who is exercising?

13. A company that produces snowboards, which are seasonal products, forecast monthly sales for 1 year to be where S is the sales in thousands of units and t is the time in months, with t = 1 corresponding to January. • Graph the function for a 1 year period • What months have maximum sales and which months have minimum sales.

14. Throughout the day, the depth of the water at the end of a dock in Bangor, Washington varies with the tides. The table shows the depths (in feet) at various times during the morning. • Use a trigonometric function to model this data. • A boat needs at least 10 feet of water to moor at the dock. During what times in the evening can it safely dock? Mathematical Modeling