+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 FunctionA function that repeats itself over and over is a horizontal direction.

Period of a FunctionThe length of one repetition of the function.

Principal AxisThe 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 axisWhat is the connection between:

y = sinx y = sinx + c

+IN GENERAL:

y = AsinBx + C

To find: Period = 360/B Principal axis y = C

AffectsAmplitude

AffectsPeriod

AffectsPrincipal Axis

+Examples:

Sketch the following graphs: y = 2sinx + 4 y = -3sin2x y = sin(0.5X) - 2

Find for each graph:MaximumMinimumAmplitudePrincipal Axis Period