8.3 – Model Periodic Behavior Essential Question: How do you write an equation for a sin or cos graph?

General Sine Waves (sin & cos): If the graphs of y = A cos Bx and y = A sin Bx are translated horizontally h units and vertically k units, then the resulting graphs have equations: y – k = A sin B(x – h) y – k = A cos B(x – h)

Pictures:

To find translation, use axis of the wave: Axis of the Wave – a horizontal line midway between the maximum and the minimum. y = Max + Min 2

To find translation: For Cosine: - Count the translation amounts (horizontally and vertically) from (0, A) to the highest point on the curve. For Sine: - Count the translation amounts (horizontally and vertically) from (0, 0) to the point where the curve intersects the axis of the wave (where curve is going up).

Example Find the equation for the given graph. - For Cosine: - For Sine: x y x y

Example What is the translation? y – 2 = 2 cos (x – 3) y = -5 + 2 sin (x + 1)

Example Give an equation of the trig. graph.

Example Sketch the graph of y = 3 + 5 sin 2(x + 1)

Example a) What is the amplitude? b) What is the period?

Example c) What is the axis of the wave? d) Give an equation for the trigonometric graph.