Transformations of sine and cosine Period, Amplitude, phase shift, and vertical shifts

Why? Sine and cosine are periodic functions. Periodic functions are those whose outputs repeat regularly over time. The outputs of sine and cosine oscilate symmetrically between some maximum and some minimum value. In practical applications, the middle value is rarely zero. Often all outputs are positive. For example the output could be the height of a point on a wheel, or the water level at some point on a pier. The middle value would be the hub of the wheel, or the water level at slack tide.

More why In practical applications, we may choose, or be forced to, start recording time, or measuring values when the output is something other than a maximum or middle value. Changing the middle value to something other than zero is a vertical shift. Starting the cycle at some point other than the middle or maximum value is a horizontal shift, called phase shift in science. This week and next we will focus on the math, after break we will focus on the applications Except for this brief reminder.

frequency Frequency is the reciprocal of “period”, so the longer the period, the smaller (lower) the frequency, and the shorter the period, the larger (higher) the frequency. Frequency: how many complete cycles within a given unit of time: usually seconds- so frequency units are hertz Wavelength is directly proportional to the period: The longer the wavelength the longer the period, and the smaller(lower) the frequency.

Radio telescope

Radio telescope array

Trig Curves Transformed Graph Sine, Cosine Functions

Copyright © 2009 Pearson Education, Inc.

Variations of the Basic Graphs We are interested in the graphs of functions in the form y = A sin B (x – h) + k and y = A cos B (x – h) + k B careful! where A, B, h, and k are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs. Copyright © 2009 Pearson Education, Inc.

Trig Curves Transformed Graph Sine, Cosine, and Tangent Functions Equation of a Sine Function Amplitude Period Complete Cycle

Trig Curves Transformed Graphing a Sine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Curves Transformed Graphing a Sine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Curves Transformed Graphing a Sine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Curves Transformed Graph Sine, Cosine, and Tangent Functions Equation of a Cosine Function Amplitude Period Complete Cycle

Trig Curves Transformed Graphing a Cosine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Curves Transformed Graphing a Cosine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Curves Transformed Graphing a Cosine Functions Draw one cycle of the function’s graph. Amplitude Period

Trig Functions Translation Translations of Trigonometric Graphs Translation of a Sine Function Amplitude Period

Trig Functions Translation Translations of Trigonometric Graphs Translation of a Sine Function Amplitude Period

Trig Functions Translation Graphing a Vertical Translation Graph the function. Amplitude Translation down 2 Period

Trig Functions Translation Graphing a Vertical Translation Graph the function. Amplitude Translation up 3 Period

Trig Functions Translation Graphing a Horizontal Translation Graph the function. Amplitude Period Translation right p

Trig Functions Translation Graphing a Horizontal Translation Graph the function. Amplitude Period Translation left p

Trig Functions Translation Graphing a Horizontal Translation Graph the function. Amplitude Period Translation right p/2

Trig Functions Translation Graphing a Reflection Graph the function. Amplitude Period Reflection over x-axis

Trig Functions Translation Graphing a Reflection Graph the function. Amplitude Period Reflection over x-axis