Sine and cosine functions are the building blocks of periodic motion. These waves show up everywhere, from sound to light to ocean tides. Understanding how to graph and manipulate them is key to modeling real-world phenomena.
The notes cover how to adjust sine and cosine graphs using , , , and vertical shift. They also explain how to identify these elements from a graph and write equations. This knowledge is crucial for analyzing and predicting cyclic behavior.
Graphs of Sine and Cosine Functions
Graphing sine and cosine variations
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General form of sine and cosine functions: y=Asin(B(x−C))+D and y=Acos(B(x−C))+D
A: Amplitude, determines the or compression of the graph
∣A∣>1 stretches the graph vertically (taller waves)
∣A∣<1 compresses the graph vertically (shorter waves)
A<0 reflects the graph over the x-axis (upside-down waves)
B: Frequency, related to the and affects the or stretch
As ∣B∣ increases, the period decreases, compressing the graph horizontally (more waves in the same space)
As ∣B∣ decreases, the period increases, stretching the graph horizontally (fewer waves in the same space)
B is also known as the in the context of
C: Phase shift, represents the horizontal shift of the graph
C>0 shifts the graph to the right (delayed start)
C<0 shifts the graph to the left (early start)
D: Vertical shift, moves the graph up or down
D>0 shifts the graph up (higher waves)
D<0 shifts the graph down (lower waves)
Period (P) is the length of one complete cycle of the function
Calculated using the formula P=∣B∣2π, where B is the frequency
Examples of periods: 2π (standard sine or cosine), π (double frequency), 4π (half frequency)
Period is measured in radians when using the
Key features of sinusoidal graphs
is the horizontal line around which the graph oscillates
Determined by the vertical shift (D)
Equation of the midline: y=D
Examples of midlines: y=0 (standard sine or cosine), y=2 (shifted up by 2), y=−1 (shifted down by 1)
Amplitude is the maximum distance between the midline and the maximum or minimum points of the graph
Determined by the absolute value of A
Examples of amplitudes: 1 (standard sine or cosine), 2 (stretched vertically), 0.5 (compressed vertically)
are the maximum and minimum points of the graph
For sine functions:
Maximum occurs at x=2∣B∣π+BC+∣B∣2πn
Minimum occurs at x=−2∣B∣π+BC+∣B∣2πn
For cosine functions:
Maximum occurs at x=BC+∣B∣2πn
Minimum occurs at x=∣B∣π+BC+∣B∣2πn
Where n is any integer
Examples of extrema: (0, 1) and (π, -1) for standard sine, (0, 1) and (π, 1) for standard cosine
Equations from sinusoidal graphs
Identify the midline (D) from the graph or context
The midline is the horizontal line that the graph oscillates around
Examples: a tide graph with a midline at the average sea level, a sound wave with a midline at atmospheric pressure
Determine the amplitude (A) by measuring the distance from the midline to the maximum or minimum
Examples: the height of a wave from the average water level to the or , the loudness of a sound from the average to the peak
Find the period (P) by measuring the length of one complete cycle
Calculate the frequency using B=P2π
Examples: the time between high tides (about 12 hours), the time for one complete rotation of a Ferris wheel
Identify the phase shift (C) by comparing the graph to the parent function
For sine, find the horizontal distance from the origin to the nearest maximum point
For cosine, find the horizontal distance from the origin to the nearest maximum or minimum point
Examples: a tide graph starting at high tide (phase shift of π/2 for sine), a Ferris wheel starting at the bottom (phase shift of π for cosine)
Substitute the values of A, B, C, and D into the general form of the function
y=Asin(B(x−C))+D for sine functions
y=Acos(B(x−C))+D for cosine functions
Example: a tide graph with an amplitude of 2 m, a period of 12 hours, a phase shift of 3 hours, and a midline at 4 m would have the equation y=2sin(6π(x−3))+4
Additional Concepts
repeat their values at regular intervals, with sine and cosine being common examples
The unit circle is a fundamental tool for understanding trigonometric functions and their relationships
are equations involving trigonometric functions that are true for all values of the variables