Graphing Trig Functions

Steps to graphing 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 dotted horizontal line along \(y = c\). This is the middle, or midline, of the graph.
2. Lightly draw two more dotted horizontal lines that are \(a\) units above and \(a\) units below the midline, or along \(y = c + a\) and \(y = c - a\). These two lines represent the top and bottom of the sine or cosine graph.
3. Lightly draw a dotted vertical line along \(x = k\). Be careful with the sign. This is starting edge of a 1-period portion of the graph.
4. Lightly draw another dotted vertical line that is 1 period to the right of the vertical line from step #3, or along \(x = k + \text{Period}\). This is the ending edge of a 1-period portion of the graph.
5. The 4 dotted lines that you drew in steps #2 - 4 should outline a rectangular box. The height of this box should be double the amplitude, or \(2a\), and the length of the box should equal the period. It should be centered on the midline. Draw the appropriate sine or cosine graph inside the box. Be sure to reflect it over the midline if the coefficient \(a\) is negative.

As you draw the appropriate sine or cosine graph, it might be helpful to locate where the 5 key points are going to be. You can find where these 5 points are going to be located by dividing the length of the period by 4. This will tell you how far apart the 5 points will be from each other.

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