Finding expressions for sine and cosine graphs.

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Determine the formula for cosine function graphed below.

 

Solution: First we write general equation of cosine function.

Y = A cos(B(x-D))+C

First we draw a mid line ( dashed line) passing horizontally  through the middle of this graph and follow  the procedure given below.

 

  1. look at the  distance of  max or min point of this graph from this midline. This will give you  amplitude  A= 4
  2. Look at the distance of mid line (dashed one) from x axis. This gives you vertical shift C =4
  3. Look at the period which is the distance between either two consecutive max points or two min points. Then use formula B=\frac{2\pi }{period}
  4. Here  distance between two max points (period) = 7-1= 6.
  5. B=\frac{2\pi }{6} = \frac{\pi }{3}
  6. Phase shift is the horz shift. Here graph is shifted to right horizontally by 1 unit  so D=1
  7. Finally plugin all the values of A,B,C and D into general cosine  fucntion and we get
  8. Y = 4 cos(\frac{\pi }{3}(x-1))+4

 

 

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