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Setting Up Trigonometric Models



Trigonometric functions are useful for many applications involving periodic behaviour. A few examples include blood pressure, populations that change seasonally, and the phases of the moon. When working with trigonometric models we will use the form below. \[y = a \sin(bx -c) + d\]

Let's take a look at some of the associated terminology.

For the functions \(y = a \sin(bx -c) + d\) or \(y = a\cos(bx-c) + d\),

\[ \begin{array}
aa= \text {the } \textbf{amplitude} \text{(height from midpoint to highest point)} \\ 
\frac{2\pi}{b} = \text{the } \textbf{period}\\ 
\frac{b}{2\pi}= \text{the } \textbf{frequency } \text{(number of complete cycles per unit time)} \\ 
\frac{c}{b} = \text{the } \textbf{phase shift } \text {and} \\ 
d  =\text{the } \textbf{vertical shift}
\end{array} \]

The following is a graphical representation of the above concepts:

trigonometric graph attributes

Example: Determine the amplitude and vertical shift of the periodic

\(y = 3 \sin(2x-\pi) + 7\).


Amplitude: 3
Vertical Shift: 7

Example: Determine the period of the function \(y = \sin(3x-\pi) + 2\).


Period = \(\frac{2\pi}{b} = \frac{2\pi}{3}\)

Example: How do the frequency and amplitude of the two periodic
functions below compare?

trigonometric functions with different frequency and amplitude

Solution: Function A has a greater amplitude than function B. Function B has a greater frequency than function A.

Example 1:

Example 2: