06Ffindingeqns

toc = Finding the Equations of Trig Graphs =

From any graph with the shape of a sine wave, we could find an equation in terms of either sin or cos. Most questions will therefore specify the form of the function to be found.

** Example 1 **
The equation of the following graph is in the form y = acos(nx) + c Find the values of a, n, c and hence state the equation of the function.

Median line: half-way between the maximum and minimum points:
 * Median: c = 1

Amplitude: distance from median to maximum point (or half the distance between max and min)
 * Amplitude: a = 2

Period: Length of one complete cycle.
 * Period = 6 p

To find n, use period = 2 p /n math \\ . \qquad \qquad \dfrac{2\pi}{n}=6\pi \\ \\ .\qquad \qquad n = \dfrac{2\pi}{6\pi}=\dfrac{1}{3} \\ \\ . \quad \centerdot \;\; n = \frac{1}{3} math

Hence, equation of graph is: math y = 2\cos \big( \frac{1}{3}x \big) + 1 math

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