4.2 Graphing Sine and Cosine Period 4.1 Review • Parent graphs f(x) = sin(x) and g(x) = cos(x) • For y = a*sin(bx - c) + d and y = a*cos(bx - c) + d, the sinusoidal axis is y = d and the amplitude is |a|. • To graph: 1)Graph SA 2)Find period 2π/b 3)Mark off each increment on xaxis period/4 4)Plot the following points (opposite if a is neg, max = d + |a|, min = d – |a|, start on y-axis and go 1 increment at a time): Sin – SA, max, SA, min Cos – Max, SA, min, SA Period • A function with period P will repeat on intervals of length P, and these intervals are referred to as periods. • Typically, sine and cosine take 2π to repeat. • The period can be altered. • With a graph, you can determine period by finding the horizontal distance between consecutive maximums or minimums. What are some things that are periodic? • Ex: Average daily temperature Graphing Calculator Investigation • Graph each of the following and determine the period of the function. Then, try to figure out how to determine period from just the equation. Y = sin(2x) y = cos(4x) - 2 Y = 2sin(x/2) + 1 y = -3cos(πx/12) Y =-4sin(πx – 1) y = 2cos(πx/2 + 2) Given y = a*sin(bx-c) + d or y = a*cos(bx-c) + d, the period of the function is 2𝜋 𝑏 Even and Odd • b must always be positive. We can use even and odd properties to make it + if it is not. • Sin is _________ Cos is _________ • Use even and odd properties to make b +: Sin(-2x) Cos(-π/2) Cos(-3x – π) Sin(5 - πx/4) Guided Practice • Find the amplitude, period, increment, S.A., domain and range and graph each function: Y = 4cos(2x) – 3 Y = -sin(x/3) + 2 Y = 2cos(3x) – 4 Y = 3sin(3x) Y = 3cos(x/2) + 1 Y = -2sin(2x) - 2 Homework • Find the amplitude, period, increment, S.A., domain and range and graph each function: Y = 2cos4x – 1 Y = -2sin(x/4) Y = -3cos2x +3 Y = 4sin3x Y = -cos(x/2) Y = 2sin4x - 2