4 5 Sinusoidal Graphs Sketching and Writing Equations

4. 5 Sinusoidal Graphs Sketching and Writing Equations

Sinusoidal Graphs of sine and cosine functions. Periodic Functions whose graphs have a repeating pattern. Period The horizontal length of each cycle in a periodic graph. Midline The horizontal line halfway between a sinusoid’s max/min values.

The Graph of y = sin x x y

Characteristics of the Sine Function • The domain is the set of all real numbers. • The range consists of all real numbers from -1 to 1, inclusive. • The sine function is an odd function (symmetric with respect to the origin). • The sine function has a period of 2π • Midline is the x-axis

Period y = sinx 3 pattern starts 2 pattern starts 1 -1 1 period Period= 2 -2 -3 1 period

The Graph of y = cos x x y

Characteristics of the Cosine Function • The domain is the set of all real numbers. • The range consists of all real numbers from -1 to 1, inclusive. • The cosine function is an even function (symmetric with respect to the y-axis). • The cosine function has a period of 2π • Midline is the x-axis

Period y = cosx 3 pattern starts 2 1 -1 1 period -2 1 period Period = 2 -3

Period 3 pattern starts 2 1 -1 1 period Period= -2 -3

Period 3 pattern starts 2 1 -1 Period = 3 -2 2 -3 1 period

Amplitude Half the distance between the maximum and minimum of a sinusoidal graph.

Amplitude 3 max. 2 1 Amplitude -1 min. -2 Amplitude = 1 -3

Amplitude max. 3 2 Amplitude 1 -1 Amplitude = 2 -2 -3 min.

Amplitude max. 3 2 Amplitude 1 -1 Amplitude = 1 -2 -3 min.

Amplitude = 2 3 2 max. 1 -1 3 -2 -3 Amplitude min.

Sinusoidal Graphs General Form of Equations Amplitude = Period =

Sinusoidal Graphs y = 3 sin 4 x y = -4 cos 5 x Amplitude = Period =

Writing Equations Answer these questions first: 1. Is it a “sin” or “cos” graph? 2. What is the amplitude? 3. What is the period? 4. Is the graph reflected? 5. Is it shifted up/down? 6. Is it shifted left/right?

Writing Equations b= Write an equation: cosine graph amplitude = 4 Period = a=4 b= =2

Writing Equations b= Write an equation: sine graph amplitude = 3 Period = 4 a=3 b=

Writing Equations b= Write an equation: a = ± 2 sine graph b= amplitude = 2 Period = 3 Reflected over x-axis

Writing Equations b= Write an equation: cosine graph amplitude = ½ Period = Reflected over x-axis a= b= Invert and multiply!

Writing Equations Write an Equation Sine graph Amplitude = 5 Period = Shifted up 2 Changing the midline causes vertical shifts a= b=

Sinusoidal Graphs “sin” Period = 3 Amplitude = 1 2 a =1 y = sin 2 x 1 -1 -2 -3 b=

Sinusoidal Graphs “sin” Period = 2 b= a =3 Amplitude = 3 3 y = 3 sinx 2 1 -1 -2 -3

Sinusoidal Graphs “cos” Period = Amplitude = 1 a =1 b= 3 2 1 -1 -2 -3

Sinusoidal Graphs “cos” Period = 3 Amplitude = 2 a = -2 3 2 1 -1 Reflected over x-axis! -2 -3 b=

Sinusoidal Graphs “cos” Period = 6 Amplitude = 3 a = -3 3 2 1 Reflected over x-axis! -1 -2 -3 b=

Graphing Sinusoidal Functions y = 3 sin 4 x 3 2 1 -1 -2 -3

Graphing Sinusoidal Functions y = 2 sin x 3 2 1 -1 -2 -3

Graphing Sinusoidal Functions y = -3 sin 2 x 3 2 1 -1 -2 -3

Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

Phase Shift • Horizontal translations of sine and cosine graphs. Left Phase Shift Right Phase Shift

Phase Shift right Amp = 3 T= P. S. =

Phase Shift left Amp = 2 T= P. S. =

Phase Shift “Cosine” Amplitude = 5 Period = Phase shift =

Phase Shift “sine” Amplitude = 3 Period = 4 Phase shift =