The amplitude of a trigonometric function is the maximum distance from the midline to the highest or lowest point of the graph.
The period of a trigonometric function is the length of one complete cycle of the graph.
The frequency of a trigonometric function is the number of complete cycles the graph completes in a given interval.
The phase shift of a trigonometric function is the horizontal translation of the graph.
The vertical shift of a trigonometric function is the vertical translation of the graph.
Graphs of the Sine and Cosine Functions
The sine and cosine functions are periodic functions that repeat their values in a regular pattern.
The sineand cosinefunctions are definedfor allrealnumbers.
Graph of the Sine Function
The sine function is defined as:
\$\$y = \sin(x)\$\$
The graph of the sine function is a wave that oscillates between -1 and 1.
The sinefunction is an oddfunction, which means that \$\sin(-x) = -\sin(x)\$.
Graph of the Cosine Function
The cosine function is defined as:
\$\$y = \cos(x)\$\$
The graph of the cosine function is a wave that oscillates between -1 and 1.
The cosinefunction is an evenfunction, which means that \$\cos(-x) = \cos(x)\$.
Transformations of the Sine and Cosine Functions
The sine and cosine functions can be transformed by changing their amplitude, period, frequency, phase shift, and vertical shift.
Amplitude
The amplitude of a trigonometric function is the maximum distance from the midline to the highest or lowest point of the graph.
The amplitude of the sine and cosine functions is determined by the coefficient of the function.
The amplitudeis always positive.
Period and Frequency
The period of a trigonometric function is the length of one complete cycle of the graph.
The frequency of a trigonometric function is the number of complete cycles the graph completes in a given interval.
The period of the sine and cosine functions is determined by the coefficient of the variable.
The periodis always positive.
Phase Shift
The phase shift of a trigonometric function is the horizontal translation of the graph.
The phase shift of the sine and cosine functions is determined by the constant added to the variable.
The phaseshiftcan be positiveor negative.
Vertical Shift
The vertical shift of a trigonometric function is the vertical translation of the graph.
The vertical shift of the sine and cosine functions is determined by the constant added to the function.
The verticalshiftcan be positiveor negative.
Graphs of Trig Functions with More than One Transformation
The sine and cosine functions can be transformed by changing their amplitude, period, frequency, phase shift, and vertical shift.
The orderof transformationsmatters.
Finding the Equation of a Function from a Given Graph
To find the equation of a trigonometric function from a given graph, follow these steps:
- Determine the type of function ( sine or cosine).
- Find the amplitude.
- Find the period.
- Find the phase shift.
- Find the vertical shift.
The orderof transformationsmatters.
