Starter Draw the graph of y = log(x+1) for -6≤ x ≤ 14. Draw in the asymptote Asymptote is at x = -1

Note 9: Periodic Functions Periodic Function A function that repeats itself over and over is a horizontal direction. Period of a Function The length of one repetition of the function. Principal Axis The horizontal line that the function oscillates about. Principal axis = maximum + minimum 2

Amplitude The vertical distance from the principal axis to the maximum or to the minimum point. Amplitude = maximum – minimum 2

Note 10: Sine and Cosine Curve Draw an accurate sketch of the Sine and Cosine Curve: x-axis from 0° to 360° - plot every 30° y-axis from -1 to 1 – plot every 0.25

Characteristics of the Sine and Cosine Curve The period is 360° The amplitude is 1 The maximum value is 1 and minimum value is -1 The domain is: 0° < x < 360° The range is: -1 < y < 1 The cosine curve is just the sine curve shifted by 90°

Investigation 1: Using technology plot the following: y = sinx y = 3sinx y = 0.5sinx For each graph: Find the maximum and minimum value Find the period and amplitude Describe the effect of a in the function y = asinx What is the amplitude of: y = 4sinx y = ⅔sinx

Investigation 2: Using technology plot the following: y = cosx y = cos2x y = cos(0.5x) For each graph: Find the maximum and minimum value Find the period and amplitude Describe the effect of b in the function y = cosbx What is the period of: y = cos4x y = cos¼x

Investigation 3: Using technology plot the following: y = sinx + 2 y = sinx – 1 For each graph: Find the maximum and minimum value Find the period and amplitude Calculate the equation of the principal axis What is the connection between: y = sinx y = sinx + c

IN GENERAL: y = AsinBx + C To find: Period = 360/B Principal axis y = C Affects Amplitude Affects Period Affects Principal Axis

Sketch the following graphs: Examples: Sketch the following graphs: y = 2sinx + 4 y = -3sin2x y = sin(0.5X) - 2 Find for each graph: Maximum Minimum Amplitude Principal Axis Period