Graphing Trig Functions

Steps to modeling a sinusoidal graph using the standard functions \(y = a \sin{\left(k\left(x - b\right)\right)} + c\) or \(y = a \cos{\left(k\left(x - b\right)\right)} + c\):

1. Lightly draw a rectangular box around a 1-period portion of the graph, matching either a sine or a cosine curve.
2. Determine the maximum and minimum \(y\)-values of the graph, which should correspond to the top and bottom of the box.
3. Determine the midline using the formula \(c = \frac{max+min}{2}\). This should correspond to a horizontal line through the middle of the box. This also represents the vertical shift of the graph.
4. Determine the amplitude using the formula \(a = \frac{max-min}{2}\). This represents the distance from the midline up to the top of the graph and also the distance from the midline down to the bottom of the graph, which is the vertical stretch. Also, remember the total distance from top to bottom is \(2a\).
5. Determine the period by finding the horizontal distance between 2 consecutive maximums or between 2 consecutive minimums. This should also correspond to the horizontal length of the box. Once you have the period, you can find the \(k\)-value using \(k = \frac{2\pi}{\text{period}}\). This corresponds to the horizontal stretch.
6. Finally, determine the phase (horizontal) shift of the graph, which is \(b\). This should correspond to the \(x\)-value of the left edge of the box.

The values that you get for \(a\), \(k\), and \(c\) will be the same regardless of whether you are trying to use a sine or cosine equation. Only the \(b\) value will be different, and maybe the sign of \(a\) if you are using a reflected graph.

Note that other texts or resources may use different letters instead of \(a\), \(k\), \(b\), and \(c\). But the general idea remains the same.