Graphs of Sine and Cosine

Graphs of Sine and Cosine Section 4.5

Sine Curve 1 π 2π -1 Key Points: 0 π 2π Value: 1 0 -1 0 0

Cosine Curve 1 π 2π -1 Key Points: 0 π 2π Value: 0 -1 1 1 0

Equations • For the rest of this section, we will be graphing: y = a Sin (bx – c) + d y = a Cos (bx – c) + d y = Sin x a = 1 c = 0 b = 1 d = 0

Graph the equation y = 2 Sin x 2 1 π 2π -1 -2 Key Points: 0 π 2π Value: 2 0 -2 0 0

Amplitude (a) • Half the distance between the maximum and minimum values of the function • Given by the value of │a │ • Graph the functions: y = 4 Sin x y = ½ Cos x y = -2 Sin x

y = 4 Sin x y = ½Cos x 4 y = -2Sin x 3 2 1 π 2π -1 -2 -3 -4

y = a Sin (bx – c) + d b gives us the period of the curve Period = y = 4 Sin 2x 4 Amplitude = Period = = π

Key Points Would having a period of π change the key points of the curve? 1 π 2π -1

Finding Key Points In General For Y = 4Sin 2x Find the period of the curve Divide the period into 4 equal parts From your starting point, add this distance 4 times for each period Period = π Distance = 0, , , ,

y = 4Sin 2x 4 1 π -1 -4

Graph the following curves • y = 4 Cos 8x • y = ½ Cos 2πx • y = -2 Sin 6x

y = 4Cos 8x Amplitude = 4 b = 8 → Period = → Distance = 4 -4

y = ½Cos 2πx Amplitude = ½ b = 2π → Period = → Distance = ½ - ½

y = -2Sin 6x Amplitude = 2 b = 6 → Period = → Distance = 2 - 2

y = a Sin (bx – c) + d • a = • b = • c = amplitude Find the period → Find the “phase shift” → horizontal shift →

y = ½ Sin (x - ) • a = • b = • c = ½ 1 → Period = → P. S. =

y = -3 Cos (2πx + 4π) • a = • b = • c = 3 2π → Period = → P. S. =

y = a Sin (bx – c) + d • a = • b = • c = • d = amplitude Find the period → Find the “phase shift” → Vertical Shift

y = • a = • b = • c = • d = 2 → Period = → P. S. = 3

y = • a = • b = • c = • d = 4 → Period = → P. S. = -2

y = 2 1 4 -2 -2 -6

Graphs of Sine and Cosine